From d19410d6cb4f4a65bf62e641ee15328019a3e9d4 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 17:17:06 +0330 Subject: [PATCH 001/114] add Advanced Heuristics Folder --- notebooks/Advanced Heuristics/index.md | 148 ++++++++++++++++++ .../Advanced Heuristics/pic/admissible.png | Bin 0 -> 74668 bytes notebooks/Advanced Heuristics/pic/c.png | Bin 0 -> 36239 bytes .../Advanced Heuristics/pic/consistency.png | Bin 0 -> 110197 bytes .../Advanced Heuristics/pic/distance.png | Bin 0 -> 77803 bytes notebooks/Advanced Heuristics/pic/example.png | Bin 0 -> 29868 bytes .../Advanced Heuristics/pic/hamilton path.png | Bin 0 -> 174794 bytes .../pic/heuristic proof.png | Bin 0 -> 77762 bytes .../Advanced Heuristics/pic/monotonic.png | Bin 0 -> 97434 bytes notebooks/Advanced Heuristics/pic/proof.png | Bin 0 -> 107414 bytes 10 files changed, 148 insertions(+) create mode 100644 notebooks/Advanced Heuristics/index.md create mode 100644 notebooks/Advanced Heuristics/pic/admissible.png create mode 100644 notebooks/Advanced Heuristics/pic/c.png create mode 100644 notebooks/Advanced Heuristics/pic/consistency.png create mode 100644 notebooks/Advanced Heuristics/pic/distance.png create mode 100644 notebooks/Advanced Heuristics/pic/example.png create mode 100644 notebooks/Advanced Heuristics/pic/hamilton path.png create mode 100644 notebooks/Advanced Heuristics/pic/heuristic proof.png create mode 100644 notebooks/Advanced Heuristics/pic/monotonic.png create mode 100644 notebooks/Advanced Heuristics/pic/proof.png diff --git a/notebooks/Advanced Heuristics/index.md b/notebooks/Advanced Heuristics/index.md new file mode 100644 index 00000000..2c3588fc --- /dev/null +++ b/notebooks/Advanced Heuristics/index.md @@ -0,0 +1,148 @@ +# Advanced Heuristic +#### Kahbod Aeini, Mohammadreza Daviran and Sepehr Kianian + +First we review some basic definitions + +Distance is a numerical measurement of how far apart objects or points are: +* [Euclidean Distance](https://en.wikipedia.org/wiki/Euclidean_distance) calculates the distance between two real-valued vectors. +* [Manhattan Distance](https://en.wikipedia.org/wiki/Taxicab_geometry) is sum of the absolute differences of their Cartesian coordinates. +

+ distance +

+* Heuristic guidance means how far is the goal state from a given state approximately. + +***Admissiblity of a heuristic function means value of the function is always a Lower Bound of the remaining cost.*** + +So an **Admissible Heuristic** is a *non-negative* function h of nodes, where **h(n)** is *never greater than the actual cost of the shortest path from node n to a goal.* thus it means that the cost to reach the goal is never overestimated. + +Effect of **Admissibility** on a Heuristic is shown in the below schema: +

+admissible +

+ +Now we define **f(n)** function as **f(n) = h(n) + g(n)** where g(n) is sum of costs from root to n node. + +***Monotonicity or Consistency is that the heuristic function holds in triangle inequality.*** Namely **f(n) is never Decreasing.** +

+consistency +

+ + +Effect of **Monotonicity** on a Heuristic is shown in the below schema: +

+monotonic +

+ +We will prove that **Consistency** implies **Admissibility** whereas the opposite is not necessarily true. +

+proof +

+ +Now we want to show an **inconsistent, admissible example!** +So consider this figure: + +

+example +

+ +If our heuristic is admissible, we have that **h(n) $\le$ h*(n)** for every node n where **h*** is the real cost to the goal. So we deduct that **h(A) $\le$ 4**, **h(C) $\le$ 3** and **h(G) $\le$ 0**. + +If we want our heuristic to be *Consistent* we should have **h(G) = 0** and **h(n) $\le$ cost(n, c) + h(c)** so in our case we have **h(A) $\le$ 1 + h(C)** and **h(C) $\le$ 3 + h(G) = 3** + +Because of the *Admissibility* **h(C) should be less than 3**, but if **h(A) > 1 + h(C)** then our heuristic is *Inconsistent!*. Hence if we assume that **h(C) = 1**, **h(G) = 0** and **h(A) = 4** our heuristic is *Admissible but Inconsistent!* + + +We also can make a non-monotonic heuristic, monotonic with some changes! +For this we only need to define a new and more appropriate heuristic like the below schema. +

+new heuristic +

+ + + +**$\overline{h}$(n’) = max(h(n’), ℏ(n’) - c(n, a, n’))** + +Above Heuristic is defined Recursively. +

+heuristic proof +

+ +**$\overline{h}$(n’) $\ge$ $\overline{h}$(n) - c(n, a, n')** + +So obviously we have **$\overline{h}$(n’) + c(n, a, n') $\ge$ $\overline{h}$(n)** + +***Heuristic Dominance*** + +For one heuristic to dominate another, all of its values must be greater than or equal to the corresponding values of the other heuristic. +So if $h_{2}$ dominates $h_{1}$(both are admissible) it means that A* algorithm expands less nodes with $h_{2}$ in the target path than $h_{1}$. +On another hand it means if a node is expanded by $h_{2}$ it will be expanded by $h_{1}$(consider that it’s not a biconditional proposition!) + +*Proof:* + +If $h_{2}$ dominate $h_{1}$, A* with $h_{2}$ expands at most the same number of nodes as when using $h_{1}$. + +A* opens all node with **f(n) < C** and some with **f(n) = C**. + + +Note that +**$h_{2}(n)$ +$\ge$ +$h_{1}(n)$** + +If node n is expanded by A* with $h_{2}$ then **$h_{2}(n)$ < C - g(n)** + +So ***$h_{1}(n)$ $\le$ $h_{2}(n)$ < C - g(n)*** and it would be expanded by A* with $h_{1}$ as well. + +We also can improve our heuristic by some methods. + +We are always trying to lower the difference between h* and h, expanded nodes and algorithm time complexity by making change in our heuristic. + +One of the main ways for improving heuristics is **Relaxing** the problem. + +In this procedure we try to enlarge the feasible answers set by simplifying the problem. + +
+ +***Relaxing*** + +One of the most common ways to find efficient heuristic function is that we convert our hard problem which is indissoluble in reasonable time, to a *more general problem with larger state space*. + +Discussing more specifically, original problem has a set of constraints. Noe we **ignore some constraints** and solve the expanded problem. Since number of constraints have decreased, we have more goal states and our state space expands. + +***Proof that Relaxing works*** + +Consider h as the answer and h' as our heuristic + +h' = min(S'), h = min(S) $\Lambda$ S $\subseteq$ S' $\Lambda$ $\forall$ s, s' if s $\subseteq$ s' => $min(s') \le min(s)$ => $h' \le h$ + + ***Example*** + + In below problem, which is an NP-hard problem (indissoluble in linear time), we want to find the Hamiltonian path with least weight. + +

+hamilton path +

+ +Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in $n^{2}$ time complexity. + +***Pattern DB*** + +In previous sectionwe introduced relaxing method which improves heuristic function by lowering constraints. Pattern DB is yet another way to improve heuristic function which is actually a subset of Relaxing method. + +In this method we choose a subset of constraints by a specific pattern and ignore the other constraints. Since we ignored some constraints we did relaxing method and hence answer of the converted problem is a heuristic gfunction. + +***Combining Pattern DB*** + +We can act further and use couple of patterns and choose couple of constraints' subsets. Therefore we have couple of heuristic functions. This helps dominancy of our heuristic function. First thing to do finding the dominant heuristic function is to choose the maximum function of the obtained heuristic functions. Since every heuristic is always less than the real answer, maximum of these functions is a lower bound of the real answer and for every set of obtained heuristic functions we can implement this method to find a dominant heuristic function. + +***Disjoint Pattern DB*** + +One of the main flaws of choosing maximum of the obtained heuristic functions is that in many cases, there is an upper bound for these functions. Therefore even by expanding subset of chosen constraints, we will not cross the upper bound. + +One solution for solving this drawback is Disjoint Pattern DB. In this method we apport set of constraints and define independent actions for the general problem. Now we can sum the obtained heuristic function to achieve a greater heuristic function, which improves the time complexity. It is proven that the sum of obtained heuristic functions is still a lower bound of real answer of the probelm. + +**Example** + +First practical problem of the first mini project of the course is an example of this method. In this problm we have an undirected graph, which each of its nodes is a florist shop and has some flower breeds. Two brothers should march from start_node towards goal_node in a way that has the least cost and they met every existing flower breeds. In this case we choose a subset of flower breeds S that reach goal_node by a specific pattern and set goal_state as satisfying path with the least cost and meeting every existing flower breeds of set S. Hitherto we used pure Pattern DB. + +Now we can have souple of flower breeds subset and combine obtained answers by them. For example if we have K flower breeds, we can solve the problem for each flower breed and calculate its heuristic function and then choose maximum of the functions. 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- distance -

-* Heuristic guidance means how far is the goal state from a given state approximately. - -***Admissiblity of a heuristic function means value of the function is always a Lower Bound of the remaining cost.*** - -So an **Admissible Heuristic** is a *non-negative* function h of nodes, where **h(n)** is *never greater than the actual cost of the shortest path from node n to a goal.* thus it means that the cost to reach the goal is never overestimated. - -Effect of **Admissibility** on a Heuristic is shown in the below schema: -

-admissible -

- -Now we define **f(n)** function as **f(n) = h(n) + g(n)** where g(n) is sum of costs from root to n node. - -***Monotonicity or Consistency is that the heuristic function holds in triangle inequality.*** Namely **f(n) is never Decreasing.** -

-consistency -

- - -Effect of **Monotonicity** on a Heuristic is shown in the below schema: -

-monotonic -

- -We will prove that **Consistency** implies **Admissibility** whereas the opposite is not necessarily true. -

-proof -

- -Now we want to show an **inconsistent, admissible example!** -So consider this figure: - -

-example -

- -If our heuristic is admissible, we have that **h(n) $\le$ h*(n)** for every node n where **h*** is the real cost to the goal. So we deduct that **h(A) $\le$ 4**, **h(C) $\le$ 3** and **h(G) $\le$ 0**. - -If we want our heuristic to be *Consistent* we should have **h(G) = 0** and **h(n) $\le$ cost(n, c) + h(c)** so in our case we have **h(A) $\le$ 1 + h(C)** and **h(C) $\le$ 3 + h(G) = 3** - -Because of the *Admissibility* **h(C) should be less than 3**, but if **h(A) > 1 + h(C)** then our heuristic is *Inconsistent!*. Hence if we assume that **h(C) = 1**, **h(G) = 0** and **h(A) = 4** our heuristic is *Admissible but Inconsistent!* - - -We also can make a non-monotonic heuristic, monotonic with some changes! -For this we only need to define a new and more appropriate heuristic like the below schema. -

-new heuristic -

- - - -**$\overline{h}$(n’) = max(h(n’), ℏ(n’) - c(n, a, n’))** - -Above Heuristic is defined Recursively. -

-heuristic proof -

- -**$\overline{h}$(n’) $\ge$ $\overline{h}$(n) - c(n, a, n')** - -So obviously we have **$\overline{h}$(n’) + c(n, a, n') $\ge$ $\overline{h}$(n)** - -***Heuristic Dominance*** - -For one heuristic to dominate another, all of its values must be greater than or equal to the corresponding values of the other heuristic. -So if $h_{2}$ dominates $h_{1}$(both are admissible) it means that A* algorithm expands less nodes with $h_{2}$ in the target path than $h_{1}$. -On another hand it means if a node is expanded by $h_{2}$ it will be expanded by $h_{1}$(consider that it’s not a biconditional proposition!) - -*Proof:* - -If $h_{2}$ dominate $h_{1}$, A* with $h_{2}$ expands at most the same number of nodes as when using $h_{1}$. - -A* opens all node with **f(n) < C** and some with **f(n) = C**. - - -Note that -**$h_{2}(n)$ -$\ge$ -$h_{1}(n)$** - -If node n is expanded by A* with $h_{2}$ then **$h_{2}(n)$ < C - g(n)** - -So ***$h_{1}(n)$ $\le$ $h_{2}(n)$ < C - g(n)*** and it would be expanded by A* with $h_{1}$ as well. - -We also can improve our heuristic by some methods. - -We are always trying to lower the difference between h* and h, expanded nodes and algorithm time complexity by making change in our heuristic. - -One of the main ways for improving heuristics is **Relaxing** the problem. - -In this procedure we try to enlarge the feasible answers set by simplifying the problem. - -
- -***Relaxing*** - -One of the most common ways to find efficient heuristic function is that we convert our hard problem which is indissoluble in reasonable time, to a *more general problem with larger state space*. - -Discussing more specifically, original problem has a set of constraints. Noe we **ignore some constraints** and solve the expanded problem. Since number of constraints have decreased, we have more goal states and our state space expands. - -***Proof that Relaxing works*** - -Consider h as the answer and h' as our heuristic - -h' = min(S'), h = min(S) $\Lambda$ S $\subseteq$ S' $\Lambda$ $\forall$ s, s' if s $\subseteq$ s' => $min(s') \le min(s)$ => $h' \le h$ - - ***Example*** - - In below problem, which is an NP-hard problem (indissoluble in linear time), we want to find the Hamiltonian path with least weight. - -

-hamilton path -

- -Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in $n^{2}$ time complexity. - -***Pattern DB*** - -In previous sectionwe introduced relaxing method which improves heuristic function by lowering constraints. Pattern DB is yet another way to improve heuristic function which is actually a subset of Relaxing method. - -In this method we choose a subset of constraints by a specific pattern and ignore the other constraints. Since we ignored some constraints we did relaxing method and hence answer of the converted problem is a heuristic gfunction. - -***Combining Pattern DB*** - -We can act further and use couple of patterns and choose couple of constraints' subsets. Therefore we have couple of heuristic functions. This helps dominancy of our heuristic function. First thing to do finding the dominant heuristic function is to choose the maximum function of the obtained heuristic functions. Since every heuristic is always less than the real answer, maximum of these functions is a lower bound of the real answer and for every set of obtained heuristic functions we can implement this method to find a dominant heuristic function. - -***Disjoint Pattern DB*** - -One of the main flaws of choosing maximum of the obtained heuristic functions is that in many cases, there is an upper bound for these functions. Therefore even by expanding subset of chosen constraints, we will not cross the upper bound. - -One solution for solving this drawback is Disjoint Pattern DB. In this method we apport set of constraints and define independent actions for the general problem. Now we can sum the obtained heuristic function to achieve a greater heuristic function, which improves the time complexity. It is proven that the sum of obtained heuristic functions is still a lower bound of real answer of the probelm. - -**Example** - -First practical problem of the first mini project of the course is an example of this method. In this problm we have an undirected graph, which each of its nodes is a florist shop and has some flower breeds. Two brothers should march from start_node towards goal_node in a way that has the least cost and they met every existing flower breeds. In this case we choose a subset of flower breeds S that reach goal_node by a specific pattern and set goal_state as satisfying path with the least cost and meeting every existing flower breeds of set S. Hitherto we used pure Pattern DB. - -Now we can have souple of flower breeds subset and combine obtained answers by them. For example if we have K flower breeds, we can solve the problem for each flower breed and calculate its heuristic function and then choose maximum of the functions. So we used Combining Pattern DB to achieve a better heuristic function. diff --git a/notebooks/Advanced Heuristics/matadata.yml b/notebooks/Advanced Heuristics/matadata.yml deleted file mode 100644 index f23db3f1..00000000 --- a/notebooks/Advanced Heuristics/matadata.yml +++ /dev/null @@ -1,25 +0,0 @@ -title: # shown on browser tab - -header: - title: Advanced Heuristics # title of your notebook - description: # short description of your notebook - -authors: - label: - position: top - content: - # list of notebook authors - - name: Mohammadreza Daviran # name of author - role: Author # change this if you want - contact: - # list of contact information - - link: https://github.com/mmdrez4 - icon: fab fa-github - # optionally add other contact information like - # - link: # contact link - # icon: # awsomefont tag for link (check: https://fontawesome.com/v5.15/icons) - -comments: - # enable comments for your post - label: false - kind: comments diff --git a/notebooks/Advanced 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+ distance +

+* Heuristic guidance means how far is the goal state from a given state approximately. + +***Admissiblity of a heuristic function means value of the function is always a Lower Bound of the remaining cost.*** + +So an **Admissible Heuristic** is a *non-negative* function h of nodes, where **h(n)** is *never greater than the actual cost of the shortest path from node n to a goal.* thus it means that the cost to reach the goal is never overestimated. + +Effect of **Admissibility** on a Heuristic is shown in the below schema: +

+admissible +

+ +Now we define **f(n)** function as **f(n) = h(n) + g(n)** where g(n) is sum of costs from root to n node. + +***Monotonicity or Consistency is that the heuristic function holds in triangle inequality.*** Namely **f(n) is never Decreasing.** +

+consistency +

+ + +Effect of **Monotonicity** on a Heuristic is shown in the below schema: +

+monotonic +

+ +We will prove that **Consistency** implies **Admissibility** whereas the opposite is not necessarily true. +

+proof +

+ +Now we want to show an **inconsistent, admissible example!** +So consider this figure: + +

+example +

+ +If our heuristic is admissible, we have that **h(n) $\le$ h*(n)** for every node n where **h*** is the real cost to the goal. So we deduct that **h(A) $\le$ 4**, **h(C) $\le$ 3** and **h(G) $\le$ 0**. + +If we want our heuristic to be *Consistent* we should have **h(G) = 0** and **h(n) $\le$ cost(n, c) + h(c)** so in our case we have **h(A) $\le$ 1 + h(C)** and **h(C) $\le$ 3 + h(G) = 3** + +Because of the *Admissibility* **h(C) should be less than 3**, but if **h(A) > 1 + h(C)** then our heuristic is *Inconsistent!*. Hence if we assume that **h(C) = 1**, **h(G) = 0** and **h(A) = 4** our heuristic is *Admissible but Inconsistent!* + + +We also can make a non-monotonic heuristic, monotonic with some changes! +For this we only need to define a new and more appropriate heuristic like the below schema. +

+new heuristic +

+ + + +**$\overline{h}$(n’) = max(h(n’), ℏ(n’) - c(n, a, n’))** + +Above Heuristic is defined Recursively. +

+heuristic proof +

+ +**$\overline{h}$(n’) $\ge$ $\overline{h}$(n) - c(n, a, n')** + +So obviously we have **$\overline{h}$(n’) + c(n, a, n') $\ge$ $\overline{h}$(n)** + +***Heuristic Dominance*** + +For one heuristic to dominate another, all of its values must be greater than or equal to the corresponding values of the other heuristic. +So if $h_{2}$ dominates $h_{1}$(both are admissible) it means that A* algorithm expands less nodes with $h_{2}$ in the target path than $h_{1}$. +On another hand it means if a node is expanded by $h_{2}$ it will be expanded by $h_{1}$(consider that it’s not a biconditional proposition!) + +*Proof:* + +If $h_{2}$ dominate $h_{1}$, A* with $h_{2}$ expands at most the same number of nodes as when using $h_{1}$. + +A* opens all node with **f(n) < C** and some with **f(n) = C**. + + +Note that +**$h_{2}(n)$ +$\ge$ +$h_{1}(n)$** + +If node n is expanded by A* with $h_{2}$ then **$h_{2}(n)$ < C - g(n)** + +So ***$h_{1}(n)$ $\le$ $h_{2}(n)$ < C - g(n)*** and it would be expanded by A* with $h_{1}$ as well. + +We also can improve our heuristic by some methods. + +We are always trying to lower the difference between h* and h, expanded nodes and algorithm time complexity by making change in our heuristic. + +One of the main ways for improving heuristics is **Relaxing** the problem. + +In this procedure we try to enlarge the feasible answers set by simplifying the problem. + +
+ +***Relaxing*** + +One of the most common ways to find efficient heuristic function is that we convert our hard problem which is indissoluble in reasonable time, to a *more general problem with larger state space*. + +Discussing more specifically, original problem has a set of constraints. Noe we **ignore some constraints** and solve the expanded problem. Since number of constraints have decreased, we have more goal states and our state space expands. + +***Proof that Relaxing works*** + +Consider h as the answer and h' as our heuristic + +h' = min(S'), h = min(S) $\Lambda$ S $\subseteq$ S' $\Lambda$ $\forall$ s, s' if s $\subseteq$ s' => $min(s') \le min(s)$ => $h' \le h$ + + ***Example*** + + In below problem, which is an NP-hard problem (indissoluble in linear time), we want to find the Hamiltonian path with least weight. + +

+hamilton path +

+ +Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in $n^{2}$ time complexity. + +***Pattern DB*** + +In previous sectionwe introduced relaxing method which improves heuristic function by lowering constraints. Pattern DB is yet another way to improve heuristic function which is actually a subset of Relaxing method. + +In this method we choose a subset of constraints by a specific pattern and ignore the other constraints. Since we ignored some constraints we did relaxing method and hence answer of the converted problem is a heuristic gfunction. + +***Combining Pattern DB*** + +We can act further and use couple of patterns and choose couple of constraints' subsets. Therefore we have couple of heuristic functions. This helps dominancy of our heuristic function. First thing to do finding the dominant heuristic function is to choose the maximum function of the obtained heuristic functions. Since every heuristic is always less than the real answer, maximum of these functions is a lower bound of the real answer and for every set of obtained heuristic functions we can implement this method to find a dominant heuristic function. + +***Disjoint Pattern DB*** + +One of the main flaws of choosing maximum of the obtained heuristic functions is that in many cases, there is an upper bound for these functions. Therefore even by expanding subset of chosen constraints, we will not cross the upper bound. + +One solution for solving this drawback is Disjoint Pattern DB. In this method we apport set of constraints and define independent actions for the general problem. Now we can sum the obtained heuristic function to achieve a greater heuristic function, which improves the time complexity. It is proven that the sum of obtained heuristic functions is still a lower bound of real answer of the probelm. + +**Example** + +First practical problem of the first mini project of the course is an example of this method. In this problm we have an undirected graph, which each of its nodes is a florist shop and has some flower breeds. Two brothers should march from start_node towards goal_node in a way that has the least cost and they met every existing flower breeds. In this case we choose a subset of flower breeds S that reach goal_node by a specific pattern and set goal_state as satisfying path with the least cost and meeting every existing flower breeds of set S. Hitherto we used pure Pattern DB. + +Now we can have souple of flower breeds subset and combine obtained answers by them. For example if we have K flower breeds, we can solve the problem for each flower breed and calculate its heuristic function and then choose maximum of the functions. 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- distance + distance

* Heuristic guidance means how far is the goal state from a given state approximately. @@ -17,32 +17,32 @@ So an **Admissible Heuristic** is a *non-negative* function h of nodes, where ** Effect of **Admissibility** on a Heuristic is shown in the below schema:

-admissible +admissible

Now we define **f(n)** function as **f(n) = h(n) + g(n)** where g(n) is sum of costs from root to n node. ***Monotonicity or Consistency is that the heuristic function holds in triangle inequality.*** Namely **f(n) is never Decreasing.**

-consistency +consistency

Effect of **Monotonicity** on a Heuristic is shown in the below schema:

-monotonic +monotonic

We will prove that **Consistency** implies **Admissibility** whereas the opposite is not necessarily true.

-proof +proof

Now we want to show an **inconsistent, admissible example!** So consider this figure:

-example +example

If our heuristic is admissible, we have that **h(n) $\le$ h*(n)** for every node n where **h*** is the real cost to the goal. So we deduct that **h(A) $\le$ 4**, **h(C) $\le$ 3** and **h(G) $\le$ 0**. @@ -55,7 +55,7 @@ Because of the *Admissibility* **h(C) should be less than 3**, but if **h(A) > 1 We also can make a non-monotonic heuristic, monotonic with some changes! For this we only need to define a new and more appropriate heuristic like the below schema.

-new heuristic +new heuristic

@@ -64,7 +64,7 @@ For this we only need to define a new and more appropriate heuristic like the be Above Heuristic is defined Recursively.

-heuristic proof +heuristic proof

**$\overline{h}$(n’) $\ge$ $\overline{h}$(n) - c(n, a, n')** @@ -120,7 +120,7 @@ h' = min(S'), h = min(S) $\Lambda$ S $\subseteq$ S' $\Lambda$ $\forall$ s, s' if In below problem, which is an NP-hard problem (indissoluble in linear time), we want to find the Hamiltonian path with least weight.

-hamilton path +hamilton path

Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in $n^{2}$ time complexity. From 16e60fc9b66771a0d1792aac30f9df139e15d88e Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 18:23:51 +0330 Subject: [PATCH 007/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 1ca6ff50..5e62a459 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -32,6 +32,8 @@ Effect of **Monotonicity** on a Heuristic is shown in the below schema:

monotonic

+ +for more information and better understanding watch this video: [watch](https://www.youtube.com/watch?v=CJmlP03ik5g) We will prove that **Consistency** implies **Admissibility** whereas the opposite is not necessarily true.

From a9db3b6c17b68a21cef083438961478fa4252a65 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 18:30:15 +0330 Subject: [PATCH 008/114] Update index.md --- notebooks/advanced_heuristics/index.md | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 5e62a459..48271f55 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -1,6 +1,9 @@ # Advanced Heuristic #### Kahbod Aeini, Mohammadreza Daviran and Sepehr Kianian - + +## Introduction +nothing yet! + First we review some basic definitions Distance is a numerical measurement of how far apart objects or points are: From faee70f4f21f8e9bb7f9e43167403255911f0210 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 18:33:34 +0330 Subject: [PATCH 009/114] Update index.md --- notebooks/advanced_heuristics/index.md | 8 ++++++++ 1 file changed, 8 insertions(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 48271f55..694c58d7 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -1,9 +1,13 @@ # Advanced Heuristic #### Kahbod Aeini, Mohammadreza Daviran and Sepehr Kianian +## Contents + ## Introduction nothing yet! +## Body + First we review some basic definitions Distance is a numerical measurement of how far apart objects or points are: @@ -151,3 +155,7 @@ One solution for solving this drawback is Disjoint Pattern DB. In this method we First practical problem of the first mini project of the course is an example of this method. In this problm we have an undirected graph, which each of its nodes is a florist shop and has some flower breeds. Two brothers should march from start_node towards goal_node in a way that has the least cost and they met every existing flower breeds. In this case we choose a subset of flower breeds S that reach goal_node by a specific pattern and set goal_state as satisfying path with the least cost and meeting every existing flower breeds of set S. Hitherto we used pure Pattern DB. Now we can have souple of flower breeds subset and combine obtained answers by them. For example if we have K flower breeds, we can solve the problem for each flower breed and calculate its heuristic function and then choose maximum of the functions. So we used Combining Pattern DB to achieve a better heuristic function. + +## Conclusion + +## References From 57da872b55c6376de438923688f97ea4a2a8d443 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 18:42:59 +0330 Subject: [PATCH 010/114] Update index.md --- notebooks/advanced_heuristics/index.md | 23 ++++++++++++++--------- 1 file changed, 14 insertions(+), 9 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 694c58d7..46298951 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -8,7 +8,8 @@ nothing yet! ## Body -First we review some basic definitions +### Distance +First we review some basic definitions: Distance is a numerical measurement of how far apart objects or points are: * [Euclidean Distance](https://en.wikipedia.org/wiki/Euclidean_distance) calculates the distance between two real-valued vectors. @@ -16,6 +17,8 @@ Distance is a numerical measurement of how far apart objects or points are:

distance

+ +### Heuristic * Heuristic guidance means how far is the goal state from a given state approximately. ***Admissiblity of a heuristic function means value of the function is always a Lower Bound of the remaining cost.*** @@ -41,7 +44,8 @@ Effect of **Monotonicity** on a Heuristic is shown in the below schema:

for more information and better understanding watch this video: [watch](https://www.youtube.com/watch?v=CJmlP03ik5g) - + +### Monotonicity implies admissibility We will prove that **Consistency** implies **Admissibility** whereas the opposite is not necessarily true.

proof @@ -60,7 +64,7 @@ If we want our heuristic to be *Consistent* we should have **h(G) = 0** and **h( Because of the *Admissibility* **h(C) should be less than 3**, but if **h(A) > 1 + h(C)** then our heuristic is *Inconsistent!*. Hence if we assume that **h(C) = 1**, **h(G) = 0** and **h(A) = 4** our heuristic is *Admissible but Inconsistent!* - +### Monotonic heuristic function We also can make a non-monotonic heuristic, monotonic with some changes! For this we only need to define a new and more appropriate heuristic like the below schema.

@@ -80,7 +84,7 @@ Above Heuristic is defined Recursively. So obviously we have **$\overline{h}$(n’) + c(n, a, n') $\ge$ $\overline{h}$(n)** -***Heuristic Dominance*** +### Heuristic Dominance For one heuristic to dominate another, all of its values must be greater than or equal to the corresponding values of the other heuristic. So if $h_{2}$ dominates $h_{1}$(both are admissible) it means that A* algorithm expands less nodes with $h_{2}$ in the target path than $h_{1}$. @@ -101,7 +105,8 @@ $h_{1}(n)$** If node n is expanded by A* with $h_{2}$ then **$h_{2}(n)$ < C - g(n)** So ***$h_{1}(n)$ $\le$ $h_{2}(n)$ < C - g(n)*** and it would be expanded by A* with $h_{1}$ as well. - + + We also can improve our heuristic by some methods. We are always trying to lower the difference between h* and h, expanded nodes and algorithm time complexity by making change in our heuristic. @@ -112,7 +117,7 @@ In this procedure we try to enlarge the feasible answers set by simplifying the


-***Relaxing*** +### Relaxing One of the most common ways to find efficient heuristic function is that we convert our hard problem which is indissoluble in reasonable time, to a *more general problem with larger state space*. @@ -134,17 +139,17 @@ h' = min(S'), h = min(S) $\Lambda$ S $\subseteq$ S' $\Lambda$ $\forall$ s, s' if Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in $n^{2}$ time complexity. -***Pattern DB*** +### Pattern DB In previous sectionwe introduced relaxing method which improves heuristic function by lowering constraints. Pattern DB is yet another way to improve heuristic function which is actually a subset of Relaxing method. In this method we choose a subset of constraints by a specific pattern and ignore the other constraints. Since we ignored some constraints we did relaxing method and hence answer of the converted problem is a heuristic gfunction. -***Combining Pattern DB*** +### Combining Pattern DB We can act further and use couple of patterns and choose couple of constraints' subsets. Therefore we have couple of heuristic functions. This helps dominancy of our heuristic function. First thing to do finding the dominant heuristic function is to choose the maximum function of the obtained heuristic functions. Since every heuristic is always less than the real answer, maximum of these functions is a lower bound of the real answer and for every set of obtained heuristic functions we can implement this method to find a dominant heuristic function. -***Disjoint Pattern DB*** +### Disjoint Pattern DB One of the main flaws of choosing maximum of the obtained heuristic functions is that in many cases, there is an upper bound for these functions. Therefore even by expanding subset of chosen constraints, we will not cross the upper bound. From 980d45504991b062f53faa037bb45543f42d2255 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 18:45:57 +0330 Subject: [PATCH 011/114] Update index.md --- notebooks/advanced_heuristics/index.md | 12 ++++++++++++ 1 file changed, 12 insertions(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 46298951..bd8bd818 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -18,6 +18,8 @@ Distance is a numerical measurement of how far apart objects or points are: distance

+
+ ### Heuristic * Heuristic guidance means how far is the goal state from a given state approximately. @@ -45,6 +47,8 @@ Effect of **Monotonicity** on a Heuristic is shown in the below schema: for more information and better understanding watch this video: [watch](https://www.youtube.com/watch?v=CJmlP03ik5g) +
+ ### Monotonicity implies admissibility We will prove that **Consistency** implies **Admissibility** whereas the opposite is not necessarily true.

@@ -84,6 +88,8 @@ Above Heuristic is defined Recursively. So obviously we have **$\overline{h}$(n’) + c(n, a, n') $\ge$ $\overline{h}$(n)** +


+ ### Heuristic Dominance For one heuristic to dominate another, all of its values must be greater than or equal to the corresponding values of the other heuristic. @@ -139,16 +145,22 @@ h' = min(S'), h = min(S) $\Lambda$ S $\subseteq$ S' $\Lambda$ $\forall$ s, s' if Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in $n^{2}$ time complexity. +
+ ### Pattern DB In previous sectionwe introduced relaxing method which improves heuristic function by lowering constraints. Pattern DB is yet another way to improve heuristic function which is actually a subset of Relaxing method. In this method we choose a subset of constraints by a specific pattern and ignore the other constraints. Since we ignored some constraints we did relaxing method and hence answer of the converted problem is a heuristic gfunction. +
+ ### Combining Pattern DB We can act further and use couple of patterns and choose couple of constraints' subsets. Therefore we have couple of heuristic functions. This helps dominancy of our heuristic function. First thing to do finding the dominant heuristic function is to choose the maximum function of the obtained heuristic functions. Since every heuristic is always less than the real answer, maximum of these functions is a lower bound of the real answer and for every set of obtained heuristic functions we can implement this method to find a dominant heuristic function. +
+ ### Disjoint Pattern DB One of the main flaws of choosing maximum of the obtained heuristic functions is that in many cases, there is an upper bound for these functions. Therefore even by expanding subset of chosen constraints, we will not cross the upper bound. From 7c38aff22dd15c450ece01a7a1c3e5f75092b331 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 18:47:37 +0330 Subject: [PATCH 012/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index bd8bd818..bc9126ea 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -68,6 +68,8 @@ If we want our heuristic to be *Consistent* we should have **h(G) = 0** and **h( Because of the *Admissibility* **h(C) should be less than 3**, but if **h(A) > 1 + h(C)** then our heuristic is *Inconsistent!*. Hence if we assume that **h(C) = 1**, **h(G) = 0** and **h(A) = 4** our heuristic is *Admissible but Inconsistent!* +
+ ### Monotonic heuristic function We also can make a non-monotonic heuristic, monotonic with some changes! For this we only need to define a new and more appropriate heuristic like the below schema. From 5badd0d587cd24aab9a0d4a3c29a0978a645cf64 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 18:56:26 +0330 Subject: [PATCH 013/114] Update index.md --- notebooks/advanced_heuristics/index.md | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index bc9126ea..20fee5a0 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -1,7 +1,14 @@ -# Advanced Heuristic +# Advanced Heuristics #### Kahbod Aeini, Mohammadreza Daviran and Sepehr Kianian ## Contents +[Text](link) + +[Text](link) + +[Text](link) + +[Text](link) ## Introduction nothing yet! From 51630dd70aef09ba0a6a9b95de0163fb859d84fd Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:00:50 +0330 Subject: [PATCH 014/114] add table of contents --- notebooks/advanced_heuristics/index.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 20fee5a0..1623efc2 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -2,13 +2,13 @@ #### Kahbod Aeini, Mohammadreza Daviran and Sepehr Kianian ## Contents -[Text](link) +[Introduction](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#introduction) -[Text](link) +[Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body) -[Text](link) +[Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) -[Text](link) +[References](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#references) ## Introduction nothing yet! From c7d6df9f24753c4a675c0097f4189717e7b9cac1 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:03:47 +0330 Subject: [PATCH 015/114] Update index.md --- notebooks/advanced_heuristics/index.md | 1 + 1 file changed, 1 insertion(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 1623efc2..20b9361d 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -5,6 +5,7 @@ [Introduction](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#introduction) [Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body) +    hello [Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) From cca9b1a131e430ffbd9debf8846cb7b6a5928ba8 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:05:34 +0330 Subject: [PATCH 016/114] Update index.md --- notebooks/advanced_heuristics/index.md | 1 + 1 file changed, 1 insertion(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 20b9361d..f9d595a3 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -5,6 +5,7 @@ [Introduction](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#introduction) [Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body) +
    hello [Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) From 28e2cbae5e8de2081fb7e34bbf7b6f0aa3d3349c Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:10:05 +0330 Subject: [PATCH 017/114] Update index.md --- notebooks/advanced_heuristics/index.md | 11 ++++++++++- 1 file changed, 10 insertions(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index f9d595a3..d87dd1f9 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -6,7 +6,16 @@ [Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body)
-    hello +        [distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance) +        [heuristic](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic) +        [monotonicity implies admissibility](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonicity-implies-admissibility) +        [monotonic heuristic function](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonic-heuristic-function) +        [heuristic dominance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic-dominance) +        [relaxing](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#relaxing) +        [pattern-db](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#pattern-db) +        [combining-pattern-db](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#combining-pattern-db) +        [disjoint Pattern db](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#disjoint-pattern-db) + [Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) From 8ebe58b9cc8986fa346dd42c87d29837640a3cd1 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:11:29 +0330 Subject: [PATCH 018/114] Update index.md --- notebooks/advanced_heuristics/index.md | 26 +++++++++++++++++--------- 1 file changed, 17 insertions(+), 9 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index d87dd1f9..0b6a4cee 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -6,15 +6,23 @@ [Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body)
-        [distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance) -        [heuristic](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic) -        [monotonicity implies admissibility](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonicity-implies-admissibility) -        [monotonic heuristic function](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonic-heuristic-function) -        [heuristic dominance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic-dominance) -        [relaxing](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#relaxing) -        [pattern-db](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#pattern-db) -        [combining-pattern-db](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#combining-pattern-db) -        [disjoint Pattern db](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#disjoint-pattern-db) +        [Distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance) + +        [Heuristic](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic) + +        [Monotonicity implies admissibility](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonicity-implies-admissibility) + +        [Monotonic heuristic function](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonic-heuristic-function) + +        [Heuristic dominance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic-dominance) + +        [Relaxing](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#relaxing) + +        [Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#pattern-db) + +        [Combining Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#combining-pattern-db) + +        [Disjoint Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#disjoint-pattern-db) [Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) From 7c6fb89bc6d193c4d58cbdb5f8b26bb973c35517 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:12:43 +0330 Subject: [PATCH 019/114] Update index.md --- notebooks/advanced_heuristics/index.md | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 0b6a4cee..2e3f8a3d 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -2,10 +2,11 @@ #### Kahbod Aeini, Mohammadreza Daviran and Sepehr Kianian ## Contents -[Introduction](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#introduction) +1.[Introduction](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#introduction) -[Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body) +2.[Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body)
+         [Distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance)         [Heuristic](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic) @@ -25,9 +26,9 @@         [Disjoint Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#disjoint-pattern-db) -[Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) +3.[Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) -[References](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#references) +4.[References](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#references) ## Introduction nothing yet! From d6332789fc1a93c1327174140b4b5d3cffd3dd13 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:13:20 +0330 Subject: [PATCH 020/114] Update index.md --- notebooks/advanced_heuristics/index.md | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 2e3f8a3d..2079c0e0 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -7,23 +7,23 @@ 2.[Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body)
-        [Distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance) +        * [Distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance) -        [Heuristic](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic) +        * [Heuristic](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic) -        [Monotonicity implies admissibility](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonicity-implies-admissibility) +        * [Monotonicity implies admissibility](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonicity-implies-admissibility) -        [Monotonic heuristic function](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonic-heuristic-function) +        * [Monotonic heuristic function](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonic-heuristic-function) -        [Heuristic dominance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic-dominance) +        * [Heuristic dominance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic-dominance) -        [Relaxing](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#relaxing) +        * [Relaxing](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#relaxing) -        [Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#pattern-db) +        * [Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#pattern-db) -        [Combining Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#combining-pattern-db) +        * [Combining Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#combining-pattern-db) -        [Disjoint Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#disjoint-pattern-db) +        * [Disjoint Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#disjoint-pattern-db) 3.[Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) From 7199db1d57b1e965a51df525ae74eda78c8dc1df Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:13:58 +0330 Subject: [PATCH 021/114] Update index.md --- notebooks/advanced_heuristics/index.md | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 2079c0e0..cb03664f 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -7,23 +7,23 @@ 2.[Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body)
-        * [Distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance) +         * [Distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance) -        * [Heuristic](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic) +         * [Heuristic](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic) -        * [Monotonicity implies admissibility](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonicity-implies-admissibility) +         * [Monotonicity implies admissibility](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonicity-implies-admissibility) -        * [Monotonic heuristic function](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonic-heuristic-function) +         * [Monotonic heuristic function](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonic-heuristic-function) -        * [Heuristic dominance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic-dominance) +         * [Heuristic dominance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic-dominance) -        * [Relaxing](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#relaxing) +         * [Relaxing](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#relaxing) -        * [Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#pattern-db) +         * [Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#pattern-db) -        * [Combining Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#combining-pattern-db) +         * [Combining Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#combining-pattern-db) -        * [Disjoint Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#disjoint-pattern-db) +         * [Disjoint Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#disjoint-pattern-db) 3.[Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) From 09d90ece8a5d125d498d071ea59a52d1e3a503ff Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:15:13 +0330 Subject: [PATCH 022/114] Update index.md --- notebooks/advanced_heuristics/index.md | 27 +++++++++++++++++--------- 1 file changed, 18 insertions(+), 9 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index cb03664f..b9ddca55 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -7,23 +7,32 @@ 2.[Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body)
-         * [Distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance) +         +* [Distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance) -         * [Heuristic](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic) +         +* [Heuristic](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic) -         * [Monotonicity implies admissibility](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonicity-implies-admissibility) +         +* [Monotonicity implies admissibility](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonicity-implies-admissibility) -         * [Monotonic heuristic function](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonic-heuristic-function) +         +* [Monotonic heuristic function](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonic-heuristic-function) -         * [Heuristic dominance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic-dominance) +         +* [Heuristic dominance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic-dominance) -         * [Relaxing](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#relaxing) +         +* [Relaxing](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#relaxing) -         * [Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#pattern-db) +         +* [Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#pattern-db) -         * [Combining Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#combining-pattern-db) +         +* [Combining Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#combining-pattern-db) -         * [Disjoint Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#disjoint-pattern-db) +         +* [Disjoint Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#disjoint-pattern-db) 3.[Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) From 80ed11da6c95523a10542e29d4c8155e0cfc18f7 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:15:41 +0330 Subject: [PATCH 023/114] Update index.md --- notebooks/advanced_heuristics/index.md | 9 --------- 1 file changed, 9 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index b9ddca55..d980cba8 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -9,32 +9,23 @@          * [Distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance) -          * [Heuristic](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic) -          * [Monotonicity implies admissibility](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonicity-implies-admissibility) -          * [Monotonic heuristic function](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#monotonic-heuristic-function) -          * [Heuristic dominance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic-dominance) -          * [Relaxing](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#relaxing) -          * [Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#pattern-db) -          * [Combining Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#combining-pattern-db) -          * [Disjoint Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#disjoint-pattern-db) - 3.[Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) 4.[References](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#references) From 8f3466036ee4ddc3a4179691cc95b01825f103f1 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:15:56 +0330 Subject: [PATCH 024/114] Create index.md From e37d8ab6d0c27e7141e648b1aa28e8d91705e19a Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:16:19 +0330 Subject: [PATCH 025/114] Update index.md --- notebooks/advanced_heuristics/index.md | 1 - 1 file changed, 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index d980cba8..797237fc 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -5,7 +5,6 @@ 1.[Introduction](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#introduction) 2.[Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body) -
         * [Distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance) From 96a3d16877fd5b226bfe7b8f7ed173e2744b066d Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:16:37 +0330 Subject: [PATCH 026/114] Update index.md --- notebooks/advanced_heuristics/index.md | 1 - 1 file changed, 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 797237fc..22b7ca63 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -5,7 +5,6 @@ 1.[Introduction](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#introduction) 2.[Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body) -          * [Distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance)          From 77a7f7a8632e93de3a90efe1393473595e3f3b3b Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:17:24 +0330 Subject: [PATCH 027/114] finish table of contents --- notebooks/advanced_heuristics/index.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 22b7ca63..e3bf79b5 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -2,9 +2,9 @@ #### Kahbod Aeini, Mohammadreza Daviran and Sepehr Kianian ## Contents -1.[Introduction](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#introduction) +1. [Introduction](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#introduction) -2.[Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body) +2. [Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body)          * [Distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance)          @@ -24,9 +24,9 @@          * [Disjoint Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#disjoint-pattern-db) -3.[Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) +3. [Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) -4.[References](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#references) +4. [References](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#references) ## Introduction nothing yet! From c2d4f2c159f72597cbdcf96fb2ecc82146e0886b Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:17:57 +0330 Subject: [PATCH 028/114] Update index.md --- notebooks/advanced_heuristics/index.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index e3bf79b5..22b7ca63 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -2,9 +2,9 @@ #### Kahbod Aeini, Mohammadreza Daviran and Sepehr Kianian ## Contents -1. [Introduction](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#introduction) +1.[Introduction](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#introduction) -2. [Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body) +2.[Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body)          * [Distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance)          @@ -24,9 +24,9 @@          * [Disjoint Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#disjoint-pattern-db) -3. [Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) +3.[Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) -4. [References](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#references) +4.[References](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#references) ## Introduction nothing yet! From 166ae0710c4cec3501dcb01e554e1eedb4c1e689 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:21:42 +0330 Subject: [PATCH 029/114] Update index.md --- notebooks/advanced_heuristics/index.md | 1 + 1 file changed, 1 insertion(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 22b7ca63..4eed2bb0 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -180,6 +180,7 @@ In previous sectionwe introduced relaxing method which improves heuristic functi In this method we choose a subset of constraints by a specific pattern and ignore the other constraints. Since we ignored some constraints we did relaxing method and hence answer of the converted problem is a heuristic gfunction. +for further information watch this video: [watch](https://www.youtube.com/watch?v=HZWV4uOJWk8)
### Combining Pattern DB From db2d38dacf64bce55be8598a628de768d4733595 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:29:30 +0330 Subject: [PATCH 030/114] Update index.md --- notebooks/advanced_heuristics/index.md | 1 + 1 file changed, 1 insertion(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 4eed2bb0..13eb42df 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -30,6 +30,7 @@ ## Introduction nothing yet! +a $\geqslant$ b ## Body From fceb3fe44259139bdba4ad36647422e3198df0b0 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:29:56 +0330 Subject: [PATCH 031/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 13eb42df..33d36dd7 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -30,7 +30,7 @@ ## Introduction nothing yet! -a $\geqslant$ b +a $\ge$ b ## Body From cf99738c8dbdab82401fdd1ef364552e2bafa3e5 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:30:16 +0330 Subject: [PATCH 032/114] Update index.md --- notebooks/advanced_heuristics/index.md | 1 + 1 file changed, 1 insertion(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 33d36dd7..5fb69bae 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -30,6 +30,7 @@ ## Introduction nothing yet! + a $\ge$ b ## Body From d2a4e6a692c5af6db48e29040ddb5b0da3021a6d Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:30:31 +0330 Subject: [PATCH 033/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 5fb69bae..e4011f94 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -31,7 +31,7 @@ ## Introduction nothing yet! -a $\ge$ b +a$\ge$b ## Body From 356bc7dbc8290ca292cb8b204db6e293d3b95d29 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:34:18 +0330 Subject: [PATCH 034/114] Update index.md --- notebooks/advanced_heuristics/index.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index e4011f94..53a80ac3 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -31,7 +31,9 @@ ## Introduction nothing yet! -a$\ge$b +$a\ge b$ + +$a<b$ ## Body From 9109e3c7c9c041d7222dcc6156523b0c3e5a4822 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:35:17 +0330 Subject: [PATCH 035/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 53a80ac3..9f42fbc8 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -31,7 +31,7 @@ ## Introduction nothing yet! -$a\ge b$ +$a \ge b$ $a<b$ From dbcd5bc517c7df66233b1c7020a04e024794b857 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:36:58 +0330 Subject: [PATCH 036/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 9f42fbc8..015a808f 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -33,7 +33,7 @@ nothing yet! $a \ge b$ -$a<b$ +$a≤b$ ## Body From 4ae8652a19dd8ca8344be8a459eab8b33c9cbd23 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:37:19 +0330 Subject: [PATCH 037/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 015a808f..54412618 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -31,7 +31,7 @@ ## Introduction nothing yet! -$a \ge b$ +$a ≥ b$ $a≤b$ From f095d72b18d37a9d81691b382e34e7e45ef35598 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:38:14 +0330 Subject: [PATCH 038/114] Update index.md --- notebooks/advanced_heuristics/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 54412618..beced65d 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -31,7 +31,7 @@ ## Introduction nothing yet! -$a ≥ b$ +a $≥$ b $a≤b$ @@ -91,7 +91,7 @@ So consider this figure: example

-If our heuristic is admissible, we have that **h(n) $\le$ h*(n)** for every node n where **h*** is the real cost to the goal. So we deduct that **h(A) $\le$ 4**, **h(C) $\le$ 3** and **h(G) $\le$ 0**. +If our heuristic is admissible, we have that **h(n) $≤$ h*(n)** for every node n where **h*** is the real cost to the goal. So we deduct that **h(A) $\le$ 4**, **h(C) $\le$ 3** and **h(G) $\le$ 0**. If we want our heuristic to be *Consistent* we should have **h(G) = 0** and **h(n) $\le$ cost(n, c) + h(c)** so in our case we have **h(A) $\le$ 1 + h(C)** and **h(C) $\le$ 3 + h(G) = 3** From 0f0e75f6b99980fc08773592383c2f8c5d9ac1ec Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:38:40 +0330 Subject: [PATCH 039/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index beced65d..5724dc55 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -31,7 +31,7 @@ ## Introduction nothing yet! -a $≥$ b +a ≥ b $a≤b$ From 9881f033c0640b67ea171128357e8a772dc3ed98 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:41:55 +0330 Subject: [PATCH 040/114] Update index.md --- notebooks/advanced_heuristics/index.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 5724dc55..29df9ca5 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -91,9 +91,9 @@ So consider this figure: example

-If our heuristic is admissible, we have that **h(n) $≤$ h*(n)** for every node n where **h*** is the real cost to the goal. So we deduct that **h(A) $\le$ 4**, **h(C) $\le$ 3** and **h(G) $\le$ 0**. +If our heuristic is admissible, we have that **h(n) ≤ h*(n)** for every node n where **h*** is the real cost to the goal. So we deduct that **h(A) ≤ 4**, **h(C) ≤ 3** and **h(G) ≤ 0**. -If we want our heuristic to be *Consistent* we should have **h(G) = 0** and **h(n) $\le$ cost(n, c) + h(c)** so in our case we have **h(A) $\le$ 1 + h(C)** and **h(C) $\le$ 3 + h(G) = 3** +If we want our heuristic to be *Consistent* we should have **h(G) = 0** and **h(n) $\le$ cost(n, c) + h(c)** so in our case we have **h(A) ≤ 1 + h(C)** and **h(C) ≤ 3 + h(G) = 3** Because of the *Admissibility* **h(C) should be less than 3**, but if **h(A) > 1 + h(C)** then our heuristic is *Inconsistent!*. Hence if we assume that **h(C) = 1**, **h(G) = 0** and **h(A) = 4** our heuristic is *Admissible but Inconsistent!* @@ -115,9 +115,9 @@ Above Heuristic is defined Recursively. heuristic proof

-**$\overline{h}$(n’) $\ge$ $\overline{h}$(n) - c(n, a, n')** +**$\overline{h}$(n’) ≥ $\overline{h}$(n) - c(n, a, n')** -So obviously we have **$\overline{h}$(n’) + c(n, a, n') $\ge$ $\overline{h}$(n)** +So obviously we have **$\overline{h}$(n’) + c(n, a, n') ≥ $\overline{h}$(n)**
@@ -136,12 +136,12 @@ A* opens all node with **f(n) < C** and some with **f(n) = C**. Note that **$h_{2}(n)$ -$\ge$ +≥ $h_{1}(n)$** If node n is expanded by A* with $h_{2}$ then **$h_{2}(n)$ < C - g(n)** -So ***$h_{1}(n)$ $\le$ $h_{2}(n)$ < C - g(n)*** and it would be expanded by A* with $h_{1}$ as well. +So ***$h_{1}(n)$ ≤ $h_{2}(n)$ < C - g(n)*** and it would be expanded by A* with $h_{1}$ as well. We also can improve our heuristic by some methods. From 25d72f8aac3d7c2dc998234633ba21cb21686e9b Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:44:27 +0330 Subject: [PATCH 041/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 29df9ca5..4a225a9d 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -33,6 +33,8 @@ nothing yet! a ≥ b +a2 + $a≤b$ ## Body From 8d8dd3d45afbb07c4544089612c87b615bc8a8db Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:47:39 +0330 Subject: [PATCH 042/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 4a225a9d..f12c1cbd 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -176,7 +176,7 @@ h' = min(S'), h = min(S) $\Lambda$ S $\subseteq$ S' $\Lambda$ $\forall$ s, s' if hamilton path

-Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in $n^{2}$ time complexity. +Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in n2 time complexity.
From 3ea76ad92b27b5a42cd3fb84abd0fe233c6b4a1c Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:48:35 +0330 Subject: [PATCH 043/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index f12c1cbd..d811f877 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -176,7 +176,7 @@ h' = min(S'), h = min(S) $\Lambda$ S $\subseteq$ S' $\Lambda$ $\forall$ s, s' if hamilton path

-Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in n2 time complexity. +Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in $n2$ time complexity.
From 26dbbf98c0b6c6a9bb71a623f31a2e56a49af816 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:49:07 +0330 Subject: [PATCH 044/114] Update index.md --- notebooks/advanced_heuristics/index.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index d811f877..201cfbc5 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -176,7 +176,8 @@ h' = min(S'), h = min(S) $\Lambda$ S $\subseteq$ S' $\Lambda$ $\forall$ s, s' if hamilton path

-Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in $n2$ time complexity. +Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in n2 + time complexity.
From 1515cad40aeac80a0cedb07269060c51a411e1ef Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:49:51 +0330 Subject: [PATCH 045/114] Update index.md --- notebooks/advanced_heuristics/index.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 201cfbc5..6c034c21 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -177,7 +177,8 @@ h' = min(S'), h = min(S) $\Lambda$ S $\subseteq$ S' $\Lambda$ $\forall$ s, s' if

Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in n2 - time complexity. + +time complexity.
From 57511578595c78215a5475881ae7a29aa8f1913c Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:53:18 +0330 Subject: [PATCH 046/114] Update index.md --- notebooks/advanced_heuristics/index.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 6c034c21..8483eb4e 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -176,8 +176,7 @@ h' = min(S'), h = min(S) $\Lambda$ S $\subseteq$ S' $\Lambda$ $\forall$ s, s' if hamilton path

-Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in n2 - +Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in $$n2$$ time complexity.
From dda4753cefa66cfbe81fa6e7562a46d6168deb93 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:54:19 +0330 Subject: [PATCH 047/114] Update index.md --- notebooks/advanced_heuristics/index.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 8483eb4e..c166cde3 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -176,8 +176,7 @@ h' = min(S'), h = min(S) $\Lambda$ S $\subseteq$ S' $\Lambda$ $\forall$ s, s' if hamilton path

-Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in $$n2$$ -time complexity. +Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in n2 time complexity.
From b642b0b9b5570c8e8d1d7ca29a061e3c2b8a1b46 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 19:58:55 +0330 Subject: [PATCH 048/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index c166cde3..4204cda1 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -137,7 +137,7 @@ A* opens all node with **f(n) < C** and some with **f(n) = C**. Note that -**$h_{2}(n)$ +**h2(n) ≥ $h_{1}(n)$** From 9703ab76b302f57ab846e79985148c2429d0d020 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 20:01:18 +0330 Subject: [PATCH 049/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 4204cda1..f4192108 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -143,7 +143,7 @@ $h_{1}(n)$** If node n is expanded by A* with $h_{2}$ then **$h_{2}(n)$ < C - g(n)** -So ***$h_{1}(n)$ ≤ $h_{2}(n)$ < C - g(n)*** and it would be expanded by A* with $h_{1}$ as well. +So ***h1(n) ≤ h2(n) < C - g(n)*** and it would be expanded by A* with $h_{1}$ as well. We also can improve our heuristic by some methods. From d07cb1c2aec28816d54006062b8fb0cb65a8121a Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 20:03:02 +0330 Subject: [PATCH 050/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index f4192108..e0d05740 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -131,7 +131,7 @@ On another hand it means if a node is expanded by $h_{2}$ it will be expanded by *Proof:* -If $h_{2}$ dominate $h_{1}$, A* with $h_{2}$ expands at most the same number of nodes as when using $h_{1}$. +If h2 dominate h1, A* with h2 expands at most the same number of nodes as when using h1. A* opens all node with **f(n) < C** and some with **f(n) = C**. From cf0a8b317a5dda21aec317b57a8d33ce64560e23 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 20:22:38 +0330 Subject: [PATCH 051/114] Update index.md --- notebooks/advanced_heuristics/index.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index e0d05740..cf480de9 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -110,7 +110,8 @@ For this we only need to define a new and more appropriate heuristic like the be -**$\overline{h}$(n’) = max(h(n’), ℏ(n’) - c(n, a, n’))** +**h$(n’) = max(h(n’), h(n’) - c(n, a, n’))** + Above Heuristic is defined Recursively.

From 2d7cee3d6353a6c3ee04c4c0bad2ff80e2f0f25a Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 20:24:56 +0330 Subject: [PATCH 052/114] Update index.md --- notebooks/advanced_heuristics/index.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index cf480de9..bad631d8 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -110,7 +110,7 @@ For this we only need to define a new and more appropriate heuristic like the be -**h$(n’) = max(h(n’), h(n’) - c(n, a, n’))** +**ℏ(n’) = max(h(n’), ℏ(n’) - c(n, a, n’))** Above Heuristic is defined Recursively. @@ -118,9 +118,9 @@ Above Heuristic is defined Recursively. heuristic proof

-**$\overline{h}$(n’) ≥ $\overline{h}$(n) - c(n, a, n')** +**ℏ(n’) ≥ ℏ(n) - c(n, a, n')** -So obviously we have **$\overline{h}$(n’) + c(n, a, n') ≥ $\overline{h}$(n)** +So obviously we have **ℏ(n’) + c(n, a, n') ≥ ℏ(n)**
From 199e89f9e788acdda8d88474eae72a02a527f5cb Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 20:30:27 +0330 Subject: [PATCH 053/114] Update index.md --- notebooks/advanced_heuristics/index.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index bad631d8..27592219 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -95,7 +95,7 @@ So consider this figure: If our heuristic is admissible, we have that **h(n) ≤ h*(n)** for every node n where **h*** is the real cost to the goal. So we deduct that **h(A) ≤ 4**, **h(C) ≤ 3** and **h(G) ≤ 0**. -If we want our heuristic to be *Consistent* we should have **h(G) = 0** and **h(n) $\le$ cost(n, c) + h(c)** so in our case we have **h(A) ≤ 1 + h(C)** and **h(C) ≤ 3 + h(G) = 3** +If we want our heuristic to be *Consistent* we should have **h(G) = 0** and **h(n) \le$ cost(n, c) + h(c)** so in our case we have **h(A) ≤ 1 + h(C)** and **h(C) ≤ 3 + h(G) = 3** Because of the *Admissibility* **h(C) should be less than 3**, but if **h(A) > 1 + h(C)** then our heuristic is *Inconsistent!*. Hence if we assume that **h(C) = 1**, **h(G) = 0** and **h(A) = 4** our heuristic is *Admissible but Inconsistent!* @@ -142,9 +142,9 @@ Note that ≥ $h_{1}(n)$** -If node n is expanded by A* with $h_{2}$ then **$h_{2}(n)$ < C - g(n)** +If node n is expanded by A* with $h_{2}$ then **h2(n)$ < C - g(n)** -So ***h1(n) ≤ h2(n) < C - g(n)*** and it would be expanded by A* with $h_{1}$ as well. +So ***h1(n) ≤ h2(n) < C - g(n)*** and it would be expanded by A* with h1$ as well. We also can improve our heuristic by some methods. From 1eb2286ea92ce5917660dacf7777fbdd5b9f4b2d Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 20:38:24 +0330 Subject: [PATCH 054/114] Update index.md --- notebooks/advanced_heuristics/index.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 27592219..041b056d 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -127,8 +127,8 @@ So obviously we have **ℏ(n’) + c(n, a, n') ≥ ℏ(n)** ### Heuristic Dominance For one heuristic to dominate another, all of its values must be greater than or equal to the corresponding values of the other heuristic. -So if $h_{2}$ dominates $h_{1}$(both are admissible) it means that A* algorithm expands less nodes with $h_{2}$ in the target path than $h_{1}$. -On another hand it means if a node is expanded by $h_{2}$ it will be expanded by $h_{1}$(consider that it’s not a biconditional proposition!) +So if h2 dominates h1(both are admissible) it means that A* algorithm expands less nodes with h2 in the target path than h1. +On another hand it means if a node is expanded by h2 it will be expanded by h1(consider that it’s not a biconditional proposition!) *Proof:* @@ -140,9 +140,9 @@ A* opens all node with **f(n) < C** and some with **f(n) = C**. Note that **h2(n) ≥ -$h_{1}(n)$** +$h1(n)$** -If node n is expanded by A* with $h_{2}$ then **h2(n)$ < C - g(n)** +If node n is expanded by A* with h2 then **h2(n)$ < C - g(n)** So ***h1(n) ≤ h2(n) < C - g(n)*** and it would be expanded by A* with h1$ as well. From ae79238fb6d74a5ce0e4125a3f6b080f5fcd78f6 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 20:44:31 +0330 Subject: [PATCH 055/114] Update index.md --- notebooks/advanced_heuristics/index.md | 10 ++++------ 1 file changed, 4 insertions(+), 6 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 041b056d..7ebed775 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -35,8 +35,6 @@ a ≥ b a2 -$a≤b$ - ## Body ### Distance @@ -95,7 +93,7 @@ So consider this figure: If our heuristic is admissible, we have that **h(n) ≤ h*(n)** for every node n where **h*** is the real cost to the goal. So we deduct that **h(A) ≤ 4**, **h(C) ≤ 3** and **h(G) ≤ 0**. -If we want our heuristic to be *Consistent* we should have **h(G) = 0** and **h(n) \le$ cost(n, c) + h(c)** so in our case we have **h(A) ≤ 1 + h(C)** and **h(C) ≤ 3 + h(G) = 3** +If we want our heuristic to be *Consistent* we should have **h(G) = 0** and **h(n) ≤ cost(n, c) + h(c)** so in our case we have **h(A) ≤ 1 + h(C)** and **h(C) ≤ 3 + h(G) = 3** Because of the *Admissibility* **h(C) should be less than 3**, but if **h(A) > 1 + h(C)** then our heuristic is *Inconsistent!*. Hence if we assume that **h(C) = 1**, **h(G) = 0** and **h(A) = 4** our heuristic is *Admissible but Inconsistent!* @@ -140,11 +138,11 @@ A* opens all node with **f(n) < C** and some with **f(n) = C**. Note that **h2(n) ≥ -$h1(n)$** +h1(n)** -If node n is expanded by A* with h2 then **h2(n)$ < C - g(n)** +If node n is expanded by A* with h2 then **h2(n) < C - g(n)** -So ***h1(n) ≤ h2(n) < C - g(n)*** and it would be expanded by A* with h1$ as well. +So ***h1(n) ≤ h2(n) < C - g(n)*** and it would be expanded by A* with h1 as well. We also can improve our heuristic by some methods. From f3accb2290fe18ab8869d5a7c63a9a7cd777a911 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 20:53:53 +0330 Subject: [PATCH 056/114] Update index.md --- notebooks/advanced_heuristics/index.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 7ebed775..0c6bd77b 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -165,7 +165,8 @@ Discussing more specifically, original problem has a set of constraints. Noe we Consider h as the answer and h' as our heuristic -h' = min(S'), h = min(S) $\Lambda$ S $\subseteq$ S' $\Lambda$ $\forall$ s, s' if s $\subseteq$ s' => $min(s') \le min(s)$ => $h' \le h$ +h' = min(S'), h = min(S) ∧ S ⊆ S' ∧ ∀ s, s': if s ⊆ s' &U+21d2; min(s') ≤ min(s) => h' ≤ h$ + ***Example*** From 4c82f090b0e477f9fea939779314b73e23ecaee0 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 20:56:24 +0330 Subject: [PATCH 057/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 0c6bd77b..6fcacd21 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -165,7 +165,7 @@ Discussing more specifically, original problem has a set of constraints. Noe we Consider h as the answer and h' as our heuristic -h' = min(S'), h = min(S) ∧ S ⊆ S' ∧ ∀ s, s': if s ⊆ s' &U+21d2; min(s') ≤ min(s) => h' ≤ h$ +h' = min(S'), h = min(S) ∧ S ⊆ S' ∧ ∀ s, s': if s ⊆ s' &#U+21d2; min(s') ≤ min(s) => h' ≤ h ***Example*** From 891787e4c3bedba4408ac73577f76eddf7bca946 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 20:58:58 +0330 Subject: [PATCH 058/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 6fcacd21..2b021b79 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -165,7 +165,7 @@ Discussing more specifically, original problem has a set of constraints. Noe we Consider h as the answer and h' as our heuristic -h' = min(S'), h = min(S) ∧ S ⊆ S' ∧ ∀ s, s': if s ⊆ s' &#U+21d2; min(s') ≤ min(s) => h' ≤ h +h' = min(S'), h = min(S) ∧ S ⊆ S' ∧ ∀ s, s': if s ⊆ s' ⇒ min(s') ≤ min(s) ⇒ h' ≤ h ***Example*** From bf7c69ff1bc5ac5c850b9a96c80854d0cc8f6ec6 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:12:11 +0330 Subject: [PATCH 059/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 2b021b79..b3118d2a 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -67,7 +67,7 @@ Now we define **f(n)** function as **f(n) = h(n) + g(n)** where g(n) is sum of c

consistency

- +A heuristic is consistent: if h(n) ≤ c(n, a, n’) + h(n’) Effect of **Monotonicity** on a Heuristic is shown in the below schema:

From 97abd5c32cac58ef94be4a1f31d0cf2344dbf01f Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:13:50 +0330 Subject: [PATCH 060/114] Update index.md --- notebooks/advanced_heuristics/index.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index b3118d2a..5d7ec890 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -67,7 +67,9 @@ Now we define **f(n)** function as **f(n) = h(n) + g(n)** where g(n) is sum of c

consistency

-A heuristic is consistent: if h(n) ≤ c(n, a, n’) + h(n’) +A heuristic is consistent if: h(n) ≤ c(n, a, n’) + h(n’) +if h is consistent, we have: f(n’) = g(n’) + h (n’) = g(n) + c(n, a, n’) + h(n’) ≥ [= f(n)] +I.e., f(n) is nondecreasing along any path. Effect of **Monotonicity** on a Heuristic is shown in the below schema:

From c464cf76a86c56445172c6b534a8c1a955ba5e02 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:19:32 +0330 Subject: [PATCH 061/114] Update index.md --- notebooks/advanced_heuristics/index.md | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 5d7ec890..32fa3425 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -67,8 +67,10 @@ Now we define **f(n)** function as **f(n) = h(n) + g(n)** where g(n) is sum of c

consistency

-A heuristic is consistent if: h(n) ≤ c(n, a, n’) + h(n’) -if h is consistent, we have: f(n’) = g(n’) + h (n’) = g(n) + c(n, a, n’) + h(n’) ≥ [= f(n)] +A heuristic is consistent if: _h(n)_ _≤_ _c(n, a, n’)_ _+_ _h(n’)_ + +if h is consistent, we have: _f(n’) = g(n’) + h (n’) = g(n) + c(n, a, n’) + h(n’) ≥ g(n) + h(n) [= f(n)]_ + I.e., f(n) is nondecreasing along any path. Effect of **Monotonicity** on a Heuristic is shown in the below schema: From da79457b09250df660b44aa821578e672e9a5ad7 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:20:41 +0330 Subject: [PATCH 062/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 32fa3425..2a219fe6 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -67,7 +67,7 @@ Now we define **f(n)** function as **f(n) = h(n) + g(n)** where g(n) is sum of c

consistency

-A heuristic is consistent if: _h(n)_ _≤_ _c(n, a, n’)_ _+_ _h(n’)_ +A heuristic is consistent if: _h(n) ≤ c(n, a, n’) + h(n’)_ if h is consistent, we have: _f(n’) = g(n’) + h (n’) = g(n) + c(n, a, n’) + h(n’) ≥ g(n) + h(n) [= f(n)]_ From e15ac0639862e7bccdc843bdec80f1dcfcab1950 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:21:15 +0330 Subject: [PATCH 063/114] Delete consistency.png --- .../advanced_heuristics/pic/consistency.png | Bin 110197 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 notebooks/advanced_heuristics/pic/consistency.png diff --git a/notebooks/advanced_heuristics/pic/consistency.png b/notebooks/advanced_heuristics/pic/consistency.png deleted file mode 100644 index 88ffed8725bbdc841fa68e26eaa3e392050ccbb9..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 110197 zcmeFZWmH>T+b-GyMOvh#MT$GcT?!PJ;2Iz}6xSlbr4%c!!CE{)f_riI0wuT?iUloR 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1f464fe51effc8ce9f647673ab80793ad4348b96 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:27:19 +0330 Subject: [PATCH 067/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 2a219fe6..6c95a095 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -65,7 +65,7 @@ Now we define **f(n)** function as **f(n) = h(n) + g(n)** where g(n) is sum of c ***Monotonicity or Consistency is that the heuristic function holds in triangle inequality.*** Namely **f(n) is never Decreasing.**

-consistency +consistency

A heuristic is consistent if: _h(n) ≤ c(n, a, n’) + h(n’)_ From 9a5c21a55fd80d4d89034b22fa867b3d9a1b9835 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:28:59 +0330 Subject: [PATCH 068/114] Update index.md --- notebooks/advanced_heuristics/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 6c95a095..d4385b89 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -65,9 +65,9 @@ Now we define **f(n)** function as **f(n) = h(n) + g(n)** where g(n) is sum of c ***Monotonicity or Consistency is that the heuristic function holds in triangle inequality.*** Namely **f(n) is never Decreasing.**

-consistency +consistency

-A heuristic is consistent if: _h(n) ≤ c(n, a, n’) + h(n’)_ +A heuristic is consistent if: _h(n) ≤ c(n, a, n’) + h(n’) if h is consistent, we have: _f(n’) = g(n’) + h (n’) = g(n) + c(n, a, n’) + h(n’) ≥ g(n) + h(n) [= f(n)]_ From 8865804824200641686d6701d3c703b2a60d4e84 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:30:18 +0330 Subject: [PATCH 069/114] Update index.md --- notebooks/advanced_heuristics/index.md | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index d4385b89..4d6068df 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -65,11 +65,12 @@ Now we define **f(n)** function as **f(n) = h(n) + g(n)** where g(n) is sum of c ***Monotonicity or Consistency is that the heuristic function holds in triangle inequality.*** Namely **f(n) is never Decreasing.**

-consistency +consistency

-A heuristic is consistent if: _h(n) ≤ c(n, a, n’) + h(n’) +A heuristic is consistent if: +_h(n) ≤ c(n, a, n’) + h(n’)_ -if h is consistent, we have: _f(n’) = g(n’) + h (n’) = g(n) + c(n, a, n’) + h(n’) ≥ g(n) + h(n) [= f(n)]_ +if h is consistent, we have: _f(n’) = g(n’) + h (n’) = g(n) + c(n, a, n’) + h(n’) ≥ g(n) + h(n) [= f(n)]_ I.e., f(n) is nondecreasing along any path. From 0c3961198bfbff8bfb954ce3272660cd5807c7cd Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:32:00 +0330 Subject: [PATCH 070/114] Update index.md --- notebooks/advanced_heuristics/index.md | 5 +---- 1 file changed, 1 insertion(+), 4 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 4d6068df..5dad4e44 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -67,11 +67,8 @@ Now we define **f(n)** function as **f(n) = h(n) + g(n)** where g(n) is sum of c

consistency

-A heuristic is consistent if: -_h(n) ≤ c(n, a, n’) + h(n’)_ - +A heuristic is consistent if: h(n) ≤ c(n, a, n’) + h(n’) if h is consistent, we have: _f(n’) = g(n’) + h (n’) = g(n) + c(n, a, n’) + h(n’) ≥ g(n) + h(n) [= f(n)]_ - I.e., f(n) is nondecreasing along any path. Effect of **Monotonicity** on a Heuristic is shown in the below schema: From 0c3007dd0b08558f619ff7e5ca45af70e24aa7c0 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:32:51 +0330 Subject: [PATCH 071/114] Update index.md --- notebooks/advanced_heuristics/index.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 5dad4e44..dd041ecc 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -68,7 +68,9 @@ Now we define **f(n)** function as **f(n) = h(n) + g(n)** where g(n) is sum of c consistency

A heuristic is consistent if: h(n) ≤ c(n, a, n’) + h(n’) -if h is consistent, we have: _f(n’) = g(n’) + h (n’) = g(n) + c(n, a, n’) + h(n’) ≥ g(n) + h(n) [= f(n)]_ + +if h is consistent, we have: f(n’) = g(n’) + h (n’) = g(n) + c(n, a, n’) + h(n’) ≥ g(n) + h(n) [= f(n)] + I.e., f(n) is nondecreasing along any path. Effect of **Monotonicity** on a Heuristic is shown in the below schema: From 3785787ecd0ef841e27a31f725a641054f74fa85 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:33:45 +0330 Subject: [PATCH 072/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index dd041ecc..145e02fd 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -85,7 +85,7 @@ for more information and better understanding watch this video: [watch](https:// ### Monotonicity implies admissibility We will prove that **Consistency** implies **Admissibility** whereas the opposite is not necessarily true.

-proof +proof

Now we want to show an **inconsistent, admissible example!** From 9b6cf09be62856963d2bba6e43c9850b7cc4fcab Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:34:48 +0330 Subject: [PATCH 073/114] Delete proof.png --- notebooks/advanced_heuristics/pic/proof.png | Bin 107414 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 notebooks/advanced_heuristics/pic/proof.png diff --git a/notebooks/advanced_heuristics/pic/proof.png b/notebooks/advanced_heuristics/pic/proof.png deleted file mode 100644 index e0f8590908f2c9852ca964a79ab359770fc0ab47..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 107414 zcmeFZWp^CG(k3WoW(JFynb~5p#SAUBn3*kRW@ct)X0Xs=wj{ThVcN4Z`{lj&5A4}K zr%qN^RCQ%WW=1>_`D90^D9Ior5Fmhofg#I&l~e-*gIoXu12=<%0bQXfZ_fr5;Ld6? z;$XG2L}#Ec9_CuI07XSGdeAu>7!)`G81%neK!-3m;s1Ls4NeON`M0i?f4lm) z`{;IdGB{4Av-#AM=%7$z11NUkcp?5rp@JPq08O%@5wnL%1TNE$?f~^aihL+=%|iLz 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-proof +proof

Now we want to show an **inconsistent, admissible example!** From fa7b75e0958cda64861fca2058186181ae78c35f Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:45:20 +0330 Subject: [PATCH 079/114] Update index.md --- notebooks/advanced_heuristics/index.md | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index ae9c7ae7..5158ae5c 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -87,6 +87,10 @@ We will prove that **Consistency** implies **Admissibility** whereas the opposit

proof

+h(n1) ≤ c(n1 ,a1 , n2) + h(n2) +≤ c(n1 ,a1 , n2) + c(n1 ,a1 , n3) + h(n3) +… +≤ Σ c(n1 ,a1 , ni+1) + 0 => h(n1) ≤ cost of (every) path from n1 to goal ≤ cost of optimal path from n1 to goal Now we want to show an **inconsistent, admissible example!** So consider this figure: From 248e74db1ee6c2ed3ad4ee829ac3824f72da242d Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:47:11 +0330 Subject: [PATCH 080/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 5158ae5c..ee1019c2 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -90,7 +90,7 @@ We will prove that **Consistency** implies **Admissibility** whereas the opposit h(n1) ≤ c(n1 ,a1 , n2) + h(n2) ≤ c(n1 ,a1 , n2) + c(n1 ,a1 , n3) + h(n3) … -≤ Σ c(n1 ,a1 , ni+1) + 0 => h(n1) ≤ cost of (every) path from n1 to goal ≤ cost of optimal path from n1 to goal +≤ Σ c(n1 ,a1 , ni+1) + 0 ⇒ h(n1) ≤ cost of (every) path from n1 to goal ≤ cost of optimal path from n1 to goal Now we want to show an **inconsistent, admissible example!** So consider this figure: From 2b85c9810ddcf9d0fede39cdcaee14a7c51e29ec Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:48:10 +0330 Subject: [PATCH 081/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index ee1019c2..cf88a19a 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -89,7 +89,9 @@ We will prove that **Consistency** implies **Admissibility** whereas the opposit

h(n1) ≤ c(n1 ,a1 , n2) + h(n2) ≤ c(n1 ,a1 , n2) + c(n1 ,a1 , n3) + h(n3) + … + ≤ Σ c(n1 ,a1 , ni+1) + 0 ⇒ h(n1) ≤ cost of (every) path from n1 to goal ≤ cost of optimal path from n1 to goal Now we want to show an **inconsistent, admissible example!** From 639961ffcf375a6c3d8bf042904eb0d9038adc05 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:49:38 +0330 Subject: [PATCH 082/114] Update index.md --- notebooks/advanced_heuristics/index.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index cf88a19a..88fc7c31 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -88,10 +88,9 @@ We will prove that **Consistency** implies **Admissibility** whereas the opposit proof

h(n1) ≤ c(n1 ,a1 , n2) + h(n2) -≤ c(n1 ,a1 , n2) + c(n1 ,a1 , n3) + h(n3) +≤ c(n1 ,a1 , n2) + c(n1 ,a1 , n3) + h(n3) ≤ … - ≤ Σ c(n1 ,a1 , ni+1) + 0 ⇒ h(n1) ≤ cost of (every) path from n1 to goal ≤ cost of optimal path from n1 to goal Now we want to show an **inconsistent, admissible example!** From be6e63fbaa97df069c7fca2a97e1e14ae5647a2b Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:50:35 +0330 Subject: [PATCH 083/114] Update index.md --- notebooks/advanced_heuristics/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 88fc7c31..68ac7335 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -87,8 +87,8 @@ We will prove that **Consistency** implies **Admissibility** whereas the opposit

proof

-h(n1) ≤ c(n1 ,a1 , n2) + h(n2) -≤ c(n1 ,a1 , n2) + c(n1 ,a1 , n3) + h(n3) ≤ +h(n1) ≤ c(n1 ,a1 , n2) + h(n2) +≤ c(n1 ,a1 , n2) + c(n1 ,a1 , n3) + h(n3) ≤ … ≤ Σ c(n1 ,a1 , ni+1) + 0 ⇒ h(n1) ≤ cost of (every) path from n1 to goal ≤ cost of optimal path from n1 to goal From 12136f585802dc8b4e524e493a3387d190d0e49f Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:53:17 +0330 Subject: [PATCH 084/114] Update index.md --- notebooks/advanced_heuristics/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 68ac7335..90faabf2 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -87,8 +87,8 @@ We will prove that **Consistency** implies **Admissibility** whereas the opposit

proof

-h(n1) ≤ c(n1 ,a1 , n2) + h(n2) -≤ c(n1 ,a1 , n2) + c(n1 ,a1 , n3) + h(n3) ≤ + h(n1) ≤ c(n1 ,a1 , n2) + h(n2) + ≤ c(n1 ,a1 , n2) + c(n1 ,a1 , n3) + h(n3) ≤ … ≤ Σ c(n1 ,a1 , ni+1) + 0 ⇒ h(n1) ≤ cost of (every) path from n1 to goal ≤ cost of optimal path from n1 to goal From bea73a9dbd8698873487cf1aeb6691bce2d25d56 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:55:07 +0330 Subject: [PATCH 085/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 90faabf2..992e4531 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -91,7 +91,7 @@ We will prove that **Consistency** implies **Admissibility** whereas the opposit ≤ c(n1 ,a1 , n2) + c(n1 ,a1 , n3) + h(n3) ≤ … -≤ Σ c(n1 ,a1 , ni+1) + 0 ⇒ h(n1) ≤ cost of (every) path from n1 to 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zXPk&hiwz}Uyr2%ADQfGN_vL&m1|Vc3MhKcEtn~7S4FoP&t$56rQfIGI z#Kd8P`5b~YB4-3Z9KxP}PC%e#L?R9w9SkvyCIaU8MWQbtYW3;UN3U}uaM(9> zdG<_Em)K8H+15iFQ=o{`b0gvaj249Y=ddW`xNDGx0}%=0N2KE586yw*SsKj<^+K4a zF2OTJU1C2)Wka~=528hAt&K^7_0Ykys?CAAD|C|}4@pb%Ea6)|d~4Uncs#2MfGNC1 zyDl)FvN9o9F_|er8}SB2{t)Z3@=0YS*}jvUwDsJhivZ-so)&e@5Ly}PgB99tk~mk- zJtom2lc=`O!(%`(;71Bi2p$6uz Date: Fri, 29 Oct 2021 22:20:16 +0330 Subject: [PATCH 088/114] Update index.md --- notebooks/advanced_heuristics/index.md | 1 + 1 file changed, 1 insertion(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 992e4531..e73ad94b 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -217,5 +217,6 @@ First practical problem of the first mini project of the course is an example of Now we can have souple of flower breeds subset and combine obtained answers by them. For example if we have K flower breeds, we can solve the problem for each flower breed and calculate its heuristic function and then choose maximum of the functions. So we used Combining Pattern DB to achieve a better heuristic function. ## Conclusion +So far we've introduced some efficient ways to make our heuristic closer to the real answer. some of them were to make our heuristic monotonic شnd Dominant, or to make a dominant heuristic based on some heuristics we had. some advanced ways were for even more basic problem, namely how to create a heuristic function for a problem. these included Relaxing problems and Patternal DBs. We Entered deeply into the Patternal DBs And introduced ways to improve them a lot. despite all these ways, there are so many problems that aren't solvable by even these ways, and there are so many ways ahead of us that you have to learn & use further. thanks for reading all of these concepts and we wish you have a great future. a bid farewell. ## References From 80d0a50cb36067ad36c01f3f0e81dc634608f10d Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 22:21:03 +0330 Subject: [PATCH 089/114] Update index.md --- notebooks/advanced_heuristics/index.md | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index e73ad94b..960ddfb7 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -220,3 +220,9 @@ Now we can have souple of flower breeds subset and combine obtained answers by So far we've introduced some efficient ways to make our heuristic closer to the real answer. some of them were to make our heuristic monotonic شnd Dominant, or to make a dominant heuristic based on some heuristics we had. some advanced ways were for even more basic problem, namely how to create a heuristic function for a problem. these included Relaxing problems and Patternal DBs. We Entered deeply into the Patternal DBs And introduced ways to improve them a lot. despite all these ways, there are so many problems that aren't solvable by even these ways, and there are so many ways ahead of us that you have to learn & use further. thanks for reading all of these concepts and we wish you have a great future. a bid farewell. ## References +https://machinelearningmastery.com/distance-measures-for-machine-learning/ +https://artint.info/2e/html/ArtInt2e.Ch3.S6.SS2.html +https://en.wikipedia.org/wiki/Consistent_heuristic +https://stackoverflow.com/questions/1568600/what-is-the-difference-between-monotonicity-and-the-admissibility-of-a-heuristic +https://stackoverflow.com/questions/46554459/intuitively-understanding-why-consistency-is-required-for-optimality-in-a-searc +https://courses.cs.washington.edu/courses/cse473/12sp/slides/04-heuristics.pdf From c9d70069b3b8b7a0214a82e5c0df5967d1193f6d Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 22:21:34 +0330 Subject: [PATCH 090/114] Update index.md --- notebooks/advanced_heuristics/index.md | 5 +++++ 1 file changed, 5 insertions(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 960ddfb7..8a85f4e8 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -221,8 +221,13 @@ So far we've introduced some efficient ways to make our heuristic closer to the ## References https://machinelearningmastery.com/distance-measures-for-machine-learning/ + https://artint.info/2e/html/ArtInt2e.Ch3.S6.SS2.html + https://en.wikipedia.org/wiki/Consistent_heuristic + https://stackoverflow.com/questions/1568600/what-is-the-difference-between-monotonicity-and-the-admissibility-of-a-heuristic + https://stackoverflow.com/questions/46554459/intuitively-understanding-why-consistency-is-required-for-optimality-in-a-searc + https://courses.cs.washington.edu/courses/cse473/12sp/slides/04-heuristics.pdf From 88155ac4441416ed0ede4a39602a43b0cb1eb7e0 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 22:30:54 +0330 Subject: [PATCH 091/114] Update index.md --- notebooks/advanced_heuristics/index.md | 8 ++++++-- 1 file changed, 6 insertions(+), 2 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 8a85f4e8..a57d1b29 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -212,12 +212,16 @@ One solution for solving this drawback is Disjoint Pattern DB. In this method we **Example** -First practical problem of the first mini project of the course is an example of this method. In this problm we have an undirected graph, which each of its nodes is a florist shop and has some flower breeds. Two brothers should march from start_node towards goal_node in a way that has the least cost and they met every existing flower breeds. In this case we choose a subset of flower breeds S that reach goal_node by a specific pattern and set goal_state as satisfying path with the least cost and meeting every existing flower breeds of set S. Hitherto we used pure Pattern DB. +

+pattern +

+ +As you can see in this problm we have an undirected graph, which each of its nodes is a florist shop and has some flower breeds. Two brothers should march from start_node towards goal_node in a way that has the least cost and they met every existing flower breeds. In this case we choose a subset of flower breeds S that reach goal_node by a specific pattern and set goal_state as satisfying path with the least cost and meeting every existing flower breeds of set S. Hitherto we used pure Pattern DB. Now we can have souple of flower breeds subset and combine obtained answers by them. For example if we have K flower breeds, we can solve the problem for each flower breed and calculate its heuristic function and then choose maximum of the functions. So we used Combining Pattern DB to achieve a better heuristic function. ## Conclusion -So far we've introduced some efficient ways to make our heuristic closer to the real answer. some of them were to make our heuristic monotonic شnd Dominant, or to make a dominant heuristic based on some heuristics we had. some advanced ways were for even more basic problem, namely how to create a heuristic function for a problem. these included Relaxing problems and Patternal DBs. We Entered deeply into the Patternal DBs And introduced ways to improve them a lot. despite all these ways, there are so many problems that aren't solvable by even these ways, and there are so many ways ahead of us that you have to learn & use further. thanks for reading all of these concepts and we wish you have a great future. a bid farewell. +So far we've introduced some efficient ways to make our heuristic closer to the real answer. some of them were to make our heuristic monotonic and Dominant, or to make a dominant heuristic based on some heuristics we had. some advanced ways were for even more basic problem, namely how to create a heuristic function for a problem. these included Relaxing problems and Patternal DBs. We Entered deeply into the Patternal DBs And introduced ways to improve them a lot. despite all these ways, there are so many problems that aren't solvable by even these ways, and there are so many ways ahead of us that you have to learn & use further. thanks for reading all of these concepts and we wish you have a great future. a bid farewell. ## References https://machinelearningmastery.com/distance-measures-for-machine-learning/ From 11e04cf6e1aa6a969256d4ef2155ae78c98ac76e Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 22:33:53 +0330 Subject: [PATCH 092/114] Add files via upload --- notebooks/advanced_heuristics/pic/pattern.png | Bin 0 -> 18144 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 notebooks/advanced_heuristics/pic/pattern.png diff --git a/notebooks/advanced_heuristics/pic/pattern.png b/notebooks/advanced_heuristics/pic/pattern.png new file mode 100644 index 0000000000000000000000000000000000000000..1e160af9ffc268c72c82c7adae71a6fe3e167185 GIT binary patch literal 18144 zcmd^n2Uycd*Kh0!qKkkiAnt;+AfQxf7OJ#hNJ8jHQwdGF29aG9gwU35=me6G1cHPP z22`Z?5=v;&2@nuziUK$8`?~MGW%v8`{qFbO=iWPc9%eHCGv{~aoHLV|IsajAaPJ-9 zJAEBJ9l*YQ`v6(24`6Q?pb0p7`0$a#hmIaOa^%>tqin~yIgYclA3uAF>m>I%zVm|T z_yh#LzaTC4z0gHr0ReF(iHnym%gf837gJWhBC94XCojAA3c!AJzsiC9efu~8dv5{! zfPH}d2Urk)ONS2cKX~N8KGs9S?*RJ_9N2f@@Zl534jer2m1y7o0|yUr9zONM}bLZ+TSC32tUl~pu>`DEu7QR;exl{DVLn4O=h%0abWHuuWEID@bD z=%h?(TcLCE@p+(%T$kN?#aF-W@?oq|11dz$+}s$H?$y*ENYqs|Wy}IGWB1A*nS&p{ z8nwjRR?5g*?#kz7Itv?lV#tQw{g6H}D-(r1fPL!i-}3sq^1ccARfxtXd~B@N+t#ry zVT-;+n9U(zZGxha$-+5hkCZvsYr!ZRH`V*8G}ZW$euKkhwZN;)hX$@E#@iqP(@WNA zLfck2rkKd)QR`BZg{pGtjX9AaZCi$xZEk$T4Mx}yCFwcs8dA41fTopl((q<`Vo5{z zXNo#$45b6}vb~KShIWv0y*xnf*16idxv*^`krys5Y$T>qLgn?*FZISXHgD-IV%bfu zU0$J02usa8=9(q2LXt*mFLdWciXr`y&BRc+(?$m)lf-alo5}(MVirJ8^;duEry!Rx zkGxFE!?WCmhSU`=g#u71?41eelcpn0I=eUN6WP~_bDlQaTYiiRsZ=#`WZ{3vSsq9z zd&T@xuT%r7A9^E$ZE^2mNgFZ@lO~b8R&2bi6JWau+S__pN(7%rrAjzQLAG)OlBrO= zto65%Lzag3GOLB@)cIT|xX9+Q^isoxDqYd_*%`W)?Si&#Zja^RvKPdf-iHPW9OG@! z&ofI#`52@?X!{Xy*HCPUO7yq_g%nJ+cuC)!@W=S$oqdBJ5bYehZZ_K^onjnf+r4}- zc&?GZ_3`gZ`?+o*fE>zNr3x{y07|UZc%aaxpKswVDo~2m-I?f`9m)9ub9!*Jddu{@ z^&a4%|AvrIa;qCTv|B%+k2^p3!eo@uvQ4QCkV^FU5g;$ESjA5T_QHE850ZnGl}eG_ z4=|hAngq(Y`~?bp&?(T3qzdV7rdp&+RK z1JInZpDa=}XDH>v6AE_(n>rh!w9OQl0$t^T&8sb5x3j8~-! zZ_t?a&do6FhhdlO5)-V#A<6Hhs zJO%SsGP)){t5(zVSu7rN7Rkmm=Y;`8l=8NcZZZra^nmQ4KcS-1n}c#VRP{Ol+jklV zHz_=S`sP5S2LrJO=r|+t>Kphz&LCBU3K$+s-f*7`3WnG4%ouEmn}09S4&e?$F;d|w;pZh5=g#kdEX1}00TFHPWg87;V= z5Ka|Pa-HYTEb7Zs6|-^0^wVXin_mn>y8W!ij&T*Q+rh8BA*LTO`jy=g+3<{Ug%XP1 z)R2?`rGHh=Iuj2>?{aG~RBSmKuCwSGqo-2zJMUT0C4^TmCTkCQKezzu><+Qo1F#W& z&dLAE>F*0b`sPwul!VEw>K#PP`f8y`C9mXepFjR^M@+FujA4%nHC2xKa%m^PeJroW z@ePvwm*$0w?6S78E3_`S{H-0fbY zUK=$og~{~9YD;!L;>5W=azl%ZJ;Cl5HYWlKxZb36g3Oa21w8taKkx0mduDkp8^Gpf zQ!YUUa@UmY|z47|6QEcw~mBbGQf94UhPI zHQ7S7564NjSG%nlT;BH*#m^CzMQ|lzm~88tG+qRZ++h*1bI@V=f|? 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In this method we pattern

-As you can see in this problm we have an undirected graph, which each of its nodes is a florist shop and has some flower breeds. Two brothers should march from start_node towards goal_node in a way that has the least cost and they met every existing flower breeds. In this case we choose a subset of flower breeds S that reach goal_node by a specific pattern and set goal_state as satisfying path with the least cost and meeting every existing flower breeds of set S. Hitherto we used pure Pattern DB. +As you can see in this problem we have an undirected graph, which each of its nodes is a florist shop and has some flower breeds. Two brothers should march from start_node towards goal_node in a way that has the least cost and they met every existing flower breeds. In this case we choose a subset of flower breeds S that reach goal_node by a specific pattern and set goal_state as satisfying path with the least cost and meeting every existing flower breeds of set S. Hitherto we used pure Pattern DB. Now we can have souple of flower breeds subset and combine obtained answers by them. For example if we have K flower breeds, we can solve the problem for each flower breed and calculate its heuristic function and then choose maximum of the functions. So we used Combining Pattern DB to achieve a better heuristic function. From 615f331337af4fda6c89affec7fe160056ebc0cc Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 22:38:36 +0330 Subject: [PATCH 094/114] Update index.md --- notebooks/advanced_heuristics/index.md | 15 ++++----------- 1 file changed, 4 insertions(+), 11 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 1b4ecbad..78a9ccf9 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -153,20 +153,13 @@ If node n is expanded by A* with h2 then **h2(n) < C - g(n So ***h1(n) ≤ h2(n) < C - g(n)*** and it would be expanded by A* with h1 as well. - -We also can improve our heuristic by some methods. - -We are always trying to lower the difference between h* and h, expanded nodes and algorithm time complexity by making change in our heuristic. - -One of the main ways for improving heuristics is **Relaxing** the problem. - -In this procedure we try to enlarge the feasible answers set by simplifying the problem. -
### Relaxing - -One of the most common ways to find efficient heuristic function is that we convert our hard problem which is indissoluble in reasonable time, to a *more general problem with larger state space*. + +We also can improve our heuristic by some methods. We are always trying to lower the difference between h* and h, expanded nodes and algorithm time complexity by making change in our heuristic. +One of the main ways for improving heuristics is **Relaxing** the problem .In this procedure we try to enlarge the feasible answers set by simplifying the problem. +So one of the most common ways to find efficient heuristic function is that we convert our hard problem which is indissoluble in reasonable time, to a *more general problem with larger state space*. Discussing more specifically, original problem has a set of constraints. Noe we **ignore some constraints** and solve the expanded problem. Since number of constraints have decreased, we have more goal states and our state space expands. From 12d111a1dcc318fa4bc8a149c4f05cdbba23a449 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 23:11:39 +0330 Subject: [PATCH 095/114] Update index.md --- notebooks/advanced_heuristics/index.md | 6 +----- 1 file changed, 1 insertion(+), 5 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 78a9ccf9..8d1d934e 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -29,11 +29,7 @@ 4.[References](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#references) ## Introduction -nothing yet! - -a ≥ b - -a2 +Heuristics are methods for solving problems in a fast way. Actually It's about using a practical method that doesn’t necessarily need to be perfect! Heuristic methods speed up the process of reaching a satisfactory solution. Moreover Advanced Heuristics are Creating a sequence of heuristics to detect complex malware and detecting then solving the problems. In this lecture note first we will review some prerequisites then we will mention some approaches to make heuristics more efficient! ## Body From 25afe1aaf014ff584b5f3342c714a426cdbbe726 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 23:16:53 +0330 Subject: [PATCH 096/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 8d1d934e..b9504b1c 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -29,7 +29,7 @@ 4.[References](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#references) ## Introduction -Heuristics are methods for solving problems in a fast way. Actually It's about using a practical method that doesn’t necessarily need to be perfect! Heuristic methods speed up the process of reaching a satisfactory solution. Moreover Advanced Heuristics are Creating a sequence of heuristics to detect complex malware and detecting then solving the problems. In this lecture note first we will review some prerequisites then we will mention some approaches to make heuristics more efficient! +Heuristics are methods for solving problems in a fast way. Actually It's about using a practical method that doesn’t necessarily need to be perfect! Heuristic methods speed up the process of reaching a satisfactory solution. Moreover Advanced Heuristics creating a sequence of heuristics to detect complex malware and detecting then solving the problems. In this lecture note first we review some prerequisites then we will mention some approaches to make heuristics more efficient! ## Body From 7cec7cc87d7cffffcc3cb8c255d6eb400d32a5e0 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 23:25:19 +0330 Subject: [PATCH 097/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index b9504b1c..2d3fde66 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -29,7 +29,7 @@ 4.[References](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#references) ## Introduction -Heuristics are methods for solving problems in a fast way. Actually It's about using a practical method that doesn’t necessarily need to be perfect! Heuristic methods speed up the process of reaching a satisfactory solution. Moreover Advanced Heuristics creating a sequence of heuristics to detect complex malware and detecting then solving the problems. In this lecture note first we review some prerequisites then we will mention some approaches to make heuristics more efficient! +Heuristics are methods for solving problems in a fast way. Actually It's about using a practical method that doesn’t necessarily need to be perfect! Heuristic methods speed up the process of reaching a satisfactory solution. Moreover Advanced Heuristics are creating a sequence of heuristics to detect complex malware and detecting then solving the problems. In this lecture note first we review some prerequisites then we will mention some approaches to make heuristics more efficient! ## Body From 92deefe36291a365efd8f20660941f3c1c7a1760 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 23:38:12 +0330 Subject: [PATCH 098/114] Update matadata.yml --- notebooks/advanced_heuristics/matadata.yml | 35 +++++++++++++++------- 1 file changed, 24 insertions(+), 11 deletions(-) diff --git a/notebooks/advanced_heuristics/matadata.yml b/notebooks/advanced_heuristics/matadata.yml index 13a82fc0..e0e98a0e 100644 --- a/notebooks/advanced_heuristics/matadata.yml +++ b/notebooks/advanced_heuristics/matadata.yml @@ -1,25 +1,38 @@ -title: # shown on browser tab +title: LN | Advanced Heuristics # shown on browser tab header: - title: advanced Heuristics # title of your notebook - description: # short description of your notebook + title: Advanced Heuristics # title of your notebook + description: making heuristics better and more efficient # short description of your notebook authors: label: position: top + text: Authors + kind: people content: # list of notebook authors - - name: Mohammadreza Daviran # name of author - role: Author # change this if you want + - name: Mohammadreza Daviran + role: Author contact: - # list of contact information - - link: https://github.com/ + - link: https://github.com/mmdrez4 + icon: fab fa-github + - link: mailto:mohammadreza.dn80@gmail.com + icon: fas fa-envelope + + + - name: Kahbod Aeini + role: Author + contact: + - link: https://github.com/kahbodaeini + icon: fab fa-github + + + - name: Sepehr Kianian + role: Author + contact: + - link: https://github.com/Sepehrkianian09 icon: fab fa-github - # optionally add other contact information like - # - link: # contact link - # icon: # awsomefont tag for link (check: https://fontawesome.com/v5.15/icons) comments: - # enable comments for your post label: false kind: comments From c066f646f0b9586a63c5a497d946083febed3360 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 12 Nov 2021 18:35:46 +0330 Subject: [PATCH 099/114] Update index.md --- notebooks/advanced_heuristics/index.md | 15 +++++---------- 1 file changed, 5 insertions(+), 10 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 2d3fde66..821311b1 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -2,10 +2,8 @@ #### Kahbod Aeini, Mohammadreza Daviran and Sepehr Kianian ## Contents -1.[Introduction](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#introduction) +[Introduction](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#introduction) -2.[Body](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#body) -         * [Distance](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#distance)          * [Heuristic](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#heuristic) @@ -24,12 +22,13 @@          * [Disjoint Pattern DB](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#disjoint-pattern-db) -3.[Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) +[Conclusion](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#conclusion) -4.[References](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#references) +[References](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#references) ## Introduction -Heuristics are methods for solving problems in a fast way. Actually It's about using a practical method that doesn’t necessarily need to be perfect! Heuristic methods speed up the process of reaching a satisfactory solution. Moreover Advanced Heuristics are creating a sequence of heuristics to detect complex malware and detecting then solving the problems. In this lecture note first we review some prerequisites then we will mention some approaches to make heuristics more efficient! +Heuristics are methods for solving problems in a fast way. Actually It's about using a practical method that doesn’t necessarily need to be perfect! Heuristic methods speed up the process of reaching a satisfactory solution. +Moreover Advanced Heuristics are creating a sequence of heuristics to detect complex malware and detecting then solving the problems. In this lecture note first we review some prerequisites then we will mention some approaches to make heuristics more efficient! ## Body @@ -219,8 +218,4 @@ https://artint.info/2e/html/ArtInt2e.Ch3.S6.SS2.html https://en.wikipedia.org/wiki/Consistent_heuristic -https://stackoverflow.com/questions/1568600/what-is-the-difference-between-monotonicity-and-the-admissibility-of-a-heuristic - -https://stackoverflow.com/questions/46554459/intuitively-understanding-why-consistency-is-required-for-optimality-in-a-searc - https://courses.cs.washington.edu/courses/cse473/12sp/slides/04-heuristics.pdf From 06dee1726afa33253bef1b2f422f95b2a59a3691 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 12 Nov 2021 18:51:35 +0330 Subject: [PATCH 100/114] Update index.md --- notebooks/advanced_heuristics/index.md | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 821311b1..1c4fd6e0 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -27,10 +27,9 @@ [References](https://github.com/mmdrez4/notes/blob/master/notebooks/advanced_heuristics/index.md#references) ## Introduction -Heuristics are methods for solving problems in a fast way. Actually It's about using a practical method that doesn’t necessarily need to be perfect! Heuristic methods speed up the process of reaching a satisfactory solution. -Moreover Advanced Heuristics are creating a sequence of heuristics to detect complex malware and detecting then solving the problems. In this lecture note first we review some prerequisites then we will mention some approaches to make heuristics more efficient! - -## Body +Heuristics are methods for solving problems in a fast way. Actually It's about using a practical method that doesn’t necessarily need to be perfect! Heuristic methods speed up the process of reaching a satisfactory solution. +In this lecture we review some prerequisites then we present a couple of approaches and Heuristics, which seeked to be flawless, but the space and time complexity of these algorithms is not sufficient. So we try to improve the Heuristics to eliminate the recent obstacles and make our heuristics more efficient. we call these methods advanced heuristics! +We'll show how to detect complex problems then solving them by advanced heuristics and compare them with the previous methods. ### Distance First we review some basic definitions: From 9bc92f8f26b7a86bdb707bc2fe61b1558366352a Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 12 Nov 2021 19:38:05 +0330 Subject: [PATCH 101/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 1c4fd6e0..b6e8fa6f 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -208,7 +208,7 @@ As you can see in this problem we have an undirected graph, which each of its no Now we can have souple of flower breeds subset and combine obtained answers by them. For example if we have K flower breeds, we can solve the problem for each flower breed and calculate its heuristic function and then choose maximum of the functions. So we used Combining Pattern DB to achieve a better heuristic function. ## Conclusion -So far we've introduced some efficient ways to make our heuristic closer to the real answer. some of them were to make our heuristic monotonic and Dominant, or to make a dominant heuristic based on some heuristics we had. some advanced ways were for even more basic problem, namely how to create a heuristic function for a problem. these included Relaxing problems and Patternal DBs. We Entered deeply into the Patternal DBs And introduced ways to improve them a lot. despite all these ways, there are so many problems that aren't solvable by even these ways, and there are so many ways ahead of us that you have to learn & use further. thanks for reading all of these concepts and we wish you have a great future. a bid farewell. +So far we've introduced some efficient ways to make our heuristic closer to the real answer. some of them were to make our heuristic monotonic and Dominant, or to make a dominant heuristic based on some heuristics we had. some advanced ways were even for more basic problem, namely how to create a heuristic function for a problem. these included Relaxing problems and Patternal DBs. We Entered deeply into the Patternal DBs And introduced ways to improve them a lot. despite all these ways, there are so many problems that aren't solvable by even these ways, and there are so many ways ahead of us that you have to learn & use further. thanks for reading all of these concepts and we wish you have a great future. a bid farewell. ## References https://machinelearningmastery.com/distance-measures-for-machine-learning/ From 5453eee7edb5be695999e7ea9d80c454e00355f9 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 12 Nov 2021 19:41:30 +0330 Subject: [PATCH 102/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index b6e8fa6f..276d9c6c 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -172,7 +172,7 @@ h' = min(S'), h = min(S) ∧ S ⊆ S' ∧ ∀ s, s': if s &Su hamilton path

-Hamilton path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in n2 time complexity. +Hamiltonian path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in n2 time complexity.
From 0fab07f528cf7b2ea30a982eceefaa655d302f61 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 12 Nov 2021 19:43:25 +0330 Subject: [PATCH 103/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 276d9c6c..14eb972d 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -111,7 +111,7 @@ For this we only need to define a new and more appropriate heuristic like the be -**ℏ(n’) = max(h(n’), ℏ(n’) - c(n, a, n’))** +**ℏ(n’) = max(h(n’), ℏ(n) - c(n, a, n’))** Above Heuristic is defined Recursively. From a61db974907225ca2012fea556508b823ff85494 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 12 Nov 2021 19:47:56 +0330 Subject: [PATCH 104/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 14eb972d..543e8fa1 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -205,7 +205,7 @@ One solution for solving this drawback is Disjoint Pattern DB. In this method we As you can see in this problem we have an undirected graph, which each of its nodes is a florist shop and has some flower breeds. Two brothers should march from start_node towards goal_node in a way that has the least cost and they met every existing flower breeds. In this case we choose a subset of flower breeds S that reach goal_node by a specific pattern and set goal_state as satisfying path with the least cost and meeting every existing flower breeds of set S. Hitherto we used pure Pattern DB. -Now we can have souple of flower breeds subset and combine obtained answers by them. For example if we have K flower breeds, we can solve the problem for each flower breed and calculate its heuristic function and then choose maximum of the functions. So we used Combining Pattern DB to achieve a better heuristic function. +Now we can have couple of flower breeds subset and combine obtained answers by them. For example if we have K flower breeds, we can solve the problem for each flower breed and calculate its heuristic function and then choose maximum of the functions. So we used Combining Pattern DB to achieve a better heuristic function. ## Conclusion So far we've introduced some efficient ways to make our heuristic closer to the real answer. some of them were to make our heuristic monotonic and Dominant, or to make a dominant heuristic based on some heuristics we had. some advanced ways were even for more basic problem, namely how to create a heuristic function for a problem. these included Relaxing problems and Patternal DBs. We Entered deeply into the Patternal DBs And introduced ways to improve them a lot. despite all these ways, there are so many problems that aren't solvable by even these ways, and there are so many ways ahead of us that you have to learn & use further. thanks for reading all of these concepts and we wish you have a great future. a bid farewell. From 6cb298e2b99018909c421c03a5580cb6365ba3bb Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 12 Nov 2021 19:51:34 +0330 Subject: [PATCH 105/114] Update index.md --- notebooks/advanced_heuristics/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 543e8fa1..bc30084c 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -178,7 +178,7 @@ Hamiltonian path is a tree which every its inner node have exactly one leaf. Hen ### Pattern DB -In previous sectionwe introduced relaxing method which improves heuristic function by lowering constraints. Pattern DB is yet another way to improve heuristic function which is actually a subset of Relaxing method. +In previous section we introduced relaxing method which improves heuristic function by lowering constraints. Pattern DB is yet another way to improve heuristic function which is actually a subset of Relaxing method. In this method we choose a subset of constraints by a specific pattern and ignore the other constraints. Since we ignored some constraints we did relaxing method and hence answer of the converted problem is a heuristic gfunction. @@ -205,7 +205,7 @@ One solution for solving this drawback is Disjoint Pattern DB. In this method we As you can see in this problem we have an undirected graph, which each of its nodes is a florist shop and has some flower breeds. Two brothers should march from start_node towards goal_node in a way that has the least cost and they met every existing flower breeds. In this case we choose a subset of flower breeds S that reach goal_node by a specific pattern and set goal_state as satisfying path with the least cost and meeting every existing flower breeds of set S. Hitherto we used pure Pattern DB. -Now we can have couple of flower breeds subset and combine obtained answers by them. For example if we have K flower breeds, we can solve the problem for each flower breed and calculate its heuristic function and then choose maximum of the functions. So we used Combining Pattern DB to achieve a better heuristic function. +Now we can have couple of flower breeds subset and combine obtained answers by them. For example if we have K flower breeds, we can solve the problem for each flower breed and calculate its heuristic function and then choose maximum of the functions. So we used Combining Pattern DB to achieve a better heuristic function. ## Conclusion So far we've introduced some efficient ways to make our heuristic closer to the real answer. some of them were to make our heuristic monotonic and Dominant, or to make a dominant heuristic based on some heuristics we had. some advanced ways were even for more basic problem, namely how to create a heuristic function for a problem. these included Relaxing problems and Patternal DBs. We Entered deeply into the Patternal DBs And introduced ways to improve them a lot. despite all these ways, there are so many problems that aren't solvable by even these ways, and there are so many ways ahead of us that you have to learn & use further. thanks for reading all of these concepts and we wish you have a great future. a bid farewell. From 639623a24e88b144026dc98df4baafe509c52633 Mon Sep 17 00:00:00 2001 From: sepehrkianian09 <59197271+sepehrkianian09@users.noreply.github.com> Date: Sat, 13 Nov 2021 00:02:36 +0330 Subject: [PATCH 106/114] fix(index.md): Fixed my problems --- notebooks/advanced_heuristics/index.md | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index bc30084c..7d1c32ab 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -163,6 +163,7 @@ Consider h as the answer and h' as our heuristic h' = min(S'), h = min(S) ∧ S ⊆ S' ∧ ∀ s, s': if s ⊆ s' ⇒ min(s') ≤ min(s) ⇒ h' ≤ h +As you may have understood, S is a subset of S'. Therefore S' is a bigger Space than S that contains all elements of S. So, if you wanna calculate the minimum element of S', besides checking S elements, you should check more elements that aren't in S. So it's probable that you visit an element in S' which is less than the minimum of S. Therefore we can say min(S') is less than or equal to min(S). ***Example*** @@ -172,7 +173,7 @@ h' = min(S'), h = min(S) ∧ S ⊆ S' ∧ ∀ s, s': if s &Su hamilton path

-Hamiltonian path is a tree which every its inner node have exactly one leaf. Hence, the problem is converting above graph to a 1-array tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-array tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in n2 time complexity. +Hamiltonian path is a tree which every its inner has have exactly one son. Hence, the problem is converting the above graph to a 1-ary tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-ary tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in n2 time complexity.
@@ -187,7 +188,7 @@ for further information watch this video: [watch](https://www.youtube.com/watch? ### Combining Pattern DB -We can act further and use couple of patterns and choose couple of constraints' subsets. Therefore we have couple of heuristic functions. This helps dominancy of our heuristic function. First thing to do finding the dominant heuristic function is to choose the maximum function of the obtained heuristic functions. Since every heuristic is always less than the real answer, maximum of these functions is a lower bound of the real answer and for every set of obtained heuristic functions we can implement this method to find a dominant heuristic function. +We can go further and use couple of patterns and choose couple of constraints' subsets. Therefore we have couple of heuristic functions. This helps dominancy of our heuristic function. First thing to do finding the dominant heuristic function is to choose the maximum function of the obtained heuristic functions. Since every heuristic is always less than the real answer, maximum of these functions is a lower bound of the real answer and for every set of obtained heuristic functions we can implement this method to find a dominant heuristic function.
@@ -203,12 +204,12 @@ One solution for solving this drawback is Disjoint Pattern DB. In this method we pattern

-As you can see in this problem we have an undirected graph, which each of its nodes is a florist shop and has some flower breeds. Two brothers should march from start_node towards goal_node in a way that has the least cost and they met every existing flower breeds. In this case we choose a subset of flower breeds S that reach goal_node by a specific pattern and set goal_state as satisfying path with the least cost and meeting every existing flower breeds of set S. Hitherto we used pure Pattern DB. +As you can see in this problem we have an undirected graph, which each of its nodes is a florist shop and has some flower breeds. Two brothers should march from their home towards their mother's place in a way that has the least cost and they met every existing flower breeds. In this case we choose a subset of flower breeds S that reach the goal_node(mother's place) by a specific pattern and set goal_state as satisfying path with the least cost and meeting every existing flower breeds of set S. Hitherto we used pure Pattern DB. Now we can have couple of flower breeds subset and combine obtained answers by them. For example if we have K flower breeds, we can solve the problem for each flower breed and calculate its heuristic function and then choose maximum of the functions. So we used Combining Pattern DB to achieve a better heuristic function. ## Conclusion -So far we've introduced some efficient ways to make our heuristic closer to the real answer. some of them were to make our heuristic monotonic and Dominant, or to make a dominant heuristic based on some heuristics we had. some advanced ways were even for more basic problem, namely how to create a heuristic function for a problem. these included Relaxing problems and Patternal DBs. We Entered deeply into the Patternal DBs And introduced ways to improve them a lot. despite all these ways, there are so many problems that aren't solvable by even these ways, and there are so many ways ahead of us that you have to learn & use further. thanks for reading all of these concepts and we wish you have a great future. a bid farewell. +At first we were facing with problems and previous Uninformed methods, which solved those problems with exponential time and space complexity. Therefore we introduced some new methods, namely Informed Methods, which used a heuristic function to solve those problems more efficient. Later we introduced some efficient ways to make our heuristic closer to the real answer. some of them were to make our heuristic monotonic and Dominant, or to make a dominant heuristic based on some heuristics we had. Some advanced ways were even for more basic problem, namely how to create a heuristic function for a problem. these included Relaxing problems and Patternal DBs. We then Entered into the Patternal DBs And introduced ways to improve them a lot. despite all these ways, there are so many problems that aren't solvable by even these ways, and there are so many ways ahead of us that you have to learn & use further. thanks for reading all of these concepts and we wish you have a great future. a bid farewell. ## References https://machinelearningmastery.com/distance-measures-for-machine-learning/ From d352360a3be1c65d0969c793ab4f4c5f9b8ad790 Mon Sep 17 00:00:00 2001 From: sepehrkianian09 <59197271+sepehrkianian09@users.noreply.github.com> Date: Sat, 13 Nov 2021 00:08:32 +0330 Subject: [PATCH 107/114] feat(index.md): Add Pattern-DB Pics --- .../pic/Combining-Pattern-DB.jpg | Bin 0 -> 6385 bytes notebooks/advanced_heuristics/pic/Pattern-Db.jpg | Bin 0 -> 9887 bytes 2 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 notebooks/advanced_heuristics/pic/Combining-Pattern-DB.jpg create mode 100644 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zffjq8VF}l!>zXZNEJdhzCNfQ9T}R&etQ(=v!Wd%oqSlJ?FDqgg@2ZmMdw9|d;KO$o zAt7TRBcT7aYX<@OZjS*TzM5kv$^-a|AP!DmzE~n+@wf?PLOOaTE^Zz%6^HxOhwn@f z1rn1Yd8r2Q^hUi;YO>_T>j7b&htB^-{mLcK_WBN>JWW^Mbk1Sa&nmbm7(ppJZ=W1m{+_M zyC%3>)70_2o^ZbZycd@Q-8xdqG5~5BK#u1w^lcP5gvxz1L9tz6@BipGXcq=jE4+8~ zGwgruSU_+XCDb2@CJb}fLhO#DpqPl{hB0UP*c-|eF#QMHs8HXZes1kdQh9@E8CdJh z$9wtIJP&_34`tx2*}|ob@?c9!;6BoQ-*Lcnr7Uv2E;ju)+3rRaZcd}4Ij?ob{B-u8 zIga^LRijV;NM2HKN^`#Kw+m=sw8Igi#94DSlP%JxbVajNcp*>4T#2ONNXN7J>Q!=t z+K=yUxce?eL56M34Mv`=72kTx2)=x=We00!9_+c>TlQSI>I{XeKWc56e9xYJvbvM~ z9gW^vslzAx~%^C%P^Se_JM(d{LoELkBsQIHHtnW>xmD&ZHDS!D3)~omrmZ8 U!Z6n7DNcPFnM?*>^V|9V0`x1fEdT%j literal 0 HcmV?d00001 From d5c33906616faf73c498907e126b92cf3a1b5631 Mon Sep 17 00:00:00 2001 From: sepehrkianian09 <59197271+sepehrkianian09@users.noreply.github.com> Date: Sat, 13 Nov 2021 00:11:43 +0330 Subject: [PATCH 108/114] feat(index.md): Add Pattern-DB pics to the note --- notebooks/advanced_heuristics/index.md | 9 +++++++++ 1 file changed, 9 insertions(+) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 7d1c32ab..eef800a3 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -184,12 +184,21 @@ In previous section we introduced relaxing method which improves heuristic funct In this method we choose a subset of constraints by a specific pattern and ignore the other constraints. Since we ignored some constraints we did relaxing method and hence answer of the converted problem is a heuristic gfunction. for further information watch this video: [watch](https://www.youtube.com/watch?v=HZWV4uOJWk8) + +

+ distance +

+
### Combining Pattern DB We can go further and use couple of patterns and choose couple of constraints' subsets. Therefore we have couple of heuristic functions. This helps dominancy of our heuristic function. First thing to do finding the dominant heuristic function is to choose the maximum function of the obtained heuristic functions. Since every heuristic is always less than the real answer, maximum of these functions is a lower bound of the real answer and for every set of obtained heuristic functions we can implement this method to find a dominant heuristic function. +

+ distance +

+
### Disjoint Pattern DB From 0dcacf015de9ac23ec44e0df0eec953c63300412 Mon Sep 17 00:00:00 2001 From: sepehrkianian09 <59197271+sepehrkianian09@users.noreply.github.com> Date: Sat, 13 Nov 2021 00:14:52 +0330 Subject: [PATCH 109/114] Fix(index.md): Fix Image dimensions --- notebooks/advanced_heuristics/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index eef800a3..a39ab873 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -186,7 +186,7 @@ In this method we choose a subset of constraints by a specific pattern and ignor for further information watch this video: [watch](https://www.youtube.com/watch?v=HZWV4uOJWk8)

- distance + distance


@@ -196,7 +196,7 @@ for further information watch this video: [watch](https://www.youtube.com/watch? We can go further and use couple of patterns and choose couple of constraints' subsets. Therefore we have couple of heuristic functions. This helps dominancy of our heuristic function. First thing to do finding the dominant heuristic function is to choose the maximum function of the obtained heuristic functions. Since every heuristic is always less than the real answer, maximum of these functions is a lower bound of the real answer and for every set of obtained heuristic functions we can implement this method to find a dominant heuristic function.

- distance + distance


From eb8baefe247b0ba47594b45fbcdeefb3b9c55cfe Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Sat, 13 Nov 2021 00:16:07 +0330 Subject: [PATCH 110/114] Update index.md --- notebooks/advanced_heuristics/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index a39ab873..0bf5d550 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -186,7 +186,7 @@ In this method we choose a subset of constraints by a specific pattern and ignor for further information watch this video: [watch](https://www.youtube.com/watch?v=HZWV4uOJWk8)

- distance + distance


@@ -196,7 +196,7 @@ for further information watch this video: [watch](https://www.youtube.com/watch? We can go further and use couple of patterns and choose couple of constraints' subsets. Therefore we have couple of heuristic functions. This helps dominancy of our heuristic function. First thing to do finding the dominant heuristic function is to choose the maximum function of the obtained heuristic functions. Since every heuristic is always less than the real answer, maximum of these functions is a lower bound of the real answer and for every set of obtained heuristic functions we can implement this method to find a dominant heuristic function.

- distance + distance


From 51b316b4d39238572bb07ba2e53ec39dad13dad7 Mon Sep 17 00:00:00 2001 From: sepehrkianian09 <59197271+sepehrkianian09@users.noreply.github.com> Date: Sat, 13 Nov 2021 00:19:11 +0330 Subject: [PATCH 111/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 0bf5d550..a3af9263 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -181,7 +181,7 @@ Hamiltonian path is a tree which every its inner has have exactly one son. Hence In previous section we introduced relaxing method which improves heuristic function by lowering constraints. Pattern DB is yet another way to improve heuristic function which is actually a subset of Relaxing method. -In this method we choose a subset of constraints by a specific pattern and ignore the other constraints. Since we ignored some constraints we did relaxing method and hence answer of the converted problem is a heuristic gfunction. +In this method we choose a subset of constraints by a specific pattern and ignore the other constraints. Since we ignored some constraints we did relaxing method and hence answer of the converted problem is a heuristic function. for further information watch this video: [watch](https://www.youtube.com/watch?v=HZWV4uOJWk8) From 70d27a2d94d50376c6687c82467530125e4abb7d Mon Sep 17 00:00:00 2001 From: sepehrkianian09 <59197271+sepehrkianian09@users.noreply.github.com> Date: Tue, 16 Nov 2021 01:01:08 +0330 Subject: [PATCH 112/114] fix(adv_heuristics): final Fix of Sep_parts --- notebooks/advanced_heuristics/index.md | 13 ++++++++----- 1 file changed, 8 insertions(+), 5 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index a3af9263..b68e6103 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -158,12 +158,15 @@ So one of the most common ways to find efficient heuristic function is that we c Discussing more specifically, original problem has a set of constraints. Noe we **ignore some constraints** and solve the expanded problem. Since number of constraints have decreased, we have more goal states and our state space expands. ***Proof that Relaxing works*** + +Consider Our problem as P. To calculate heuristics in relaxing way, we omit some Constraints and turn P into a more general Problem called P'. +then, Consider S as the set of answers that are valid for P, and S' as the set of answers that are valid for P'. then, if you consider h as our answer and h' as our heuristic function, we can say: -Consider h as the answer and h' as our heuristic - -h' = min(S'), h = min(S) ∧ S ⊆ S' ∧ ∀ s, s': if s ⊆ s' ⇒ min(s') ≤ min(s) ⇒ h' ≤ h +h' = min(S'), h = min(S) ∧ S ⊆ S' + +And as you know that: ∀ s, s': if s ⊆ s' ⇒ min(s') ≤ min(s) -As you may have understood, S is a subset of S'. Therefore S' is a bigger Space than S that contains all elements of S. So, if you wanna calculate the minimum element of S', besides checking S elements, you should check more elements that aren't in S. So it's probable that you visit an element in S' which is less than the minimum of S. Therefore we can say min(S') is less than or equal to min(S). +We can conclude that: h' ≤ h ***Example*** @@ -173,7 +176,7 @@ As you may have understood, S is a subset of S'. Therefore S' is a bigger Space hamilton path

-Hamiltonian path is a tree which every its inner has have exactly one son. Hence, the problem is converting the above graph to a 1-ary tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-ary tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in n2 time complexity. +Hamiltonian path is a rooted tree which every its inner has have exactly one son. Hence, the problem is converting the above graph to a 1-ary tree, which covers all nodes of the graph with existing edges and minimum sum of edges' weight. Now we ignore being 1-ary tree constraint to convert original problem to a more general one. Accordingly our problem is now a Minimum Spaning Tree problem which is soluable in n2 time complexity.
From 202f9e749d49c0ad09b64c3ade675103146eee92 Mon Sep 17 00:00:00 2001 From: sepehrkianian09 <59197271+sepehrkianian09@users.noreply.github.com> Date: Tue, 16 Nov 2021 01:05:42 +0330 Subject: [PATCH 113/114] fix(adv_heuristics): Fix final parts of Sep --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index b68e6103..27b294b9 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -208,7 +208,7 @@ We can go further and use couple of patterns and choose couple of constraints' s One of the main flaws of choosing maximum of the obtained heuristic functions is that in many cases, there is an upper bound for these functions. Therefore even by expanding subset of chosen constraints, we will not cross the upper bound. -One solution for solving this drawback is Disjoint Pattern DB. In this method we apport set of constraints and define independent actions for the general problem. Now we can sum the obtained heuristic function to achieve a greater heuristic function, which improves the time complexity. It is proven that the sum of obtained heuristic functions is still a lower bound of real answer of the probelm. +One solution for solving this drawback is Disjoint Pattern DB. In this method we split set of constraints into some disjoint subsets of constraints, and define independent actions for the general problem. Now we can sum the obtained heuristic function to achieve a greater heuristic function, which improves the time complexity. It is proven that the sum of obtained heuristic functions is still a lower bound of real answer of the problem. **Example** From 8ec8e00c59911264ddc247e75bf3c37903247e43 Mon Sep 17 00:00:00 2001 From: sepehrkianian09 <59197271+sepehrkianian09@users.noreply.github.com> Date: Tue, 16 Nov 2021 01:08:58 +0330 Subject: [PATCH 114/114] fix(adv_heuristics): Fix final parts of Sep --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 27b294b9..ac93f08a 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -160,7 +160,7 @@ Discussing more specifically, original problem has a set of constraints. Noe we ***Proof that Relaxing works*** Consider Our problem as P. To calculate heuristics in relaxing way, we omit some Constraints and turn P into a more general Problem called P'. -then, Consider S as the set of answers that are valid for P, and S' as the set of answers that are valid for P'. then, if you consider h as our answer and h' as our heuristic function, we can say: +then, Consider S as the set of answers that are valid for P, and S' as the set of answers that are valid for P'. so, if you consider h as our answer and h' as our heuristic function, we can say: h' = min(S'), h = min(S) ∧ S ⊆ S'

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z#uBCAIqsr-dz~fH8O#vaXp0T5;tcf#tFBr~26Tp3UyUuYNi8Qa4Uwa5wjHknlfQ6n)UpgK)ll;I{eC9S%GUshbrPN2* z%wM7G0>)N}y2shB7p_90fU`#`Nctqn<}b-}FzJ`R>vfK^tcZP7k9TPhMrNXTk1CUo zR0@WV$(?XYa+wmXr>fWASNO;%e+{6_%J|=7 z|MwC9&ibze{*}PL68KjF|4QIr3H<*gF#qQa>!q)!^Vw}{G?br_o|$ev^k(#b0B`9Q AQ~&?~ literal 0 HcmV?d00001 From 1741692885cd1c3628d2baf316ee4c6b55760d28 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:37:22 +0330 Subject: [PATCH 077/114] Add files via upload From b88ceace65f1cfb1dc74f16d6a8162018c44a05c Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 21:38:17 +0330 Subject: [PATCH 078/114] Update index.md --- notebooks/advanced_heuristics/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 145e02fd..ae9c7ae7 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -85,7 +85,7 @@ for more information and better understanding watch this video: [watch](https:// ### Monotonicity implies admissibility We will prove that **Consistency** implies **Admissibility** whereas the opposite is not necessarily true.

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z?IOdrm{bCddBvOhG{Y#kq(czo!1K_~%^AKEPaXcEx$+yqivw8obFO6s`i`k%4IW#4LsvTJ-$GsCsv*-^rzM z^9Qca576fOXbxwj03l0qgzTb}{~YXZwe71)Z?3M(^cKh+aK)ba_mpFRbK%Y?UsFey5rrn6W{)WT Date: Fri, 29 Oct 2021 17:33:30 +0330 Subject: [PATCH 005/114] Create matadata.yml --- notebooks/advanced_heuristics/matadata.yml | 25 ++++++++++++++++++++++ 1 file changed, 25 insertions(+) create mode 100644 notebooks/advanced_heuristics/matadata.yml diff --git a/notebooks/advanced_heuristics/matadata.yml b/notebooks/advanced_heuristics/matadata.yml new file mode 100644 index 00000000..13a82fc0 --- /dev/null +++ b/notebooks/advanced_heuristics/matadata.yml @@ -0,0 +1,25 @@ +title: # shown on browser tab + +header: + title: advanced Heuristics # title of your notebook + description: # short description of your notebook + +authors: + label: + position: top + content: + # list of notebook authors + - name: Mohammadreza Daviran # name of author + role: Author # change this if you want + contact: + # list of contact information + - link: https://github.com/ + icon: fab fa-github + # optionally add other contact information like + # - link: # contact link + # icon: # awsomefont tag for link (check: https://fontawesome.com/v5.15/icons) + +comments: + # enable comments for your post + label: false + kind: comments From 6e33d3609eacf5aa712b8aef8e2a75ecff5ed5f1 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 17:57:10 +0330 Subject: [PATCH 006/114] Update index.md --- notebooks/advanced_heuristics/index.md | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md index 2c3588fc..1ca6ff50 100644 --- a/notebooks/advanced_heuristics/index.md +++ b/notebooks/advanced_heuristics/index.md @@ -7,7 +7,7 @@ Distance is a numerical measurement of how far apart objects or points are: * [Euclidean Distance](https://en.wikipedia.org/wiki/Euclidean_distance) calculates the distance between two real-valued vectors. * [Manhattan Distance](https://en.wikipedia.org/wiki/Taxicab_geometry) is sum of the absolute differences of their Cartesian coordinates.

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z?IOdrm{bCddBvOhG{Y#kq(czo!1K_~%^AKEPaXcEx$+yqivw8obFO6s`i`k%4IW#4LsvTJ-$GsCsv*-^rzM z^9Qca576fOXbxwj03l0qgzTb}{~YXZwe71)Z?3M(^cKh+aK)ba_mpFRbK%Y?UsFey5rrn6W{)WT Date: Fri, 29 Oct 2021 17:27:37 +0330 Subject: [PATCH 004/114] create advanced_heuristics folder --- notebooks/advanced_heuristics/index.md | 148 ++++++++++++++++++ .../advanced_heuristics/pic/admissible.png | Bin 0 -> 74668 bytes notebooks/advanced_heuristics/pic/c.png | Bin 0 -> 36239 bytes .../advanced_heuristics/pic/consistency.png | Bin 0 -> 110197 bytes .../advanced_heuristics/pic/distance.png | Bin 0 -> 77803 bytes notebooks/advanced_heuristics/pic/example.png | Bin 0 -> 29868 bytes .../advanced_heuristics/pic/hamilton path.png | Bin 0 -> 174794 bytes .../pic/heuristic proof.png | Bin 0 -> 77762 bytes .../advanced_heuristics/pic/monotonic.png | Bin 0 -> 97434 bytes notebooks/advanced_heuristics/pic/proof.png | Bin 0 -> 107414 bytes 10 files changed, 148 insertions(+) create mode 100644 notebooks/advanced_heuristics/index.md create mode 100644 notebooks/advanced_heuristics/pic/admissible.png create mode 100644 notebooks/advanced_heuristics/pic/c.png create mode 100644 notebooks/advanced_heuristics/pic/consistency.png create mode 100644 notebooks/advanced_heuristics/pic/distance.png create mode 100644 notebooks/advanced_heuristics/pic/example.png create mode 100644 notebooks/advanced_heuristics/pic/hamilton path.png create mode 100644 notebooks/advanced_heuristics/pic/heuristic proof.png create mode 100644 notebooks/advanced_heuristics/pic/monotonic.png create mode 100644 notebooks/advanced_heuristics/pic/proof.png diff --git a/notebooks/advanced_heuristics/index.md b/notebooks/advanced_heuristics/index.md new file mode 100644 index 00000000..2c3588fc --- /dev/null +++ b/notebooks/advanced_heuristics/index.md @@ -0,0 +1,148 @@ +# Advanced Heuristic +#### Kahbod Aeini, Mohammadreza Daviran and Sepehr Kianian + +First we review some basic definitions + +Distance is a numerical measurement of how far apart objects or points are: +* [Euclidean Distance](https://en.wikipedia.org/wiki/Euclidean_distance) calculates the distance between two real-valued vectors. +* [Manhattan Distance](https://en.wikipedia.org/wiki/Taxicab_geometry) is sum of the absolute differences of their Cartesian coordinates. +

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z?IOdrm{bCddBvOhG{Y#kq(czo!1K_~%^AKEPaXcEx$+yqivw8obFO6s`i`k%4IW#4LsvTJ-$GsCsv*-^rzM z^9Qca576fOXbxwj03l0qgzTb}{~YXZwe71)Z?3M(^cKh+aK)ba_mpFRbK%Y?UsFey5rrn6W{)WT Date: Fri, 29 Oct 2021 17:20:01 +0330 Subject: [PATCH 002/114] Create matadat.yml --- notebooks/Advanced Heuristics/matadata.yml | 25 ++++++++++++++++++++++ 1 file changed, 25 insertions(+) create mode 100644 notebooks/Advanced Heuristics/matadata.yml diff --git a/notebooks/Advanced Heuristics/matadata.yml b/notebooks/Advanced Heuristics/matadata.yml new file mode 100644 index 00000000..f23db3f1 --- /dev/null +++ b/notebooks/Advanced Heuristics/matadata.yml @@ -0,0 +1,25 @@ +title: # shown on browser tab + +header: + title: Advanced Heuristics # title of your notebook + description: # short description of your notebook + +authors: + label: + position: top + content: + # list of notebook authors + - name: Mohammadreza Daviran # name of author + role: Author # change this if you want + contact: + # list of contact information + - link: https://github.com/mmdrez4 + icon: fab fa-github + # optionally add other contact information like + # - link: # contact link + # icon: # awsomefont tag for link (check: https://fontawesome.com/v5.15/icons) + +comments: + # enable comments for your post + label: false + kind: comments From 5d71e0f49c31825ec2a3ccfe27b0cfcdd8b58067 Mon Sep 17 00:00:00 2001 From: mmdrez4 <59529379+mmdrez4@users.noreply.github.com> Date: Fri, 29 Oct 2021 17:25:13 +0330 Subject: [PATCH 003/114] Delete notebooks/Advanced Heuristics directory --- notebooks/Advanced Heuristics/index.md | 148 ------------------ notebooks/Advanced Heuristics/matadata.yml | 25 --- .../Advanced Heuristics/pic/admissible.png | Bin 74668 -> 0 bytes notebooks/Advanced Heuristics/pic/c.png | Bin 36239 -> 0 bytes .../Advanced Heuristics/pic/consistency.png | Bin 110197 -> 0 bytes .../Advanced Heuristics/pic/distance.png | Bin 77803 -> 0 bytes notebooks/Advanced Heuristics/pic/example.png | Bin 29868 -> 0 bytes .../Advanced 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diff --git a/notebooks/Advanced Heuristics/index.md b/notebooks/Advanced Heuristics/index.md deleted file mode 100644 index 2c3588fc..00000000 --- a/notebooks/Advanced Heuristics/index.md +++ /dev/null @@ -1,148 +0,0 @@ -# Advanced Heuristic -#### Kahbod Aeini, Mohammadreza Daviran and Sepehr Kianian - -First we review some basic definitions - -Distance is a numerical measurement of how far apart objects or points are: -* [Euclidean Distance](https://en.wikipedia.org/wiki/Euclidean_distance) calculates the distance between two real-valued vectors. -* [Manhattan Distance](https://en.wikipedia.org/wiki/Taxicab_geometry) is sum of the absolute differences of their Cartesian coordinates. -