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Add documentation for transitive reduction #313

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May 29, 2023
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Add documentation for transitive reduction #313

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Add documentation for transitive reduction
nrkramer May 27, 2023
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nrkramer May 27, 2023
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17 changes: 17 additions & 0 deletions README.md
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Expand Up @@ -122,6 +122,7 @@ If you are interested, please contact us at zigrazor@gmail.com or contribute to
- [Cycle Detection](#cycle-detection)
- [Bellman-Ford](#bellman-ford)
- [Floyd Warshall](#floyd-warshall)
- [Transitive Reduction](#transitive-reduction)
- [Kruskal Algorithm](#kruskal-algorithm)
- [Borůvka's Algorithm](#borůvkas-algorithm)
- [Graph Slicing based on connectivity](#graph-slicing-based-on-connectivity)
Expand Down Expand Up @@ -432,6 +433,22 @@ We initialize the solution matrix same as the input graph matrix as a first step
1) k is not an intermediate vertex in shortest path from i to j. We keep the value of dist[i][j] as it is.
2) k is an intermediate vertex in shortest path from i to j. We update the value of dist[i][j] as dist[i][k] + dist[k][j] if dist[i][j] > dist[i][k] + dist[k][j]

### Transitive Reduction

[Transitive Reduction](https://en.wikipedia.org/wiki/Transitive_reduction)

This algorithm is used to construct a directed graph with the same reachability and satisfies transitive closure, with as few edges as possible. More concretely, it creates a minimum equivalent graph with as few edges as possible, removing "short-circuit" paths through the graph.

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This is done by iterating through each node-pair, checking to see if two edges exist that leads out of the first node OR out of the last node, removing the node-pair edge if it exists.

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Expected: 80; Actual: 184

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In pseudocode:
foreach x in graph.vertices
foreach y in graph.vertices
foreach z in graph.vertices
delete edge xz if edges xy and yz exist

Our implementation has if gates that do early checking for edges in multiple places, which gives it a slightly faster runtime than the cubic pseudocode here.

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Code scanning / Markdownlint (reported by Codacy)

Expected: 80; Actual: 157

Expected: 80; Actual: 157

### Kruskal Algorithm

[Kruskal Algorithm](https://en.wikipedia.org/wiki/Kruskal%27s_algorithm) can be used to find the minimum spanning forest of an undirected edge-weighted graph. Time Complexity O(E log E) = O(E log V) where V is number of vertices and E is number of edges in graph. The main speed limitation for this algorithm is sorting the edges.
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