Dynamic Programming is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of subproblems, so that we do not have to re-compute them when needed later. This simple optimization reduces time complexities from exponential to polynomial. For example, if we write simple recursive solution for Fibonacci Numbers, we get exponential time complexity and if we optimize it by storing solutions of subproblems, time complexity reduces to linear
Ugly Numbers Ugly numbers are numbers whose only prime factors are 2, 3 or 5. The sequence 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, … shows the first 11 ugly numbers. By convention, 1 is included.
Given a number n, the task is to find n’th Ugly number.
Input : n = 7 Output : 8
Input : n = 10 Output : 12
Input : n = 15 Output : 24
Input : n = 150 Output : 5832
Here is a time efficient solution with O(n) extra space. The ugly-number sequence is 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, … because every number can only be divided by 2, 3, 5, one way to look at the sequence is to split the sequence to three groups as below: (1) 1×2, 2×2, 3×2, 4×2, 5×2, … (2) 1×3, 2×3, 3×3, 4×3, 5×3, … (3) 1×5, 2×5, 3×5, 4×5, 5×5, … We can find that every subsequence is the ugly-sequence itself (1, 2, 3, 4, 5, …) multiply 2, 3, 5. Then we use similar merge method as merge sort, to get every ugly number from the three subsequence. Every step we choose the smallest one, and move one step after.