Online Programming Server

There are two key attributes that a problem must have in order for dynamic programming to be applicable: optimal substructure and overlapping subproblems. However, when the overlapping problems are much smaller than the original problem, the strategy is called "divide and conquer" rather than "dynamic programming". This is why mergesort, quicksort, and finding all matches of a regular expression are not classified as dynamic programming problems.

Optimal substructure means that the solution to a given optimization problem can be obtained by the combination of optimal solutions to its subproblems. Consequently, the first step towards devising a dynamic programming solution is to check whether the problem exhibits such optimal substructure. Such optimal substructures are usually described by means of recursion. For example, given a graph G=(V,E), the shortest path p from a vertex u to a vertex v exhibits optimal substructure: take any intermediate vertex w on this shortest path p. If p is truly the shortest path, then the path p₁ from u to w and p₂ from w to v are indeed the shortest paths between the corresponding vertices (by the simple cut-and-paste argument described in CLRS). Hence, one can easily formulate the solution for finding shortest paths in a recursive manner, which is what the Bellman-Ford algorithm does.

Overlapping subproblems means that the space of subproblems must be small, that is, any recursive algorithm solving the problem should solve the same subproblems over and over, rather than generating new subproblems. For example, consider the recursive formulation for generating the Fibonacci series: F_i = F_i-1 + F_i-2, with base case F₁=F₂=1. Then F₄₃ = F₄₂ + F₄₁, and F₄₂ = F₄₁ + F₄₀. Now F₄₁ is being solved in the recursive subtrees of both F₄₃ as well as F₄₂. Even though the total number of subproblems is actually small (only 43 of them), we end up solving the same problems over and over if we adopt a naive recursive solution such as this. Dynamic programming takes account of this fact and solves each subproblem only once. Note that the subproblems must be only slightly smaller (typically taken to mean a constant additive factor) than the larger problem; when they are a multiplicative factor smaller the problem is no longer classified as dynamic programming.