# Introduction-to-algorithms — dynamic programmimng

Dynamic programming typically applies to optimization problems in which we make a set of choices in order to arrive at an optimal solution. As we make each choice, subproblems of the same form often arise. Dynamic programming is effective when a given subproblem may arise form more than one partial set of choices.

• The key technique is to store the solution to each such subproblem in case it should reappear.

Divide-and-conquer algorithms portion the problem into disjoint subproblems, solve the subproblems recursively, and then combine their solutions to solve the original problem — that is, when subproblems share subsubproblems.

• In this context, a divide-and-conquer algorithm does more work than necessary, repeatedly, solving the common subsubproblems.

A dynamic-programming algorithm solves each subsubproblem just once and then saves its answer in a table, thereby avoiding the work of recomputing the answer every time it solves each subsubproblem.

We typically apply dynamic programming to optimization problems. Such problems can have many possible solutions. Each solution has a value, and we wish to find a solution with the optimal (minimum or maximum) value. We call such a solution an optimal solution to the problem, as opposed to the optimal solution, since there may be several solutions that achieve the optimal value.

When developing a dynamic-programming algorithm, we follow a sequence of four steps:

1. Characterize the structure of an optimal solution.
2. Recursively define the value of an optimal solution.
3. Compute the value of an optimal solution, typically in a bottom-up fashion.
4. Construct an optimal solution from computed information.

Step 1-3 from the basis of a dynamic-programming solution to a problem.

• If we need only the value of an optimal solution, and not the solution itself, the we can omit step 4.
• When we do perform step 4, we sometimes maintain additional information during step 3 so that we can easily construct an optimal solution.

Two common operations are indexing and assigning to an index position. Bothe of these operations take the same amount of time no matter how large the list becomes. When an operation like this is independent of the size of the list they are $O(1)$.