$\text{Dynamic Programming}$

$\text{Overview}$

$\text{The DP approach}$

Dynamic programming is most suited for optimization problems: this is clear from the type of problems in the list above, which all involve finding a solution that is optimal in some sense, most often the maximum or minimum of some function. Such problems should have two main properties in order to justify the use of dynamic programming:

1. $\text{Optimal substructure}$
   The optimal solution to a problem consists of optimal solutions to its subproblems. Mind that the subproblems of the same problem should be $\mathit{\text{independent}}$ of each other, i.e. the optimal solution to one of its subproblems should not depend on the optimal solution to another of its subproblems.
2. $\text{Overlapping subproblems}$
   Subproblems share subsubproblems, and the same subsubproblems are encountered multiple times when solving different subproblems. Dynamic programming takes advantage of this property by solving each part of the problem only once and then storing the solution in a table where it can be looked up when needed. In other words, the slogan of dynamic programming is a resounding $\text{DRY}$: Don't Repeat Yourself, and compute each problem only once!

A good test for the second property is to see if the recursive solution has repeated calls for the same inputs. Note that this is different from the type of subproblems that divide-and-conquer algorithms solve, where recursive calls typically generate new (but of the same type as the original) subproblems each time. A divide-and-conquer approach for a problem whose subproblems overlap would be inefficient, since it would solve the same subproblem each time it reappears in the recursion tree.