Overlapping Subproblems

When solving problems using dynamic programming, an important concept to understand is overlapping subproblems. Overlapping subproblems occur when the solution to a larger problem can be expressed in terms of solutions to smaller subproblems that are repeatedly solved.

To illustrate this concept, let's consider an example. Imagine that you are a basketball coach and you want to calculate the total number of points scored by a player in a given game. Instead of considering the entire game as one large problem, you can break it down into smaller subproblems by considering the points scored in each quarter.

By solving the subproblems of each quarter separately and combining their solutions, you can obtain the total number of points scored in the game. This approach allows you to avoid redundant calculations and improve the efficiency of your solution.

Dynamic programming takes advantage of overlapping subproblems by storing the solutions to the subproblems in a data structure, such as an array or a matrix. This allows us to avoid recomputing the solutions to the subproblems when they are encountered again.

The key idea is that if a problem can be divided into smaller subproblems, and the same subproblems are encountered multiple times, we can store the solutions to the subproblems and reuse them to solve the larger problem.

By recognizing and solving overlapping subproblems efficiently, dynamic programming enables us to solve complex problems in a more efficient and optimized manner.