What is the difference between top-down and bottom-up approaches in Dynamic Programming?

In dynamic programming, both top-down and bottom-up approaches are used to solve problems by breaking them down into smaller subproblems. The main difference between these approaches lies in the order in which the subproblems are solved.

1. Top-down approach (also known as memoization or recursive approach):
In the top-down approach, the problem is divided into smaller subproblems, and the solution to each subproblem is stored in a memoization table or cache. Initially, the solution to the main problem is not known, so the algorithm recursively solves the subproblems, starting from the top (main problem) and moving towards the base cases. Whenever a subproblem is encountered, the algorithm checks if its solution is already present in the memoization table. If so, it retrieves the solution from the table; otherwise, it computes the solution and stores it in the table for future use. This approach avoids redundant computations by reusing the solutions of previously solved subproblems.

2. Bottom-up approach (also known as tabulation or iterative approach):
In the bottom-up approach, the problem is solved by iteratively solving the subproblems in a bottom-up manner, starting from the base cases and gradually building up to the main problem. The solutions to the subproblems are stored in a table or array. The algorithm starts with the base cases and computes the solutions for all subproblems in a systematic order, typically in a loop. It uses the solutions of previously solved subproblems to solve the current subproblem. This approach avoids recursion and relies on iteration to solve the problem.

In summary, the top-down approach starts from the main problem and recursively solves smaller subproblems, storing their solutions in a memoization table. On the other hand, the bottom-up approach starts from the base cases and iteratively solves subproblems, storing their solutions in a table. Both approaches aim to solve the problem by breaking it down into smaller subproblems, but they differ in the order in which the subproblems are solved.