What is memoization and how does it relate to dynamic programming?

Memoization is a technique used in dynamic programming to optimize the time complexity of a problem by storing the results of expensive function calls and reusing them when the same inputs occur again. Memoization is a form of caching that speeds up the computations by avoiding redundant computations and reusing previously computed results.

The key steps involved in memoization are:

1. Check if the result is already computed: Before computing the result, check if the result has already been computed and stored in a cache.

2. Compute the result if not already computed: If the result is not already computed, compute it and store it in the cache.

3. Return the result: Return the computed result.

Memoization is often used in recursive algorithms where the same inputs are used repeatedly. In recursive algorithms, the same function may be called with the same arguments multiple times, resulting in redundant computations. Memoization can be used to store the results of previous function calls and to avoid the redundant computations.

The use of memoization in dynamic programming is closely related to the concept of optimal substructure. Optimal substructure is a property of some problems where the optimal solution to the problem can be obtained by combining the optimal solutions to its subproblems. Dynamic programming involves breaking down a problem into smaller subproblems and solving each subproblem only once. By storing the solutions to the subproblems in a cache, memoization can be used to avoid redundant computations and speed up the computations.

Memoization is particularly useful in problems where the time complexity is exponential or factorial, such as the Fibonacci sequence or the binomial coefficient. In such problems, memoization can be used to reduce the time complexity from exponential or factorial to polynomial time.

In summary, memoization is a technique used in dynamic programming to optimize the time complexity of a problem by storing the results of expensive function calls and reusing them when the same inputs occur again. Memoization is a form of caching that speeds up the computations by avoiding redundant computations and reusing previously computed results. Memoization is closely related to the concept of optimal substructure and is particularly useful in problems where the time complexity is exponential or factorial.