what is Dynamic Programming?

simpler subproblems and solving each subproblem only once, storing the results to avoid redundant computations. It is applicable to optimization problems where the solution can be expressed as a combination of optimal solutions to subproblems.

Key characteristics of dynamic programming include:

Optimal Substructure: The problem can be divided into smaller, overlapping subproblems, and the optimal solution to the main problem can be constructed from the optimal solutions of its subproblems.

Memoization or Tabulation: Dynamic programming approaches typically use memoization (top-down) or tabulation (bottom-up) techniques to store the solutions to subproblems in a table or array. This prevents redundant computations by reusing the results of previously solved subproblems.

Dynamic programming can be applied to a wide range of problems, including:

Fibonacci Sequence: Computing Fibonacci numbers efficiently using memoization or tabulation.

Shortest Path Problems: Finding the shortest path in a graph using algorithms like Floyd-Warshall or Dijkstra's algorithm.

Longest Common Subsequence (LCS): Finding the longest subsequence that is common to two sequences, often used in bioinformatics and text comparison.

Knapsack Problem: Maximizing the value of items to be placed in a knapsack without exceeding its weight capacity.

Matrix Chain Multiplication: Optimally multiplying a chain of matrices to minimize the number of scalar multiplications.

Dynamic programming can significantly improve the efficiency of algorithms compared to naive recursive approaches by avoiding redundant com

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