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