Dynamic Programming (DP) is an algorithmic technique for solving complex problems by breaking them down into simpler subproblems and storing the results to avoid redundant calculations. It is an optimization approach that solves problems exhibiting the properties of overlapping subproblems and optimal substructure.
History:
The term "Dynamic Programming" was coined by American mathematician Richard Bellman in the 1950s. Bellman developed this technique while working on mathematical problems at the RAND Corporation. Despite its name, DP is not related to computer programming but rather refers to the dynamic nature of the problem-solving approach.- Overlapping Subproblems: DP is applicable when a problem can be broken down into smaller subproblems, and these subproblems are reused multiple times. Instead of solving the same subproblems repeatedly, DP solves each subproblem only once and stores the result for future reference.
- Optimal Substructure: A problem has optimal substructure if its optimal solution can be constructed from the optimal solutions of its subproblems. This property allows DP to make decisions at each step based on the solutions of smaller subproblems, leading to an optimal overall solution.
How it Works:
DP solves problems by combining the solutions of subproblems. It typically involves the following steps:- Characterize the structure of an optimal solution: Identify how the problem can be broken down into subproblems and express the solution in terms of the solutions to smaller subproblems.
- Recursively define the value of an optimal solution: Write a recursive function or formula that expresses the solution to the problem in terms of solutions to smaller subproblems.
- Compute the value of an optimal solution in a bottom-up manner: Start with the base cases and iteratively compute the solutions to larger subproblems using the recursive formula until the desired solution is obtained.
- Construct an optimal solution from the computed information: Use the computed values to construct the optimal solution to the original problem.
DP can be implemented using two main approaches:
- Memoization (Top-Down): Start with the original problem and recursively solve subproblems. Store the results of subproblems in a memoization table to avoid redundant calculations.
- Tabulation (Bottom-Up): Solve subproblems iteratively in a bottom-up manner, filling up a table with solutions to subproblems. The final solution is obtained from the table.
- Fibonacci sequence calculation
- Knapsack problem
- Longest common subsequence
- Shortest path in a graph
- Matrix chain multiplication
DP has wide applications in various fields, including computer science, mathematics, economics, and bioinformatics. It is particularly useful in optimization problems where a large problem can be decomposed into smaller subproblems, and the solutions to these subproblems can be combined to solve the original problem efficiently.
By leveraging the principles of overlapping subproblems and optimal substructure, DP avoids redundant calculations and provides an efficient way to solve complex problems. However, it requires careful problem formulation and identification of the appropriate subproblems to apply the technique effectively.