Dynamic programming is a problem-solving technique that involves breaking down a complex problem into smaller overlapping subproblems and solving each subproblem only once. It is particularly useful when a problem has overlapping subproblems and exhibits the property of optimal substructure.
The main idea behind dynamic programming is to store the solutions to the subproblems in a table or memoization array, so that when a subproblem is encountered again, its solution can be looked up rather than recomputed. This approach reduces the time complexity of the algorithm by avoiding redundant computations.
Dynamic programming typically involves the following steps:
- Characterize the structure of an optimal solution: Identify the elements that make up the problem and define the relationships between them.
- Define the value of an optimal solution recursively: Express the value of the problem in terms of the values of its smaller subproblems.
- Compute the value of an optimal solution in a bottom-up fashion: Solve the subproblems in a specific order, typically from the smallest subproblems up to the original problem, using the memoization table to avoid redundant computations.
- Construct an optimal solution from the computed information: Based on the memoization table or the computed values, trace back and build the final solution.
Dynamic programming is often used to solve optimization problems, such as finding the shortest path in a graph, finding the maximum value in a sequence, or solving knapsack problems. It is also applied in various fields, including computer science, operations research, economics, and bioinformatics, among others.
Some classic examples of dynamic programming algorithms are the Fibonacci sequence, the Knapsack problem, the Longest Common Subsequence problem, and Dijkstra\’s algorithm for finding the shortest path in a graph.