Dynamic programming is a problem-solving technique used in computer science and mathematics to efficiently solve problems that can be broken down into smaller overlapping subproblems. It is especially useful for optimization problems and problems where the same subproblems are encountered multiple times. The key idea in dynamic programming is to solve each subproblem only once and store its solution to avoid redundant calculations, resulting in improved time and space complexity. Dynamic programming typically involves the following steps: 1. **Define the Problem:** Clearly state the problem you want to solve and break it down into smaller subproblems. Identify the relationships between these subproblems. 2. **Recurrence Relation:** Express the solution to a subproblem in terms of the solutions to smaller subproblems. This is often done through a recurrence relation or recursive formula. 3. **Memoization:** Store the solutions to subproblems in a data structure (usually an array or a dictionary) to avoid recomputation. This process is called memoization. 4. **Bottom-Up or Top-Down Approach:** Dynamic programming can be implemented in two main approaches: - **Bottom-Up (Iterative) Approach:** Start by solving the smallest subproblems and work your way up to the larger problem, filling a table or array as you go. - **Top-Down (Recursive) Approach:** Start with the original problem and recursively break it down into subproblems, using memoization to store and reuse solutions to subproblems. 5. **Optimal Solution:** Once you've solved all the subproblems, the final solution to the original problem can be found in the memoization table or through a recurrence relation. Dynamic programming is often used to solve problems related to optimization, such as finding the shortest path in a graph, the maximum value of a sequence, or the best way to allocate resources. Common examples of problems that can be solved using dynamic programming include: - **Fibonacci sequence calculation:** Finding the nth Fibonacci number. - **Longest common subsequence:** Finding the longest subsequence that two sequences have in common. - **Shortest path problems:** Finding the shortest path between two points in a graph (e.g., Dijkstra's algorithm, Bellman-Ford algorithm). - **Knapsack problem:** Determining the most valuable combination of items to include in a knapsack given weight constraints. - **Matrix chain multiplication:** Finding the most efficient way to multiply a chain of matrices. - **Edit distance:** Determining the minimum number of operations (insertions, deletions, substitutions) required to transform one string into another. Dynamic programming is a powerful technique for solving complex problems efficiently, but it does require careful analysis of the problem and an understanding of how to express its subproblems and their relationships. When applied correctly, dynamic programming can lead to significant improvements in algorithm performance and scalability.