Algorithm Design Questions Medium
Dynamic programming is a problem-solving technique that involves breaking down a complex problem into smaller overlapping subproblems and solving them in a bottom-up manner. It is particularly useful for solving optimization problems where the goal is to find the best solution among a set of possible solutions.
The concept of dynamic programming is based on the principle of optimal substructure, which states that an optimal solution to a problem can be constructed from optimal solutions to its subproblems. By solving and storing the solutions to these subproblems, dynamic programming avoids redundant computations and improves efficiency.
In the context of optimization problems, dynamic programming involves defining a recursive relationship between the optimal solution of the original problem and the optimal solutions of its subproblems. This relationship is often represented using a recurrence relation or a recurrence equation.
To solve an optimization problem using dynamic programming, the following steps are typically followed:
1. Characterize the structure of the problem: Identify the subproblems and their relationships. Determine the parameters that define the subproblems and the objective function that needs to be optimized.
2. Define the recurrence relation: Express the optimal solution of the original problem in terms of the optimal solutions of its subproblems. This relation should be based on the principle of optimal substructure.
3. Formulate the base cases: Identify the simplest subproblems that can be solved directly without further decomposition. These base cases serve as the starting point for the dynamic programming algorithm.
4. Design the dynamic programming algorithm: Use the recurrence relation and the base cases to construct a bottom-up algorithm that solves the subproblems in a systematic manner. This algorithm typically involves filling up a table or an array to store the solutions to the subproblems.
5. Compute the optimal solution: Once the dynamic programming algorithm has been implemented, the optimal solution to the original problem can be obtained by combining the solutions of the subproblems according to the recurrence relation.
By using dynamic programming, optimization problems can be solved efficiently by avoiding redundant computations and leveraging the optimal substructure property. This technique has applications in various fields such as computer science, operations research, economics, and engineering, where finding the best solution among a set of possibilities is crucial.