Dynamic programming is a method for efficiently solving optimization problems by breaking them down into smaller subproblems and storing their solutions. It is used in many areas, such as computer science, operations research, and financial mathematics. The goal of this glossary entry is to explain dynamic programming in more detail, to demonstrate its advantages and applications, and to describe the various techniques.

## Dynamic Programming: Definition and Basics

**Dynamic programming** is an approach for efficiently solving optimization problems, which is based on decomposing the problem into smaller, more easily solvable subproblems. The solutions to these subproblems are stored and reused when needed to solve larger problems, in order to save time and resources. Dynamic programming can be used in many different areas, such as computer science, operations research, artificial intelligence, financial mathematics, and more.

## Principles of Dynamic Programming

Dynamic programming is based on two fundamental principles:

- Â
**Overlap of subproblems:**The original optimization problem can be divided into smaller subproblems, which occur repeatedly in the solution of the overall problem. The solutions to these subproblems can be stored in a table and reused when needed to reduce computation time. Â **Optimal substructure:**The optimal solution to the overall problem can be constructed from the optimal solutions of the subproblems. This means that the optimal solution to a problem depends on the optimal solution to its subproblems.

## Advantages of Dynamic Programming

Dynamic programming offers several advantages over other optimization methods:

- Â
**Efficiency:**By storing and reusing the solutions of subproblems, computation time can be significantly reduced, especially for problems with many overlapping subproblems. Â **Simplicity:**Many optimization problems can be intuitively broken down into smaller subproblems, making the approach easily understandable. Â **Flexibility:**Dynamic programming can be applied to a wide variety of problems, from simple recursive functions to complex multidimensional problems.

## Techniques of Dynamic Programming

There are two basic techniques for applying dynamic programming to an optimization problem:

- Â
**Top-Down Approach:**This approach begins with the original problem and divides it into smaller subproblems, which are solved recursively. The solutions to the subproblems are stored in a memory (e.g., an array or a hash table) for reuse when needed. This approach is also known as "memoization." Â **Bottom-Up Approach:**In this approach, the subproblems are solved in ascending order of size, starting with the smallest subproblems. The solutions are stored in a table, and the optimal solution to the overall problem is achieved by combining the solutions of the subproblems. This approach often requires less memory space than the top-down approach and is usually easier to implement.

## Applications of Dynamic Programming

Dynamic programming can be applied to a variety of optimization problems, including:

- Â
**Sequence alignment:**In bioinformatics, dynamic programming is used to determine the optimal alignment of sequences, such as DNA or protein sequences. Â **Travelling Salesman Problem:**In this well-known combinatorial optimization problem, the goal is to find the shortest route for a salesman who must visit a certain number of cities and return to the starting point. Dynamic programming can be used to solve this problem. Â **Resource allocation:**Dynamic programming can be used to optimize the allocation of resources, such as investments in projects or planning of production processes. Â **Graph theory:**In graph theory, dynamic programming can be used to solve problems such as shortest paths, maximum flows, and minimum spanning trees. Â **Game theory:**Dynamic programming can be used to analyze games with complete information to find optimal strategies for the players.

## Conclusion

Dynamic programming is a powerful method for efficiently solving optimization problems. By breaking down problems into smaller subproblems and storing the solutions to these subproblems, time and resources can be saved. Dynamic programming is applied in many different areas, such as computer science, operations research, artificial intelligence, and financial mathematics.