## Table of contents

## Introduction

Dynamic Programming (DP) is a popular algorithmic technique used in solving optimization problems. It has wide applications in various fields such as computer science, operations research, economics, and engineering. In this article, we will provide a comprehensive overview of DP, its concepts, and how it is used in solving problems.

## What is Dynamic Programming (DP)?

It is a technique used in solving optimization problems by breaking them down into smaller subproblems, solving each subproblem only once, and then storing the results for future use. The idea is to reduce the time complexity of a problem by solving each subproblem only once, rather than solving the same subproblem repeatedly.

## Basic Concepts of Dynamic Programming

### Overlapping Subproblems

It is based on the concept of overlapping subproblems. It is assumed that the subproblems encountered in an optimization problem are not independent, but rather they share substructures. As a result, a subproblem can be solved once and the solution reused whenever that subproblem is encountered again.

### Optimal Substructure

It relies on optimal substructure, which means that the optimal solution to a problem can be constructed from the optimal solutions to its subproblems. Therefore, the optimal solution of a problem can be obtained by solving subproblems recursively and combining their solutions.

## Steps Involved in Dynamic Programming

The following are the steps involved in solving an optimization problem using DP:

- Define the problem and identify the subproblems.
- Formulate a recursive relation for the problem.
- Create a memoization table and initialize it.
- Fill in the memoization table by solving subproblems recursively.

## Applications of Dynamic Programming

It has wide applications in various fields such as computer science, operations research, economics, and engineering. Some of the common applications include:

- Shortest path algorithms
- Knapsack problem
- Longest common subsequence problem
- Sequence alignment problem
- Coin change problem

## Advantages of Dynamic Programming

It has the following advantages:

- It reduces the time complexity of an optimization problem.
- It solves each subproblem only once, saving computational resources.
- It is a simple and easy-to-understand technique.

## Disadvantages of Dynamic Programming

It has the following disadvantages:

- It requires a significant amount of memory to store the results of the subproblems.
- It may not be suitable for problems with a large number of subproblems.

## Conclusion

In conclusion, dynamic programming is a powerful algorithmic technique used in solving optimization problems. Its applications are wide and varied, and it has numerous advantages and disadvantages. Understanding the basic concepts and steps involved in solving complex optimization problems.

## FAQs

- What is the difference between DP and recursion? DP is a technique that uses recursion to solve problems by breaking them down into smaller subproblems and solving each subproblem only once. Recursion, on the other hand, is a method of solving a problem where the solution is based on solving smaller instances of the same problem.
- Is DP suitable for all types of optimization problems? No, DP may not be suitable for problems with a large number of subproblems, as it requires a significant amount of memory to store the results of the subproblems.
- What is memoization? Memoization is a technique used in dynamic programming where the results of subproblems are stored in a table to avoid redundant calculations.
- What are the advantages of dynamic programming? Dynamic programming reduces the time complexity of an optimization problem, solves each subproblem only once, and is a simple and easy-to-understand technique.