Introduction
 The dynamic Programming method is a famous algorithmic principle used widely in several applications.
Definition
 Dynamic programming is a technique for solving complex problems by breaking them down into smaller & simpler subproblems and solving each subproblem only once, saving the results in a table and reusing them when needed.
Characteristics
 Dynamic programming is a useful algorithmic technique that is often used to solve optimization problems.
 dynamic programming is a powerful and flexible technique that can be used to solve a wide range of optimization problems, providing optimal solutions in a computationally efficient manner.
 This approach can behave as an efficient algorithm for problems that would be infeasible to solve by brute force.
 This algorithm is used to solve complex problems efficiently, often with significantly reduced time complexity compared to brute force methods.
Working Principle/Mechanism
 The steps for solving a problem using dynamic programming are:


Identify the problem: In this, the problem can be broken down into smaller subproblems, and the solutions of the subproblems can be combined to solve the larger problem.

Define the subproblems and the relationship between them: In this step, we can identify the variables and parameters that will be used to define the subproblems and the way in which they are related to each other.

Solve the subproblems: This means finding the optimal solution to each subproblem using a recursive or iterative algorithm.

Store the solutions: The solution of the subproblems are stored in a table. This is known as memoization/tabulation.

Use the solutions: Use the solutions of the subproblems to solve the larger problem. In other words, by combining the solutions of the subproblems to find the optimal solution to the original problem.

Advantage

One of the main advantages of dynamic programming is that it can significantly reduce the computational time required to solve complex problems by breaking them down into smaller subproblems.

Dynamic programming guarantees that the solution obtained is the optimal one, unlike other methods that may only provide approximate solutions.

Dynamic programming can trade off space complexity for time complexity, allowing for efficient use of memory and computation resources.

Dynamic programming can be applied to a wide range of problems in various fields, such as computer science, operations research, economics, and engineering.

Dynamic programming breaks down complex problems into smaller subproblems, making it easier to solve them. This approach is particularly useful when dealing with problems that have a recursive structure.

Dynamic programming uses memoization, which is a technique that stores the results of previous computations to avoid repetitive calculations. This can significantly speed up the computation time.
Disadvantage
 Here are some of the disadvantages of dynamic programming:


High memory usage: Dynamic programming algorithms often require a large amount of memory to store intermediate results. This can be a problem when dealing with large problems, as it can lead to high memory usage and slow down the computation.

Difficulty in identifying subproblems: Identifying the appropriate subproblems to solve can be a challenging task, especially for complex problems with many possible subproblems. It requires a deep understanding of the problem structure and may involve trial and error.

Time complexity: Dynamic programming algorithms can have a high time complexity, especially when dealing with problems that have a large number of subproblems or a complex recursive structure. In some cases, it may be necessary to use heuristics or approximation techniques to reduce the computation time.

Not always applicable: Dynamic programming is not always applicable to all types of problems. Some problems may not have the necessary properties, such as optimal substructure or overlapping subproblems, required for dynamic programming to be effective.

Example
 Some common dynamic programming algorithms are the Fibonacci sequence, the Knapsack problem, and the Longest Common Subsequence problem.
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