Dynamic Programming

Dynamic Programming (DP) is an algorithmic technique used to solve optimization problems by breaking down the problem into smaller subproblems and solving them in a way that optimizes the overall solution. The key idea behind DP is to store the solutions to subproblems and use them to solve larger subproblems.

Dynamic Programming is a method of solving problems by breaking them down into smaller, simpler subproblems and solving them in an optimal way. It involves using previously solved subproblems to build up the solution to the larger problem. This technique is used in many fields, including computer science, mathematics, and engineering, and can significantly reduce the computation time and space complexity of an algorithm.

Figure: DP (by Mehmet Akif CIFCI)

In DP, the problem is divided into overlapping subproblems, and the solution to each subproblem is stored in a table. The table can then solve larger subproblems until the entire problem is solved. DP is usually used when the subproblems are dependent on each other and the solution to the larger problem can be constructed from the solutions of smaller subproblems.

One of the most famous examples of the DP algorithm is the Fibonacci sequence. The Fibonacci sequence is defined as follows:

def fib(n):

    if n == 0:
        return 0
    elif n == 1:
        return 1

    dp = [0] * (n+1)
    dp[0] = 0
    dp[1] = 1

    for i in range(2, n+1):
        dp[i] = dp[i-1] + dp[i-2]
    return dp[n]

In this implementation, the fib function takes an integer n as input and returns the nth Fibonacci number using dynamic programming. The function first initializes the base cases for n=0 and n=1. It then creates a DP table dp of size n+1 to store the solutions to subproblems. The DP table is initialized with the base cases.

The function then uses a loop to fill in the DP table by solving subproblems. Each entry in the DP table is the sum of the two previous entries, which are already stored in the table. Finally, the function returns the solution by returning the nth entry in the DP table.

Note that this is just one example of how to implement dynamic programming in Python, and there are many variations and optimizations that can be made depending on the specific problem being solved.

To find the nth Fibonacci number, we can use the DP algorithm to calculate the sequence incrementally. We start with the base cases F(0) and F(1), and use them to calculate F(2), F(3), and so on until we reach the nth Fibonacci number.

DP is commonly used in many other optimization problems, such as shortest path, knapsack problem, and sequence alignment. It can significantly reduce the computation time and space complexity of an algorithm.

In summary, Dynamic Programming is an algorithmic technique that is used to solve optimization problems by breaking them down into smaller subproblems and solving them in a way that optimizes the overall solution.