Dynamic Programming

Dynamic programming is an algorithmic technique that breaks down a complex problem into simpler overlapping subproblems and solves them in an optimal way. It is particularly useful when the problem exhibits optimal substructure and overlapping subproblems.

Optimal Substructure

A problem has optimal substructure if an optimal solution to the problem contains optimal solutions to its subproblems. In other words, we can solve the problem by solving its subproblems and then using the optimal solutions of the subproblems to construct the optimal solution of the original problem.

Overlapping Subproblems

A problem has overlapping subproblems if it can be broken down into subproblems that are reused multiple times. In dynamic programming, we avoid redundant recomputation of subproblems by storing the results of solved subproblems and reusing them when needed.

Let's understand dynamic programming through an example. Consider the calculation of Fibonacci numbers.

The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones. It starts with 0 and 1. The first few numbers in the Fibonacci sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on.

Here's a C++ code snippet that calculates the Fibonacci number at a given position using dynamic programming:

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In the above code, we define a recursive function fibonacci that takes an integer n as input and calculates the Fibonacci number at position n. We use dynamic programming to avoid redundant computation of the same subproblems. The function returns the Fibonacci number at position n.

Let's say we want to calculate the Fibonacci number at position 6. The code calls the fibonacci function with input n equal to 6. The function calculates the Fibonacci number by recursively calling itself with n - 1 and n - 2, and using the stored results of the subproblems to construct the final result.

Dynamic programming is a powerful technique that can significantly improve the efficiency of solving complex problems. It is used in various algorithms and applications, such as solving optimization problems, finding shortest paths, and more.

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#include <iostream>

using namespace std;

​

int fibonacci(int n) {

  if (n <= 1) {

    return n;

  }

  return fibonacci(n - 1) + fibonacci(n - 2);

}

​

int main() {

  int n = 6;

  cout << "The Fibonacci number at position " << n << " is " << fibonacci(n) << endl;

  return 0;

}