Dynamic Programming

Dynamic Programming is a powerful algorithmic technique that breaks down complex problems into simpler overlapping subproblems, solves each subproblem only once, and stores the intermediate results in a table or an array to avoid redundant computations.

In other words, dynamic programming is a way to solve optimization problems by efficiently solving smaller instances of the problem and using the solutions to build up the final solution.

To understand how dynamic programming works, let's take a look at a classic example: the Fibonacci sequence.

The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones. Here's a simple implementation of the Fibonacci sequence using dynamic programming in C++:

1#include <iostream>
2#include <vector>
3
4using namespace std;
5
6int fibonacci(int n) {
7  vector<int> fib(n + 1);
8  fib[0] = 0;
9  fib[1] = 1;
10
11  for (int i = 2; i <= n; i++) {
12    fib[i] = fib[i - 1] + fib[i - 2];
13  }
14
15  return fib[n];
16}
17
18int main() {
19  int n = 10;
20  int result = fibonacci(n);
21  cout << "The fibonacci sequence at position " << n << " is " << result << endl;
22
23  return 0;
24}

In this example, we use dynamic programming to efficiently compute the Fibonacci sequence at a given position n. We build up the sequence by calculating the sum of the two previous numbers and storing the results in a vector. By using dynamic programming, we avoid redundant calculations and improve the efficiency of the algorithm.

Dynamic programming is not limited to solving the Fibonacci sequence. It can be used to solve a wide range of problems that exhibit overlapping subproblems and optimal substructure. Some other famous algorithms that use dynamic programming include the Knapsack problem, the Longest Common Subsequence problem, and the Edit Distance problem.

By understanding dynamic programming and its applications, you can develop efficient algorithms for solving complex optimization problems. This technique is particularly useful in scenarios where brute-force solutions would be impractical or too slow.

Next, we will explore another important algorithmic technique: Greedy Algorithms.

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#include <iostream>

#include <vector>

​

using namespace std;

​

int fibonacci(int n) {

  vector<int> fib(n + 1);

  fib[0] = 0;

  fib[1] = 1;

​

  for (int i = 2; i <= n; i++) {

    fib[i] = fib[i - 1] + fib[i - 2];

  }

​

  return fib[n];

}

​

int main() {

  int n = 10;

  int result = fibonacci(n);

  cout << "The fibonacci sequence at position " << n << " is " << result << endl;

​

  return 0;

}