Dynamic programming is a powerful technique used to solve complex problems by breaking them down into smaller, overlapping subproblems. It allows us to efficiently solve problems by storing and reusing computed results, rather than recomputing them.

In dynamic programming, we divide a problem into smaller subproblems and solve each subproblem only once. We then combine the results of the subproblems to solve the original problem.

Dynamic programming is particularly useful when a problem has overlapping subproblems, meaning that multiple subproblems share subproblems in common.

For example, let's say we want to calculate the nth Fibonacci number. We can use dynamic programming to store the results of calculating the Fibonacci numbers up to n, so that we can reuse them when calculating subsequent Fibonacci numbers.

Here's an example of calculating the nth Fibonacci number using dynamic programming in Java:

1public class Fibonacci {
2  public static int fib(int n) {
3    int[] memo = new int[n + 1];
4    return fibHelper(n, memo);
5  }
6
7  public static int fibHelper(int n, int[] memo) {
8    if (n <= 1) {
9      return n;
10    }
11
12    if (memo[n] != 0) {
13      return memo[n];
14    }
15
16    memo[n] = fibHelper(n - 1, memo) + fibHelper(n - 2, memo);
17    return memo[n];
18  }
19
20  public static void main(String[] args) {
21    int n = 10;
22    int result = fib(n);
23    System.out.println("The " + n + "th Fibonacci number is: " + result);
24  }
25}

In this example, we store the calculated Fibonacci numbers in the memo array to avoid redundant calculations. The fibHelper function checks if the result for the current fib(n) has already been computed and stored in the memo array. If it has, it returns the stored result. Otherwise, it calculates and stores the result before returning it.

Dynamic programming is a powerful technique that can significantly improve the efficiency of solving complex problems. By breaking down problems into smaller, overlapping subproblems and reusing computed results, we can achieve faster and more scalable solutions.