Memoization

In the context of Dynamic Programming, memoization is a technique used to optimize the solutions to subproblems by storing their results so that we don't need to recompute them.

When solving problems using dynamic programming, we often encounter cases where the same subproblem is solved multiple times. This can result in redundant computations and can lead to inefficient solutions.

To address this issue, we can use memoization to store the results of solving subproblems and reuse these results when needed.

For example, let's consider the problem of computing the Fibonacci numbers. The Fibonacci sequence is defined as follows:

1F(0) = 0
2F(1) = 1
3F(n) = F(n-1) + F(n-2) for n > 1

The naive recursive approach to compute the Fibonacci numbers has exponential time complexity since it solves the same subproblems multiple times. However, by using memoization, we can avoid redundant computations and solve the problem efficiently.

Here's an implementation of the Fibonacci function using memoization in Java:

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

In this implementation, we use an array memo to store the results of solving subproblems. Before solving a subproblem, we check if its result is already stored in memo and return it if it exists. Otherwise, we compute the result and store it in memo for future use.

By using memoization, we can significantly improve the performance of our dynamic programming solutions by avoiding redundant computations and reusing previously computed results.

xxxxxxxxxx

class Main {

  public static void main(String[] args) {

    // replace with your Java logic here

    int result = fibonacci(6);

    System.out.println("The 6th Fibonacci number is: " + result);

  }

​

  private static int fibonacci(int n) {

    int[] memo = new int[n + 1];

    return fibonacciHelper(n, memo);

  }

​

  private static int fibonacciHelper(int n, int[] memo) {

    if (n <= 1) {

      return n;

    }

​

    if (memo[n] != 0) {

      return memo[n];

    }

​

    memo[n] = fibonacciHelper(n - 1, memo) + fibonacciHelper(n - 2, memo);

    return memo[n];

  }

}