Top-Down Approach

The Top-Down Approach is another technique used in dynamic programming to solve problems by breaking them down into smaller subproblems. In contrast to the Bottom-Up Approach, which starts with the smallest subproblems and builds up to the larger problem, the Top-Down Approach starts with the larger problem and breaks it down into smaller subproblems.

The Top-Down Approach is often implemented using recursion and memoization. Recursion allows us to express the problem in terms of smaller subproblems, while memoization allows us to store the solutions to the subproblems to avoid redundant computations.

For example, let's consider a classic problem: finding the n-th Fibonacci number. The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding numbers. By using the Top-Down Approach, we can express the problem in terms of smaller subproblems, such as finding the (n-1)-th and (n-2)-th Fibonacci numbers, and recursively combine their solutions to find the n-th Fibonacci number.

Here's an example of a Java code implementation of the Top-Down Approach for calculating the Fibonacci sequence:

1<<code>>

In this example, the fibonacci method takes an integer n as input and recursively calculates the n-th Fibonacci number. We first check if n is less than or equal to 1, in which case we return n as the base case. Otherwise, we recursively call fibonacci(n - 1) and fibonacci(n - 2) to find the (n-1)-th and (n-2)-th Fibonacci numbers, and return their sum as the solution to the n-th Fibonacci number.

The Top-Down Approach is particularly useful when solving problems that can be expressed in a recursive manner and have overlapping subproblems. By breaking down the problem into smaller subproblems, it allows us to solve them in a more manageable and efficient way.

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class Main {

  public static void main(String[] args) {

    int n = 10;

    int result = fibonacci(n);

    System.out.println("The Fibonacci number at position " + n + " is: " + result);

  }

​

  private static int fibonacci(int n) {

    if (n <= 1) {

      return n;

    }

​

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

  }

}