Java Data Structures - Fibonacci sequence

Dynamic programming approach is similar to divide and conquer in breaking down the problem into smaller and yet smaller possible sub-problems. But unlike, divide and conquer, these sub-problems are not solved independently. Rather, results of these smaller sub-problems are remembered and used for similar or overlapping sub-problems.

Dynamic programming is used where we have problems, which can be divided into similar sub-problems, so that their results can be re-used. Mostly, these algorithms are used for optimization. Before solving the in-hand sub-problem, dynamic algorithm will try to examine the results of the previously solved sub-problems. The solutions of sub-problems are combined in order to achieve the best solution.

So we can say that −

• The problem should be able to be divided into smaller overlapping sub-problem.

• An optimum solution can be achieved by using an optimum solution of smaller sub-problems.

• Dynamic algorithms use memorization.

Comparison

In contrast to greedy algorithms, where local optimization is addressed, dynamic algorithms are motivated for an overall optimization of the problem.

In contrast to divide and conquer algorithms, where solutions are combined to achieve an overall solution, dynamic algorithms use the output of a smaller sub-problem and then try to optimize a bigger sub-problem. Dynamic algorithms use memorization to remember the output of already solved sub-problems.

Dynamic programming can be used in both top-down and bottom-up manner. And of course, most of the times, referring to the previous solution output is cheaper than re-computing in terms of CPU cycles.

The Fibonacci sequence in Java

Following is the solution of the Fibonacci sequence in Java using dynamic programming technique.

Example

import java.util.Scanner;

public class Fibonacci {
   public static int fibonacci(int num) {
      int fib[] = new int[num + 1];
      fib[0] = 0;
      fib[1] = 1;
      
      for (int i = 2; i < num + 1; i++) {
         fib[i] = fib[i - 1] + fib[i - 2];
      }
      return fib[num];
   }
   public static void main(String[] args) {
	   Scanner sc = new Scanner(System.in);
      System.out.println("Enter a number :");
      int num = sc.nextInt();
      
      for (int i = 1; i <= num; i++) {
         System.out.print(" "+fibonacci(i));
      }
   }
}

Output

1 1 2 3 5 8 13 21 34 55 89 144