1. 22C:021 Computer Science Data Structures Problem Set 5: Problem 2

2. What needs to be done? • Implement a Fibonacci Number Generator • A naive Fibonacci number generator • Implement an optimized Fibonacci number generator • That remembers results of sub-problems already solved • Evaluate Performance

3. Fibonacci Number Generator: Naive public static int fibonacci(int n) { if((n == 1) || (n == 2)) return 1; else return fibonacci (n-1) + fibonacci (n-2); }

4. Fibonacci Number Generator: Optimized • We need: • An array “answers” that stores results of solved sub-problems • A type that can handle large numbers • Fibonacci numbers grow fast, integers and longs run out of range • A way to check if a sub-problem has already been solved • Only need to recurse when necessary

5. The BigInteger Data Type • Available under java.util namespace • Can store really large values • Check Java Documentation for more Details about this type • http://java.sun.com/j2se/1.4.2/docs/api/java/math/BigInteger.html

6. The “answers” array • What should be the size of the “answers” array • The size n of this array should be equal to the Fibonacci number we've been asked to generate • This array stores the nth Fibonacci number at n-1th Location private static BigInteger[] answers;

7. Initialize the “answers” array // Initializing answers int number = <fibonacci number to generate>; answers = new BigInteger[number]; // The first two numbers in series are 1 answers[0] = new BigInteger("1"); answers[1] = new BigInteger("1"); // Set all others to zeros to mark them as // not-computed for(int i = 2; i < number; i++) answers[i] = new BigInteger("0");

8. Optimized Fibonacci Number Generator public static BigInteger fastFibonacci(int n) { if((n == 1) || (n == 2)) return answers[0]; if(answers[n-1].compareTo(zero) != 0) return answers[n-1]; if(answers[n-2].compareTo(zero) == 0) answers[n-2] = fastFibonacci(n-1); if(answers[n-3].compareTo(zero) == 0) answers[n-3] = fastFibonacci(n-2); return answers[n-2].add(answers[n-3]); }

9. Optimized Fibonacci Number Generator • When asked to generate a number • We check if the number requested has already been computed and return it if it has been. • Then, we check if the required numbers have already been computed (n – 1 and n - 2) • If they have, we straightaway use their values • If they haven't, we call the generator number on each of the missing required numbers • Once we have both the value, we add them and return the sum

10. Performance Comparisons • How fast is the optimized version? • To generate the 46th Fibonacci Number • The unoptimized version takes 885490.7 ms = Approx 15 minutes • The optimized version takes 0.145 ms

11. Performance Comparison

12. Other Performance Stats • The Optimized version takes • For 100th Number: 0.329 ms • For 1000th Number: 5.172 ms • The Naive version takes • So long that I did not evaluate ...