Problem Set 5: Problem 2

1
Problem Set 5 Problem 2

• 22C021 Computer Science
• Data Structures

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

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Initialize the answers array

• // Initializing answers
• int number ltfibonacci number to generategt
• answers new BigIntegernumber
• // The first two numbers in series are 1
• answers0 new BigInteger("1")
• answers1 new BigInteger("1")
• // Set all others to zeros to mark them as
• // not-computed
• for(int i 2 i lt number i)
• answersi new BigInteger("0")

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Optimized Fibonacci Number Generator

• public static BigInteger fastFibonacci(int n)
• if((n 1) (n 2))
• return answers0
• if(answersn-1.compareTo(zero) ! 0)
• return answersn-1
• if(answersn-2.compareTo(zero) 0)
• answersn-2 fastFibonacci(n-1)
• if(answersn-3.compareTo(zero) 0)
• answersn-3 fastFibonacci(n-2)
• return answersn-2.add(answersn-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

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Performance Comparison
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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 ...