Introduction: Efficient Algorithms for the Problem of Computing Fibonocci Numbers

1. Introduction: Efficient Algorithms for the Problem of Computing Fibonocci Numbers Analysis of Algorithms Prepared by John Reif, Ph.D.

2. Readings • Main Reading Selection: • CLR, Chapter 1

3. Goal • Devise Algorithms to solve problems on a TI calculator • assume each step(a multiplication, addition, or control branch) takes 1 m sec = 10-6 sec on 16 bit words

4. Time Efficiency

5. Fibonocci Sequence • 0, 1, 1, 2, 3, 5, 8, 13… • Recursive Definition • Fn = if n < 1 then n else Fn-1 + Fn-2 • Problem • Compute Fn for n=109 fast.

6. Fibonacci Growth • Can show as n  ∞ • Golden ratio • So Fn ~ .45 ∙ 2 .7n is .7n bit number grows exponentially!

7. Obvious Method n if n < 1 • Fn = > .01 n2 steps > 1016m sec for n = 109 = 1010 sec ~ 317 years! Fn-1 + Fn-2 if n > 1

8. Wanted! • An efficient algorithm for Fn • Weapons: • Special properties of computational problems = combinatorics (in this case)

9. Theorem: • Proof by Induction • Basis Step • Holds by definition for n=1 • Inductive Step • Assume holds for some n>0

10. Inductive Step

11. Powering Trick 1 1 1 0 • Fix M = • To compute Mn when n is a power of 2 For i=1 to log n do

12. General Case of Powering • Decompose n = 2 j1 + 2 j2 + ... + 2 jkas sum of powers of 2 • Example • Compute

13. Algorithm

14. Bounding Mults • = 2 log n matrix products on symmetric matrices of size 2 x 2 • Each matrix product costs 6 integer mults • Total Cost = 12 (log n) integer mults > 360 for n = 109 integer mults

15. Time Analysis • Fn ~ .45 2 .7n • So Fn is m = .7n bit integer • New method to compute Fn requires multiplying m bit number • BUT • Grammar School Method for Mult takes m2bool ops = m2/16 steps ~ 1016 steps for n = 109 ~ 1016msec ~ 1010 seconds ~ 317 years!

16. Fast Multiplication of m bit integers a,b by • “Divide and Conquer” • seems to take 4 multiplications, but…

17. 3 Mult Trick

18. Recursive Mult Algorithm • Cost

19. Schonhage-Strassen Integer Multiplication Algorithm • This is feasible to compute!

20. Improved Algorithm • Computed Fn for n = 109using 360 = 12 log n integer mults each taking ~ 3 days • Total time to compute Fn • ~ 3 years with 1 msec/step

21. How Did We Get Even MoreSpeed-Up? • New trick • a 2 log n mult algorithm (rather than n adds) • Old Trick • use known efficient algorithms for integer multiplication(rather than Grammar School mult) • Also used careful analysis of efficiency of algorithms to compare methods

22. Problem (to be pondered later…) • How fast can 109 integers be sorted? • What algorithms would be used?

23. Introduction: Efficient Algorithms for the Problem of Computing Fibonocci Numbers Analysis of Algorithms Prepared by John Reif, Ph.D.