1. Prepared by John Reif, Ph.D.
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 = Fn-1 + Fn-2 if n > 1
> .01 n2 steps
> 1016 m 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
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 j jk as 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 m2 bool ops
= m2/16 steps
~ 1016 steps for n = 109
~ 1016 msec
~ 1010 seconds
~ 317 years!

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

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 = 109 using 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 More Speed-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. Prepared by John Reif, Ph.D.
Introduction: Efficient Algorithms for the Problem of Computing Fibonocci Numbers
Analysis of Algorithms
Prepared by
John Reif, Ph.D.