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.

Readings Main Reading Selection: CLR, Chapter 1

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

Time Efficiency

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.

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

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

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

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

Inductive Step

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

General Case of Powering Decompose n = 2 j1 + 2 j2 + ... + 2 jk as sum of powers of 2 Example Compute

Algorithm

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

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!

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

3 Mult Trick

Recursive Mult Algorithm Cost

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

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

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

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

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.