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This analysis focuses on developing efficient algorithms for computing Fibonacci numbers. The study evaluates different methods and techniques to solve the problem, considering time efficiency and computational complexity. The findings highlight the importance of utilizing special properties of computational problems and optimizing algorithms for faster computation.
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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
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 = > .01 n2 steps > 1016m 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
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
General Case of Powering • Decompose n = 2 j1 + 2 j2 + ... + 2 jkas sum of powers of 2 • Compute: • Example:
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 m2bool ops = m2/16 steps ~ 1016 steps for n = 109 ~ 1016msec ~ 1010 seconds ~ 317 years!
Fast Multiplication of m bit integers a,b by • “Divide and Conquer” • seems to take 4 multiplications, but…
Recursive Mult Algorithm • Cost
Schonhage-Strassen Integer Multiplication Algorithm • This is feasible to compute!
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
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
Problem (to be pondered later…) • How fast can 109 integers be sorted? • What algorithms would be used?
Introduction: Efficient Algorithms for the Problem of Computing Fibonocci Numbers Analysis of Algorithms Prepared by John Reif, Ph.D.