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Recall that Fibonacci numbers form a sequenceFnwhereF0= 0,F1= 1, andFn=Fn−1+Fn−2. The standard algorithm for finding the n-th Fibonacci number takes O(n) time. The goal of this question is to design a significantly faster algorithm for this problem. (a) Prove by induction that for alln≥1:[1 11 0]n=[Fn+1FnFnFn−1]. (b) Use the first part to design an algorithm that findsFninO(logn) time.

Recall that Fibonacci numbers form a sequenceFnwhereF0= 0,F1= 1, andFn=Fn−1+Fn−2. The standard algorithm for finding the n-th Fibonacci number takes O(n) time. The goal of this question is to design a significantly faster algorithm for this problem. (a) Prove by induction that for alln≥1:[1 11 0]n=[Fn+1FnFnFn−1]. (b) Use the first part to design an algorithm that findsFninO(logn) time.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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Recall that Fibonacci numbers form a sequenceFnwhereF0= 0,F1= 1, andFn=Fn−1+Fn−2. The standard algorithm for finding the n-th Fibonacci number takes O(n) time. The goal of this question is to design a significantly faster algorithm for this problem.

(a) Prove by induction that for alln≥1:[1 11 0]n=[Fn+1FnFnFn−1].

(b) Use the first part to design an algorithm that findsFninO(logn) time.

n Fn+1 Fn 1 Fn 1 Fn-1]
Transcribed Image Text:n Fn+1 Fn 1 Fn 1 Fn-1]
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