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Chapter 1: Q14E (page 49)

Suppose you want to compute the nth Fibonacci number Fn , modulo an integer p. Can you find an efficient way to do this?

Short Answer

Expert verified

The final running time after computing each step of fib3is O(log(n)*M(logp)).

Step by step solution

01

Introduction

Fibonacci Series – This series can be defined as every element of the series is the sum of two previous number of the series. Series will start from 0and1.

02

nth Fibonacci number formulae

Matrix multiplication for nthFibonacci number is

FnFn+1=0111nF0F1

Let’s assume that the running time for multiplying matrix of n-bit numbers is x(n).

By problem0.4,fib3 involvesO(logn) arithmetic operations for multiplication.

So,

For every arithmetic operation with modp, then the answer of the final step is long aslogp bits.

Therefore, the final running time after computing each step offib3 is O(log(n)*M(logp)).

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Most popular questions from this chapter

Is the difference of 530,000and6123,456a multiple of31 ?

Prove that the grade-school multiplication algorithm (page 24), when applied to binary numbers, always gives the right answer.

A positive integer N is a power if it is of the formqk , where q,role="math" localid="1658399000008" k are positive integers and k>1.

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(c) Give an efficient algorithm for determining whether a positive integerN is a power. Analyze its running time.

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