Chapter 1: Q14E (page 49)

Suppose you want to compute the nth Fibonacci number $F_{n}$ , 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 $fib 3$ is $O (\log (n) * M (logp)) .$

Step by step solution

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 $0 and 1$ .

nth Fibonacci number formulae

Matrix multiplication for $n^{th}$ Fibonacci number is

$[\begin{matrix} Fn \\ Fn + 1 \end{matrix}] = {[\begin{matrix} 0 & 1 \\ 1 & 1 \end{matrix}]}^{n} [\begin{matrix} F 0 \\ F 1 \end{matrix}]$

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

By problem $0.4, fib 3$ involves $O (\log n)$ arithmetic operations for multiplication.

So,

For every arithmetic operation with $\mod p$ , then the answer of the final step is long as $\log p$ bits.

Therefore, the final running time after computing each step of $fib 3$ is $O (\log (n) * M (logp)) .$

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Suppose you want to compute the nth Fibonacci number $F_{n}$ , modulo an integer $p$ . Can you find an efficient way to do this?

Short Answer

Step by step solution

Introduction

nth Fibonacci number formulae

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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

Step by step solution

Introduction

nth Fibonacci number formulae

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Computer Science Textbooks

Game Design in Computer Science

Databases

Computer Network

Computer Organisation and Architecture

Data Structures

Big Data

Study anywhere. Anytime. Across all devices.

Suppose you want to compute the nth Fibonacci number $F_{n}$ , modulo an integer $p$ . Can you find an efficient way to do this?