Design and Analysis of Algorithms

Appendix D

Solutions of Recurrence Relations

D.1 INTRODUCTION

A recurrence relation for a sequence is an equation that defines the present term by previous terms.

Example: a_n = a_{n − 1} + a_{n − 2}

The initial conditions specify the terms before the recurrence relation takes effect.

Example: a₁ = 2; a₂ = 3

Many counting problems can be modelled by recurrence relations.

Example 1

How many bit strings of length n have no “00”? All strings begin with a 1 or a 0. There are total 2ⁿ strings of length n. Out of these, we accept the following strings:

1. Starts with a 1; remaining n − 1; bits can be any string with no 00.
2. Starts with a 0 and 2nd bit is a 1; remaining n − 2 bits can be any string with no 00.
3. 2 strings of length 1 and 3 strings of length 2 without ...