A Practical Introduction to Data Structures and Algorithm Analysis

3Algorithm AnalysisHow long will it take to process the company payroll once we complete our plannedmerger? Should I buy a new payroll program from vendor X or vendor Y? If aparticular program is slow, is it badly implemented or is it solving a hard problem?Questions like these ask us to consider the difficulty of a problem, or the relativeefficiency of two or more approaches to solving a problem.This chapter introduces the motivation, basic notation, and fundamental techniquesof algorithm analysis. We focus on a methodology known as asymptoticalgorithm analysis, or simply asymptotic analysis. Asymptotic analysis attemptsto estimate the resource consumption of an algorithm. It allows us to compare therelative costs of two or more algorithms for solving the same problem. Asymptoticanalysis also gives algorithm designers a tool for estimating whether a proposedsolution is likely to meet the resource constraints for a problem before they implementan actual program. After reading this chapter, you should understand• the concept of a growth rate, the rate at which the cost of an algorithm growsas the size of its input grows;• the concept of upper and lower bounds for a growth rate, and how to estimatethese bounds for a simple program, algorithm, or problem; and• the difference between the cost of an algorithm (or program) and the cost ofa problem.The chapter concludes with a brief discussion of the practical difficulties encounteredwhen empirically measuring the cost of a program, and some principles forcode tuning to improve program efficiency.3.1 IntroductionHow do you compare two algorithms for solving some problem in terms of efficiency?One way is to implement both algorithms as computer programs and then57