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Sec. 3.4 Asymptotic Analysis 633.4 Asymptotic AnalysisDespite the larger constant for the curve labeled 10n in Figure 3.1, 2n 2 crossesit at the relatively small value of n = 5. What if we double the value of theconstant in front of the linear equation? As shown in the graph, 20n is surpassedby 2n 2 once n = 10. The additional factor of two for the linear growth rate doesnot much matter. It only doubles the x-coordinate for the intersection point. Ingeneral, changes to a constant factor in either equation only shift where the twocurves cross, not whether the two curves cross.When you buy a faster computer or a faster compiler, the new problem sizethat can be run in a given amount of time for a given growth rate is larger by thesame factor, regardless of the constant on the running-time equation. The timecurves for two algorithms with different growth rates still cross, regardless of theirrunning-time equation constants. For these reasons, we usually ignore the constantswhen we want an estimate of the growth rate for the running time or otherresource requirements of an algorithm. This simplifies the analysis and keeps usthinking about the most important aspect: the growth rate. This is called asymptoticalgorithm analysis. To be precise, asymptotic analysis refers to the study ofan algorithm as the input size “gets big” or reaches a limit (in the calculus sense).However, it has proved to be so useful to ignore all constant factors that asymptoticanalysis is used for most algorithm comparisons.It is not always reasonable to ignore the constants. When comparing algorithmsmeant to run on small values of n, the constant can have a large effect. For example,if the problem is to sort a collection of exactly five records, then an algorithmdesigned for sorting thousands of records is probably not appropriate, even if itsasymptotic analysis indicates good performance. There are rare cases where theconstants for two algorithms under comparison can differ by a factor of 1000 ormore, making the one with lower growth rate impractical for most purposes due toits large constant. Asymptotic analysis is a form of “back of the envelope” estimationfor algorithm resource consumption. It provides a simplified model of therunning time or other resource needs of an algorithm. This simplification usuallyhelps you understand the behavior of your algorithms. Just be aware of the limitationsto asymptotic analysis in the rare situation where the constant is important.3.4.1 Upper BoundsSeveral terms are used to describe the running-time equation for an algorithm.These terms — and their associated symbols — indicate precisely what aspect ofthe algorithm’s behavior is being described. One is the upper bound for the growthof the algorithm’s running time. It indicates the upper or highest growth rate thatthe algorithm can have.