Algorithms

2.1 ■ Elementary Sorts
257
on your computer to learn the extent to which its conclusion about insertion sort and
selection sort is robust.
Property D is intentionally a bit vague—the value of the small constant factor is left
unstated and the assumption that the costs of compares and exchanges are similar is left
unstated—so that it can apply in a broad variety of situations. When possible, we try to
capture essential aspects of the performance of each of the algorithms that we study in
statements like this. As discussed in Chapter 1, each Property that we consider needs to
be tested scientifically in a given situation, perhaps supplemented with a more refined
hypothesis based upon a related Proposition (mathematical truth).
For practical applications, there is one further step, which is crucial: run experiments
to validate the hypothesis on the data at hand. We defer consideration of this step to
Section 2.5 and the exercises. In this case, if your sort keys are not distinct and/or
not randomly ordered, Property D might not hold. You can randomly order an array
with StdRandom.shuffle(), but applications with significant numbers of equal keys
involve more careful analysis.
Our discussions of the analyses of algorithms are intended to be starting points, not
final conclusions. If some other question about performance of the algorithms comes
to mind, you can study it with a tool like SortCompare. Many opportunities to do so
are presented in the exercises.
We do not dwell further on the comparative performance of insertion sort and selection
sort because we are much more interested in algorithms that can run a hundred or
a thousand or a million times faster than either. Still, understanding these elementary
algorithms is worthwhile for several reasons:
■ They help us work out the ground rules.
■ They provide performance benchmarks.
■ They often are the method of choice in some specialized situations.
■ They can serve as the basis for developing better algorithms.
For these reasons, we will use the same basic approach and consider elementary algorithms
for every problem that we study throughout this book, not just sorting. Programs
like SortCompare play a critical role in this incremental approach to algorithm
development. At every step along the way, we can use such a program to help evaluate
whether a new algorithm or an improved version of a known algorithm provides the
performance gains that we expect.