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1.3. Analysis of Algorithms

Though the analysis of sorting and merge-sort that we considered in §1.2 demonstrates the intrinsic “difficulty” of the sorting problem, there are many important questions related to sorting (and to mergesort) that it does not address at all. How long might an implementation of mergesort be expected to run on a particular computer? How might its running time compare to other O(NlogN) methods? (There are many.) How does it compare to sorting methods that are fast on average, but perhaps not in the worst case? How does it compare to sorting methods that are not based on compares among elements? To answer such questions, a more detailed analysis is required. In this section we briefly describe the process of doing such an analysis.

To analyze an algorithm, we must first identify the resources of primary interest so that the detailed analysis may be properly focused. We describe the process in terms of studying the running time since it is the resource most relevant here. A complete analysis of the running time of an algorithm involves the following steps:

• Implement the algorithm completely.
• Determine the time required for each basic operation.
• Identify unknown quantities that can be used to describe the frequency of execution of the basic operations.
• Develop a realistic model for the input to the program.
• Analyze the unknown quantities, assuming the modeled input.
• Calculate the total running time by multiplying the time by the frequency for each operation, then adding all the products.

The first step in the analysis is to carefully implement the algorithm on a particular computer. We reserve the term program to describe such an implementation. One algorithm corresponds to many programs. A particular implementation not only provides a concrete object to study, but also can give useful empirical data to aid in or to check the analysis. Presumably the implementation is designed to make efficient use of resources, but it is a mistake to overemphasize efficiency too early in the process. Indeed, a primary application for the analysis is to provide informed guidance toward better implementations.

The next step is to estimate the time required by each component instruction of the program. In principle and in practice, we can often do so with great precision, but the process is very dependent on the characteristics of the computer system being studied. Another approach is to simply run the program for small input sizes to “estimate” the values of the constants, or to do so indirectly in the aggregate, as described in Exercise 1.7. We do not consider this process in detail; rather we focus on the “machine-independent” parts of the analysis in this book.

Indeed, to determine the total running time of the program, it is necessary to study the branching structure of the program in order to express the frequency of execution of the component instructions in terms of unknown mathematical quantities. If the values of these quantities are known, then we can derive the running time of the entire program simply by multiplying the frequency and time requirements of each component instruction and adding these products. Many programming environments have tools that can simplify this task. At the first level of analysis, we concentrate on quantities that have large frequency values or that correspond to large costs; in principle the analysis can be refined to produce a fully detailed answer. We often refer to the “cost” of an algorithm as shorthand for the “value of the quantity in question” when the context allows.

The next step is to model the input to the program, to form a basis for the mathematical analysis of the instruction frequencies. The values of the unknown frequencies are dependent on the input to the algorithm: the problem size (usually we name that N) is normally the primary parameter used to express our results, but the order or value of input data items ordinarily affects the running time as well. By “model,” we mean a precise description of typical inputs to the algorithm. For example, for sorting algorithms, it is normally convenient to assume that the inputs are randomly ordered and distinct, though the programs normally work even when the inputs are not distinct. Another possibility for sorting algorithms is to assume that the inputs are random numbers taken from a relatively large range. These two models can be shown to be nearly equivalent. Most often, we use the simplest available model of “random” inputs, which is often realistic. Several different models can be used for the same algorithm: one model might be chosen to make the analysis as simple as possible; another model might better reflect the actual situation in which the program is to be used.

The last step is to analyze the unknown quantities, assuming the modeled input. For average-case analysis, we analyze the quantities individually, then multiply the averages by instruction times and add them to find the running time of the whole program. For worst-case analysis, it is usually difficult to get an exact result for the whole program, so we can only derive an upper bound, by multiplying worst-case values of the individual quantities by instruction times and summing the results.

This general scenario can successfully provide exact models in many situations. Knuth’s books  are based on this precept. Unfortunately, the details in such an exact analysis are often daunting. Accordingly, we typically seek approximate models that we can use to estimate costs.

The first reason to approximate is that determining the cost details of all individual operations can be daunting in the context of the complex architectures and operating systems on modern computers. Accordingly, we typically study just a few quantities in the “inner loop” of our programs, implicitly hypothesizing that total cost is well estimated by analyzing just those quantities. Experienced programmers regularly “profile” their implementations to identify “bottlenecks,” which is a systematic way to identify such quantities. For example, we typically analyze compare-based sorting algorithms by just counting compares. Such an approach has the important side benefit that it is machine independent. Carefully analyzing the number of compares used by a sorting algorithm can enable us to predict performance on many different computers. Associated hypotheses are easily tested by experimentation, and we can refine them, in principle, when appropriate. For example, we might refine comparison-based models for sorting to include data movement, which may require taking caching effects into account.

Exercise 1.11 Run experiments on two different computers to test the hypothesis that the running time of mergesort divided by the number of compares that it uses approaches a constant as the problem size increases.

Approximation is also effective for mathematical models. The second reason to approximate is to avoid unnecessary complications in the mathematical formulae that we develop to describe the performance of algorithms. A major theme of this book is the development of classical approximation methods for this purpose, and we shall consider many examples. Beyond these, a major thrust of modern research in the analysis of algorithms is methods of developing mathematical analyses that are simple, sufficiently precise that they can be used to accurately predict performance and to compare algorithms, and able to be refined, in principle, to the precision needed for the application at hand. Such techniques primarily involve complex analysis and are fully developed in our book .