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1.8. Randomized Algorithms

The analysis of the average-case performance of quicksort depends on the input being randomly ordered. This assumption is not likely to be strictly valid in many practical situations. In general, this situation reflects one of the most serious challenges in the analysis of algorithms: the need to properly formulate models of inputs that might appear in practice.

Fortunately, there is often a way to circumvent this difficulty: “randomize” the inputs before using the algorithm. For sorting algorithms, this simply amounts to randomly permuting the input file before the sort. (See Chapter 7 for a specific implementation of an algorithm for this purpose.) If this is done, then probabilistic statements about performance such as those made earlier are completely valid and will accurately predict performance in practice, no matter what the input.

Often, it is possible to achieve the same result with less work, by making a random choice (as opposed to a specific arbitrary choice) whenever the algorithm could take one of several actions. For quicksort, this principle amounts to choosing the element to be used as the partitioning element at random, rather than using the element at the end of the array each time. If this is implemented with care (preserving randomness in the subarrays) then, again, it validates the probabilistic analysis given earlier. (Also, the cutoff for small subarrays should be used, since it cuts down the number of random numbers to generate by a factor of about M.) Many other examples of randomized algorithms may be found in [23] and [25]. Such algorithms are of interest in practice because they take advantage of randomness to gain efficiency and to avoid worst-case performance with high probability. Moreover, we can make precise probabilistic statements about performance, further motivating the study of advanced techniques for deriving such results.

THE example of the analysis of quicksort that we have been considering perhaps illustrates an idealized methodology: not all algorithms can be as smoothly dealt with as this. A full analysis like this one requires a fair amount of effort that should be reserved only for our most important algorithms. Fortunately, as we will see, there are many fundamental methods that do share the basic ingredients that make analysis worthwhile, where we can

  • Specify realistic input models.
  • Derive mathematical models that describe costs.
  • Develop concise, accurate solutions.
  • Use the solutions to compare variants and compare with other algorithms, and help adjust values of algorithm parameters.

In this book, we consider a wide variety of such methods, concentrating on mathematical techniques validating the second and third of these points.

Most often, we skip the parts of the methodology outlined above that are program-specific (dependent on the implementation), to concentrate either on algorithm design, where rough estimates of the running time may suffice, or on the mathematical analysis, where the formulation and solution of the mathematical problem involved are of most interest. These are the areas involving the most significant intellectual challenge, and deserve the attention that they get.

As we have already mentioned, one important challenge in analysis of algorithms in common use on computers today is to formulate models that realistically represent the input and that lead to manageable analysis problems. We do not dwell on this problem because there is a large class of combinatorial algorithms for which the models are natural. In this book, we consider examples of such algorithms and the fundamental structures upon which they operate in some detail. We study permutations, trees, strings, tries, words, and mappings because they are all both widely studied combinatorial structures and widely used data structures and because “random” structures are both straightforward and realistic.

In Chapters 2 through 5, we concentrate on techniques of mathematical analysis that are applicable to the study of algorithm performance. This material is important in many applications beyond the analysis of algorithms, but our coverage is developed as preparation for applications later in the book. Then, in Chapters 6 through 9 we apply these techniques to the analysis of some fundamental combinatorial algorithms, including several of practical interest. Many of these algorithms are of basic importance in a wide variety of computer applications, and so are deserving of the effort involved for detailed analysis. In some cases, algorithms that seem to be quite simple can lead to quite intricate mathematical analyses; in other cases, algorithms that are apparently rather complicated can be dealt with in a straightforward manner. In both situations, analyses can uncover significant differences between algorithms that have direct bearing on the way they are used in practice.

It is important to note that we teach and present mathematical derivations in the classical style, even though modern computer algebra systems such as Maple, Mathematica, or Sage are indispensable nowadays to check and develop results. The material that we present here may be viewed as preparation for learning to make effective use of such systems.

Much of our focus is on effective methods for determining performance characteristics of algorithm implementations. Therefore, we present programs in a widely used programming language (Java). One advantage of this approach is that the programs are complete and unambiguous descriptions of the algorithms. Another is that readers may run empirical tests to validate mathematical results. Generally our programs are stripped-down versions of the full Java implementations in the Sedgewick and Wayne Algorithms text [30]. To the extent possible, we use standard language mechanisms, so people familiar with other programming environments may translate them. More information about many of the programs we cover may be found in [30].

The basic methods that we cover are, of course, applicable to a much wider class of algorithms and structures than we are able to discuss in this introductory treatment. We cover only a few of the large number of combinatorial algorithms that have been developed since the advent of computers in mid-20th century. We do not touch on the scores of applications areas, from image processing to bioinformatics, where algorithms have proved effective and have been investigated in depth. We mention only briefly approaches such as amortized analysis and the probabilistic method, which have been successfully applied to the analysis of a number of important algorithms. Still, it is our hope that mastery of the introductory material in this book is good preparation for appreciating such material in the research literature in the analysis of algorithms. Beyond the books by Knuth, Sedgewick and Wayne, and Cormen, Leiserson, Rivest, and Stein cited earlier, other sources of information about the analysis of algorithms and the theory of algorithms are the books by Gonnet and Baeza-Yates [11], by Dasgupta, Papadimitriou, and Vazirani [7], and by Kleinberg and Tardos [16].

Equally important, we are led to analytic problems of a combinatorial nature that allow us to develop general mechanisms that may help to analyze future, as yet undiscovered, algorithms. The methods that we use are drawn from the classical fields of combinatorics and asymptotic analysis, and we are able to apply classical methods from these fields to treat a broad variety of problems in a uniform way. This process is described in full detail in our book Analytic Combinatorics [10]. Ultimately, we are not only able to directly formulate combinatorial enumeration problems from simple formal descriptions, but also we are able to directly derive asymptotic estimates of their solution from these formulations.

In this book, we cover the important fundamental concepts while at the same time developing a context for the more advanced treatment in [10] and in other books that study advanced methods, such as Szpankowski’s study of algorithms on words [32] or Drmota’ study of trees [8]. Graham, Knuth, and Patashnik [12] is a good source of more material relating to the mathematics that we use; standard references such as Comtet [5] (for combinatorics) and Henrici [14] (for analysis) also have relevant material. Generally, we use elementary combinatorics and real analysis in this book, while [10] is a more advanced treatment from a combinatorial point of view, and relies on complex analysis for asymptotics.

Properties of classical mathematical functions are an important part of our story. The classic Handbook of Mathematical Functions by Abramowitz and Stegun [1] was an indispensable reference for mathematicians for decades and was certainly a resource for the development of this book. A new reference that is intended to replace it was recently published, with associated online material [24]. Indeed, reference material of this sort is increasingly found online, in resources such as Wikipedia and Mathworld [35]. Another important resource is Sloane’s On-Line Encyclopedia of Integer Sequences [31].

Our starting point is to study characteristics of fundamental algorithms that are in widespread use, but our primary purpose in this book is to provide a coherent treatment of the combinatorics and analytic methods that we encounter. When appropriate, we consider in detail the mathematical problems that arise naturally and may not apply to any (currently known!) algorithm. In taking such an approach we are led to problems of remarkable scope and diversity. Furthermore, in examples throughout the book we see that the problems we solve are directly relevant to many important applications.

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