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Special-Purpose Sorting Methods

Sorting methods are critical components of many apps systems, and it is not unusual for special measures to be taken to make a sort as fast as possible or capable of handling huge files. We might encounter high-performance enhancements to a computer system, or special-purpose hardware specifically designed for sorting, or simply a new computer system based on some new architectural design. In such cases, the implicit assumptions that we have been making about the relative costs of operations on the data to be sorted may not be valid. In this chapter, we examine examples of sorting methods that are designed to run efficiently on various different kinds of machines. We consider several different examples of the restrictions imposed by high-performance hardware, and several methods that are useful in practice for implementing high-performance sorts.

Any new computer architecture is eventually going to need to support an efficient sorting method. Indeed, sorting has historically served as one testbed for evaluating new architectures, because it is so important and so well understood. We want to learn not just which known algorithms run best on a new machine and why, but also whether specific characteristics of a new machine can be exploited in some new algorithm. To develop a new algorithm, we define an abstract machine that encapsulates the essential properties of the real machine; design and analyze algorithms for the abstract machine; then implement, test, and refine both the best algorithms and the model. We draw on our past experience, including the many methods for general-purpose machines that we have seen in Chapters 6 through 10, but the abstract machines impose limitations that help us to focus on the true costs and make it clear that different algorithms are appropriate for different machines.

At one end of the spectrum, we shall consider low-level models where the only allowed operation is the compare–exchange operation. At the other end of the spectrum, we shall consider high-level models where we read and write large blocks of data to a slow external medium or among independent parallel processors.

First, we examine a version of mergesort known as Batcher's odd–even mergesort. It is based on a divide-and-conquer merge algorithm that uses only compare–exchange operations, with perfect-shuffle and perfect-unshuffle operations for data movement. These are of interest in their own right and apply to many problems other than sorting. Next, we examine Batcher's method as a sorting network. A sorting network is a simple abstraction for low-level sorting hardware. Networks consist of interconnected comparators, which are modules capable of performing compare–exchange operations.

Another important abstract sorting problem is the external-sorting problem, where the file to be sorted is far too large to fit in memory. The cost of accessing individual records can be prohibitive, so we shall use an abstract model, where records are transferred to and from external devices in large blocks. We consider two algorithms for external sorting and use the model to compare them.

Finally, we consider parallel sorting, for the case when the file to be sorted is distributed among independent parallel processors. We define a simple parallel-machine model, then examine how Batcher's method provides an effective solution. Our use of the same basic algorithm to solve a high-level problem and a low-level problem is a convincing example of the power of abstraction.

The different abstract machines in this chapter are simple but are worthy of study because they encapsulate specific constraints that can be critical in particular sorting apps. Low-level sorting hardware has to consist of simple components; external sorts generally require access to huge data files in blocks, with sequential access more efficient than random access; and parallel sorting involves communications constraints among processors. On the one hand, we cannot do justice to detailed machine models that fully correspond to particular real machines; on the other hand, the abstractions that we do consider lead us not only to theoretical formulations that provide information about essential limitations on performance but also to interesting algorithms that are of direct practical utility.

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