Jeffrey Scott Vitter
Abstract
Several new methods are presented for selecting n records at random without replacement from a file containing N records. Each algorithm selects the records for the sample in a sequential manner—in the same order the records appear in the file. The algorithms are online in that the records for the sample are selected iteratively with no preprocessing. The algorithms require a constant amount of space and are short and easy to implement. The main result of this paper is the design and analysis of Algorithm D, which does the sampling in O(n) time, on the average; roughly n uniform random variates are generated, and approximately n exponentiation operations (of the form ab, for real numbers a and b) are performed during the sampling. This solves an open problem in the literature. CPU timings on a large mainframe computer indicate that Algorithm D is significantly faster than the sampling algorithms in use today.
References
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Digital Library
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Index Terms
Faster methods for random sampling
Reviews
Robert M. Lynch
The Vitter paper describes several new sequential algorithms for randomly sampling :In records sequentially from a file containing :IN records. It is presented in nine sections, including the Appendix. The main text of the paper focuses on describing the algorithms and presenting performance details, and the Appendix presents the implementation of the algorithms in an pseudocode resembling PASCAL. Each of the algorithms can be implemented without the requirement of preprocessing records, and each requires a constant amount of space.:P In Section 1, the scope of the paper is presented. In Section 1, Algorithm S is described. Algorithm S was introduced by Fan et al. [1] and Jones [2], and it was again presented by Knuth [3]. Briefly, with Algorithm S, the next record in a file is either selected or skipped based upon the value of the generated uniform random variate. This select:Uskip process continues until the desired number of records :In is selected.:P In Section 4, Algorithm D is discussed at length, and it is this algorithm that is the focus of the paper. A naive implementation is presented in Section 4, but in Section 5, optimizations are given to improve its performance. These improvements reduce by 1/2 the number of variates that must be generated and the number of exponential computations that must be performed.:P In Section 6, algorithm performance is presented and in Section 7, empirical comparisons for a FORTRAN 77 implementation on an IBM 3081 are discussed. Run:Utime comparisons in microseconds are l6:In for Algorithm S and 55:In for Algorithm D. Further, for Algorithm S the expected number of random variates is [:IN±1):In]/(:IN±1) with an generated run:U time of :I0(:IN). Algorithm D has an expected number of generated variates to be :In and an expected average run:Utime of :I0(:In). Both the empirical and expected comparisons make D the algorithm of choice. Section 8 brings it all together and discusses work that is related to algorithms.:P The paper is well written and well organized. Statisticians who are not of a computer science will have little difficulty following the paper. Further leaning, the implementation of the algorithms appears relatively straightfoward. As pointed out by Vitter, and by the paper's title, Algorithm D is a major improvement over sampling algorithms in use today. :6HR EFERENCES [1] F AN, C.T.; M ULLER, M.E.; and R EZUCHA, I. Development of sampling plans by using seguential (item:Uby:Uitem) selection techniques and digital computers, :IJ. Am. Stat. :1A57 (June 1961), 387-402. See :ICR :1A4, 1 (Jan.:UFeb. 1963), Rev. 3,748. [2]J ONES, T.G. A note on sampling a tape file, :ICommun. ACM :1A5 (June 1962), 343. [3] K NUTH, D.E. :IThe art of computer programming, Vol. 2:seminumerical algorithms, Addison:UWesley, Reading, MA, 1981. See :1A22, 9 (Sept. 1981), Rev. 38,390.
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