Reservoir-sampling algorithms of time complexity O ( n (1 + log( N / n )))

One-pass algorithms for sampling n records without replacement from a population of unknown size n are known as reservoir-sampling algorithms. In this article, Vitter's reservoir-sampling algorithm, algorithm Z, is modified to give a more efficient algorithm, algorithm K. Additionally, two new algorithms, algorithm L and algorithm M, are proposed. If the time for scanning the population is ignored, all the four algorithms have expected CPU time O ( n (1 + log( N / n ))), which is optimum up to a constant factor. Expressions of the expected CPU time for the algorithms are presented. Among the four, algorithm L is the simplest, and algorithm M is the most efficient when n and N / n are large and N is O ( n 2 ).