Reference Work Entry

Encyclopedia of Algorithms

pp 1-99

Tail Bounds for Occupancy Problems

1995; Kamath, Motwani, Palem, Spirakis
  • Paul SpirakisAffiliated withDepartment of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern UniversityComputer Engineering and Informatics, Research and Academic Computer Technology Institute, Patras University

Keywords and Synonyms

Balls and bins   

Problem Definition

Consider a random allocation of m balls to n bins where each ball is placed in a bin chosen uniformly and independently. The properties of the resulting distribution of balls among bins have been the subject of intensive study in the probability and statistics literature [3,4]. In computer science, this process arises naturally in randomized algorithms and probabilistic analysis. Of particular interest is the occupancy problem where the random variable under consideration is the number of empty bins.

In this entry a series of bounds are presented (reminiscent of the Chernoff bound for binomial distributions) on the tail of the distribution of the number of empty bins; the tail bounds are successively tighter, but each new bound has a more complex closed form. Such strong bounds do not seem to have appeared in the earlier literature.

Key Results

The following notation in pres ...

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