Power of d choices with simple tabulation

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We consider the classic d-choice paradigm of Azar et al. [STOC'94] in which m balls are put into n bins sequentially as follows: For each ball we are given a choice of d bins chosen according to d hash functions and the ball is placed in the least loaded of these bins, breaking ties arbitrarily. The interest is in the number of balls in the fullest bin after all balls have been placed. In this paper we suppose that the d hash functions are simple tabulation hash functions which are easy to implement and can be evaluated in constant time. Generalising a result by Dahlgaard et al. [SODA'16] we show that for an arbitrary constant d ≥ 2 the expected maximum load is at mostlg lg lg d n + O(1). We further show that by using a simple tie-breaking algorithm introduced by Vöcking [J.ACM'03] the expected maximum load is reduced tod lg lg lg ϕ n d + O(1) where ϕd is the rate of growth of the d-ary Fibonacci numbers. Both of these expected bounds match those known from the fully random setting. The analysis by Dahlgaard et al. relies on a proof by P tra cu and Thorup [J.ACM'11] concerning the use of simple tabulation for cuckoo hashing. We require a generalisation to d > 2 hash functions, but the original proof is an 8-page tour de force of ad-hoc arguments that do not appear to generalise. Our main technical contribution is a shorter, simpler and more accessible proof of the result by P tra cu and Thorup, where the relevant parts generalise nicely to the analysis of d choices.

Original languageEnglish
Title of host publication45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
EditorsChristos Kaklamanis, Daniel Marx, Ioannis Chatzigiannakis, Donald Sannella
Number of pages14
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Publication date1 Jul 2018
Article number5
ISBN (Electronic)9783959770767
DOIs
Publication statusPublished - 1 Jul 2018
Event45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 - Prague, Czech Republic
Duration: 9 Jul 201813 Jul 2018

Conference

Conference45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
LandCzech Republic
ByPrague
Periode09/07/201813/07/2018
SponsorAvast, RSJ
SeriesLeibniz International Proceedings in Informatics, LIPIcs
Volume107
ISSN1868-8969

    Research areas

  • Balls, Bins, Load Balancing, Phrases Hashing, Simple Tabulation

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