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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2019.92

Beating Fredman-Komlós for Perfect k-Hashing

Authors: Venkatesan Guruswami and Andrii Riazanov

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract

We say a subset C subseteq {1,2,...,k}^n is a k-hash code (also called k-separated) if for every subset of k codewords from C, there exists a coordinate where all these codewords have distinct values. Understanding the largest possible rate (in bits), defined as (log_2 |C|)/n, of a k-hash code is a classical problem. It arises in two equivalent contexts: (i) the smallest size possible for a perfect hash family that maps a universe of N elements into {1,2,...,k}, and (ii) the zero-error capacity for decoding with lists of size less than k for a certain combinatorial channel. A general upper bound of k!/k^{k-1} on the rate of a k-hash code (in the limit of large n) was obtained by Fredman and Komlós in 1984 for any k >= 4. While better bounds have been obtained for k=4, their original bound has remained the best known for each k >= 5. In this work, we present a method to obtain the first improvement to the Fredman-Komlós bound for every k >= 5, and we apply this method to give explicit numerical bounds for k=5, 6.

Cite as

Venkatesan Guruswami and Andrii Riazanov. Beating Fredman-Komlós for Perfect k-Hashing. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 92:1-92:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{guruswami_et_al:LIPIcs.ICALP.2019.92,
  author =	{Guruswami, Venkatesan and Riazanov, Andrii},
  title =	{{Beating Fredman-Koml\'{o}s for Perfect k-Hashing}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{92:1--92:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.92},
  URN =		{urn:nbn:de:0030-drops-106687},
  doi =		{10.4230/LIPIcs.ICALP.2019.92},
  annote =	{Keywords: Coding theory, perfect hashing, hash family, graph entropy, zero-error information theory}
}

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Beating Fredman-Komlós for Perfect k-Hashing

Abstract

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