eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-07-04
92:1
92:14
10.4230/LIPIcs.ICALP.2019.92
article
Beating Fredman-Komlós for Perfect k-Hashing
Guruswami, Venkatesan
1
https://orcid.org/0000-0001-7926-3396
Riazanov, Andrii
1
Computer Science Department, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA, USA, 15213
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol132-icalp2019/LIPIcs.ICALP.2019.92/LIPIcs.ICALP.2019.92.pdf
Coding theory
perfect hashing
hash family
graph entropy
zero-error information theory