Optimal monotone encodings

Noga Alon, Rani Hod

Research output: Chapter in Book/Report/Conference proceedingConference contribution


Moran, Naor and Segev have asked what is the minimal r = r(n, k) for which there exists an (n,k)-monotone encoding of length r, i.e., a monotone injective function from subsets of size up to k of {1, 2, ..., n} to r bits. Monotone encodings are relevant to the study of tamper-proof data structures and arise also in the design of broadcast schemes in certain communication networks. To answer this question, we develop a relaxation of k-superimposed families, which we call α-fraction k-multi-user tracing ((k, α)-FUT families). We show that r(n, k) = Θ(k log(n/k)) by proving tight asymptotic lower and upper bounds on the size of (k, α)-FUT families and by constructing an (n,k)-monotone encoding of length O(k log(n/k)). We also present an explicit construction of an (n, 2)-monotone encoding of length 2log n + O(1), which is optimal up to an additive constant.

Original languageEnglish (US)
Title of host publicationAutomata, Languages and Programming - 35th International Colloquium, ICALP 2008, Proceedings
Number of pages13
EditionPART 1
StatePublished - 2008
Externally publishedYes
Event35th International Colloquium on Automata, Languages and Programming, ICALP 2008 - Reykjavik, Iceland
Duration: Jul 7 2008Jul 11 2008

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 1
Volume5125 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other35th International Colloquium on Automata, Languages and Programming, ICALP 2008

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science


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