Incremental codes

Yevgeniy Dodis, Shai Halevi

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

Abstract

We introduce the notion of incremental codes. Unlike a regular code of a given rate, which is an unordered set of elements with a large minimum distance, an incremental code is an ordered vector of elements each of whose prefixes is a good regular code (of the corresponding rate). Additionally, while the quality of a regular code is measured by its minimum distance, we measure the quality of an incremental code C by its competitive ratio A: the minimum distance of each prefix of C has to be at most a factor of A smaller than the minimum distance of the best regular code of the same rate. We first consider incremental codes over an arbitrary compact metric space M, and construct a 2-competitive code for M. When M is finite, the construction takes time O(|M|2), exhausts the entire space, and is NP-hard to improve in general. We then concentrate on 2 specific spaces: the real interval [0, 1] and, most importantly, the Hamming space Fn. For the interval [0, 1] we construct an optimal (infinite) code of competitive ratio ln 4 ≈ 1.386. For the Hamming space Fn (where the generic 2-competitive constructive is not efficient), we show the following. If |F| ≥ q, we construct optimal (and efficient) 1-competitive code that exhausts Fn (has rate 1). For small alphabets (|F| < q), we show that 1-competitive codes do not exist and provide several efficient constructions of codes achieving constant competitive ratios. In particular, our best construction has rate (1−o(1)) and competitive ratio (2+o(1)), essentially matching the bounds in the generic construction.

Original languageEnglish (US)
Title of host publicationApproximation, Randomization, and Combinatorial Optimization
Subtitle of host publicationAlgorithms and Techniques - 4th International Workshop on Approximation, Algorithms for Combinatorial Optimization Problems, APPROX 2001 and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2001, Proceedings
EditorsLuca Trevisan, Klaus Jansen, Michel Goemans, Jose D. P. Rolim
PublisherSpringer Verlag
Pages75-90
Number of pages16
ISBN (Electronic)3540424709
StatePublished - Jan 1 2015
Event4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2001 and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2001 - Berkeley, United States
Duration: Aug 18 2001Aug 20 2001

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume2129
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2001 and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2001
CountryUnited States
CityBerkeley
Period8/18/018/20/01

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Dodis, Y., & Halevi, S. (2015). Incremental codes. In L. Trevisan, K. Jansen, M. Goemans, & J. D. P. Rolim (Eds.), Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 4th International Workshop on Approximation, Algorithms for Combinatorial Optimization Problems, APPROX 2001 and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2001, Proceedings (pp. 75-90). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2129). Springer Verlag.