Tight bounds for Lp samplers, finding duplicates in streams, and related problems

Hossein Jowhari, Mert Sǎglam, Gábor Tardos

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

78 Citations (Scopus)

Abstract

In this paper, we present near-optimal space bounds for Lp- samplers. Given a stream of updates (additions and subtraction) to the coordinates of an underlying vector x ε Rn, a perfect L p sampler outputs the i-th coordinate with probability |x i|p/||x||pp. In SODA 2010, Monemizadeh and Woodru showed polylog space upper bounds for approximate Lp-samplers and demonstrated various applications of them. Very recently, Andoni, Krauthgamer and Onak improved the upper bounds and gave a O(ε-p log3 n) space ε relative error and constant failure rate Lp-sampler for p ε [1; 2]. In this work, we give another such algorithm requiring only O(ε-p log 3 n) space for p ε (1; 2). For p ε (0; 1), our space bound is O(ε-p log3 n), while for the p = 1 case we have an O(log(1=ε) ε-1 log2 n) space algorithm. We also give a O(log2 n) bits zero relative error L0-sampler, improving the O(log3 n) bits algorithm due to Frahling, Indyk and Sohler. As an application of our samplers, we give better upper bounds for the problem of finding duplicates in data streams. In case the length of the stream is longer than the alphabet size, L1 sampling gives us an O(log2 n) space algorithm, thus improving the previous O(log3 n) bound due to Gopalan and Radhakrishnan. In the second part of our work, we prove an (log 2 n) lower bound for sampling from 0, ±1 vectors (in this special case, the parameter p is not relevant for Lp sampling). This matches the space of our sampling algorithms for constant ε > 0. We also prove tight space lower bounds for the finding duplicates and heavy hitters problems. We obtain these lower bounds using reductions from the communication complexity problem augmented indexing.

Original languageEnglish
Title of host publicationPODS'11 - Proceedings of the 30th Symposium on Principles of Database Systems
Pages49-58
Number of pages10
DOIs
Publication statusPublished - Jul 15 2011
Event30th Symposium on Principles of Database Systems, PODS'11 - Athens, Greece
Duration: May 13 2011May 15 2011

Publication series

NameProceedings of the ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems

Other

Other30th Symposium on Principles of Database Systems, PODS'11
CountryGreece
CityAthens
Period5/13/115/15/11

Keywords

  • Finding duplicates
  • Sampling
  • Streaming algorithms

ASJC Scopus subject areas

  • Software
  • Information Systems
  • Hardware and Architecture

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  • Cite this

    Jowhari, H., Sǎglam, M., & Tardos, G. (2011). Tight bounds for Lp samplers, finding duplicates in streams, and related problems. In PODS'11 - Proceedings of the 30th Symposium on Principles of Database Systems (pp. 49-58). (Proceedings of the ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems). https://doi.org/10.1145/1989284.1989289