Optimal succinct rank data structure via approximate nonnegative tensor decomposition

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

3 Scopus citations

Abstract

Given an n-bit array A, the succinct rank data structure problem asks to construct a data structure using space n + r bits for r ≪ n, supporting rank queries of form rank(u) = Íui=01 A[i]. In this paper, we design a new succinct rank data structure with r = n/(log n)Ω(t) + n1c and query time O(t) for some constant c > 0, improving the previous best-known by Pǎtraşcu, which has r = n/(logtn )Ω(t) + Õ(n3/4) bits of redundancy. For r > n1c, our space-time tradeoff matches the cell-probe lower bound by Pǎtraşcu and Viola, which asserts that r must be at least n/(log n)O(t). Moreover, one can avoid an n1c-bit lookup table when the data structure is implemented in the cell-probe model, achieving r = ⌈n/(log n)Ω(t)⌉. It matches the lower bound for the full range of parameters. En route to our new data structure design, we establish an interesting connection between succinct data structures and approximate nonnegative tensor decomposition. Our connection shows that for specific problems, to construct a space-efficient data structure, it suffices to approximate a particular tensor by a sum of (few) nonnegative rank-1 tensors. For the rank problem, we explicitly construct such an approximation, which yields an explicit construction of the data structure.

Original languageEnglish (US)
Title of host publicationSTOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing
EditorsMoses Charikar, Edith Cohen
PublisherAssociation for Computing Machinery
Pages955-966
Number of pages12
ISBN (Electronic)9781450367059
DOIs
StatePublished - Jun 23 2019
Externally publishedYes
Event51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 - Phoenix, United States
Duration: Jun 23 2019Jun 26 2019

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019
Country/TerritoryUnited States
CityPhoenix
Period6/23/196/26/19

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Partial sum
  • Spillover representation
  • Succinct data structure
  • Tensor decomposition

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