Preprocess, set, query!

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9 Scopus citations

Abstract

Thorup and Zwick (J. ACM 52(1):1-24, 2005 and STOC'01) in their seminal work introduced the notion of distance oracles. Given an n-vertex weighted undirected graph with m edges, they show that for any integer k≥1 it is possible to preprocess the graph in Õ(mn1/k) time and generate a compact data structure of size O(kn 1+1/k ). For each pair of vertices, it is then possible to retrieve an estimated distance with multiplicative stretch 2k-1 in O(k) time. For k=2 this gives an oracle of O(n 1.5) size that produces in constant time estimated distances with stretch 3. Recently, Pǎtraşcu and Roditty (In: Proc. of 51st FOCS, 2010) broke the theoretical status-quo in the field of distance oracles and obtained a distance oracle for sparse unweighted graphs of O(n 5/3) size that produces in constant time estimated distances with stretch 2. In this paper we show that it is possible to break the stretch 2 barrier at the price of non-constant query time in unweighted undirected graphs. We present a data structure that produces estimated distances with 1+ε stretch. The size of the data structure is O(nm 1-ε′) and the query time is Õ(mn1-ε′). Using it for sparse unweighted graphs we can get a data structure of size O(n 1.87) that can supply in O(n 0.87) time estimated distances with multiplicative stretch 1.75.

Original languageEnglish
Pages (from-to)516-528
Number of pages13
JournalAlgorithmica
Volume67
Issue number4
DOIs
StatePublished - Dec 2013

Bibliographical note

Funding Information:
L. Roditty’s work is supported by Israel Science Foundation (grant no. 822/10).

Keywords

  • Data structures
  • Distance oracles
  • Graph algorithms

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