Article

Approximate distance oracles for unweighted graphs in Õ (n²) time

Authors:

Surender Baswana,

Sandeep SenAuthors Info & Claims

SODA '04: Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms

Pages 271 - 280

Published: 11 January 2004 Publication History

Get Access

Abstract

Let G(V, E) be an undirected weighted graph with |V| = n, |E| = m. Recently Thorup and Zwick introduced a remarkable data-structure that stores all pairs approximate distance information implicitly in o(n²) space, and yet answers any approximate distance query in constant time. They named this data-structure approximate distance oracle because of this feature. Given an integer k < 1, a (2k-1)-approximate distance oracle requires O(kn^1+1/k) space and answers a (2k-1)-approximate distance query in O(k) time. Thorup and Zwick showed that a (2k - 1)-approximate distance oracle can be computed in O(kmn^1/k) time, and posed the following question : Can (2k - 1)-approximate distance oracle be computed in Õ(n²) time?In this paper, we answer their question in affirmative for unweighted graphs. We present an algorithm that computes (2k -1)-approximate distance oracle for a given unweighted graph in Õ(n²) time. One of the new ideas used in the improved algorithm also leads to the first linear time algorithm for computing an optimal size (2, 1)-spanner of an unweighted graph.

References

[1]

S. Baswana and S. Sen. A simple linear time algorithm for computing a (2k - 1)-spanner of O(n1+1/k) size in weighted graphs. In Proceedings of the 30th International Colloquium on Automata, Languages and Programming (ICALP), pages 384--396, 2003.

Digital Library

Google Scholar

[2]

E. Cohen and U. Zwick. All-pairs small stretch paths. Journal of Algorithms, 38:335--353, 2001.

Digital Library

Google Scholar

[3]

D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation, 9:251--280, 1990.

Digital Library

Google Scholar

[4]

D. Dor, S. Halperin, and U. Zwick. All pairs almost shortest paths. Siam Journal on Computing, 29:1740--1759, 2000.

Digital Library

Google Scholar

[5]

M. Elkin. Computing almost shortest paths. In Proceedings of the 20th Annual ACM Symposium on Principles of Distributed Computing, pages 53--62, 2001.

Digital Library

Google Scholar

[6]

M. Elkin and D. Peleg. (1 +, ε, β)-spanner construction for general graphs. In Proceedings of 33rd ACM Symposium on Theory of Computing (STOC), pages 173--182, 2001.

Digital Library

Google Scholar

[7]

M. L. Fredman, J. Komlós, and E. Szemerédi. Storing a sparse table with o(1) worst case time. Journal of ACM, 31:538--544, 1984.

Digital Library

Google Scholar

[8]

S. Halperin and U. Zwick. Unpublished result. 1996.

Google Scholar

[9]

M. Thorup and U. Zwick. Approximate distance oracle. In Proceedings of 33rd ACM Symposium on Theory of Computing (STOC), pages 183--192, 2001.

Digital Library

Google Scholar

[10]

V. V. Vazirani. In Approximation Algorithms. Springer Verlag, 2001.

Digital Library

Google Scholar

[11]

U. Zwick. All-pairs shortest paths in weighted directed graphs - exact and almost exact algorithms. In Proceedings of the 39th Annual Symposium on Foundations of Computer Science (FOCS), pages 310--319, 1998.

Digital Library

Google Scholar

Cited By

View all

D'Andrea AD'Emidio MFrigioni DLeucci SProietti G(2015)Path-Fault-Tolerant Approximate Shortest-Path TreesPost-Proceedings of the 22nd International Colloquium on Structural Information and Communication Complexity - Volume 943910.1007/978-3-319-25258-2_16(224-238)Online publication date: 14-Jul-2015
https://dl.acm.org/doi/10.1007/978-3-319-25258-2_16
Woodruff D(2010)Additive spanners in nearly quadratic timeProceedings of the 37th international colloquium conference on Automata, languages and programming10.5555/1880918.1880970(463-474)Online publication date: 6-Jul-2010
https://dl.acm.org/doi/10.5555/1880918.1880970
Sen S(2009)Approximating Shortest Paths in GraphsProceedings of the 3rd International Workshop on Algorithms and Computation10.1007/978-3-642-00202-1_3(32-43)Online publication date: 18-Feb-2009
https://dl.acm.org/doi/10.1007/978-3-642-00202-1_3
Show More Cited By

Approximate distance oracles for unweighted graphs in Õ (n²) time
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
2. Theory of computation

Recommendations

Approximate distance oracles

Let G = (V,E) be an undirected weighted graph with |V| = n and |E| = m. Let k ≥ 1 be an integer. We show that G = (V,E) can be preprocessed in O(kmn^1/k) expected time, constructing a data structure of size O(kn^1+1/k), such that any subsequent distance ...
Approximate distance oracles for unweighted graphs in expected O(n²) time

Let G = (V, E) be an undirected graph on n vertices, and let δ(u, v) denote the distance in G between two vertices u and v. Thorup and Zwick showed that for any positive integer t, the graph G can be preprocessed to build a data structure that can ...
Approximate distance oracles with improved query time
SODA '13: Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms

Given an undirected graph G with m edges, n vertices, and non-negative edge weights, and given an integer k ≥ 2, we show that a (2k − 1)-approximate distance oracle for G of size O(kn^1+1/k) and with O(log k) query time can be constructed in O(min{kmn^1/k,...

Comments

Information & Contributors

Information

Published In

SODA '04: Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms

January 2004

1113 pages

ISBN:089871558X

Program Chair:
Ian Munro

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 11 January 2004

Check for updates

Qualifiers

Article

Acceptance Rates

Overall Acceptance Rate 411 of 1,322 submissions, 31%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

13
Total Citations
View Citations
403
Total Downloads

Downloads (Last 12 months)5
Downloads (Last 6 weeks)0

Reflects downloads up to 26 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

View all

D'Andrea AD'Emidio MFrigioni DLeucci SProietti G(2015)Path-Fault-Tolerant Approximate Shortest-Path TreesPost-Proceedings of the 22nd International Colloquium on Structural Information and Communication Complexity - Volume 943910.1007/978-3-319-25258-2_16(224-238)Online publication date: 14-Jul-2015
https://dl.acm.org/doi/10.1007/978-3-319-25258-2_16
Woodruff D(2010)Additive spanners in nearly quadratic timeProceedings of the 37th international colloquium conference on Automata, languages and programming10.5555/1880918.1880970(463-474)Online publication date: 6-Jul-2010
https://dl.acm.org/doi/10.5555/1880918.1880970
Sen S(2009)Approximating Shortest Paths in GraphsProceedings of the 3rd International Workshop on Algorithms and Computation10.1007/978-3-642-00202-1_3(32-43)Online publication date: 18-Feb-2009
https://dl.acm.org/doi/10.1007/978-3-642-00202-1_3
Gudmundsson JLevcopoulos CNarasimhan GSmid M(2008)Approximate distance oracles for geometric spannersACM Transactions on Algorithms10.1145/1328911.13289214:1(1-34)Online publication date: 28-Mar-2008
https://dl.acm.org/doi/10.1145/1328911.1328921
Derbel BGavoille C(2008)Fast deterministic distributed algorithms for sparse spannersTheoretical Computer Science10.1016/j.tcs.2008.02.019399:1-2(83-100)Online publication date: 1-Jun-2008
https://dl.acm.org/doi/10.1016/j.tcs.2008.02.019
Derungs JJacob RWidmayer P(2007)Approximate shortest paths guided by a small indexProceedings of the 10th international conference on Algorithms and Data Structures10.5555/2394893.2394959(553-564)Online publication date: 15-Aug-2007
https://dl.acm.org/doi/10.5555/2394893.2394959
Derbel BMosbah MZemmari A(2006)Fast distributed graph partition and applicationProceedings of the 20th international conference on Parallel and distributed processing10.5555/1898953.1899057(125-125)Online publication date: 25-Apr-2006
https://dl.acm.org/doi/10.5555/1898953.1899057
Derbel BGavoille C(2006)Fast deterministic distributed algorithms for sparse spannersProceedings of the 13th international conference on Structural Information and Communication Complexity10.1007/11780823_9(100-114)Online publication date: 2-Jul-2006
https://dl.acm.org/doi/10.1007/11780823_9
Baswana SKavitha TMehlhorn KPettie S(2005)New constructions of (α, β)-spanners and purely additive spannersProceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms10.5555/1070432.1070526(672-681)Online publication date: 23-Jan-2005
https://dl.acm.org/doi/10.5555/1070432.1070526
Thorup MZwick U(2005)Approximate distance oraclesJournal of the ACM10.1145/1044731.104473252:1(1-24)Online publication date: 1-Jan-2005
https://dl.acm.org/doi/10.1145/1044731.1044732
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Abstract

References

Cited By

Recommendations