Article

Small hop-diameter sparse spanners for doubling metrics

Authors:

T-H. Hubert Chan,

Anupam GuptaAuthors Info & Claims

SODA '06: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm

Pages 70 - 78

Published: 22 January 2006 Publication History

Abstract

Given a metric M = (V, d), a graph G = (V, E) is a t-spanner for M if every pair of nodes in V has a "short" path (i.e., of length at most t times their actual distance) between them in the spanner. Furthermore, this spanner has a hop diameter bounded by D if every such short path also uses at most D edges. We consider the problem of constructing sparse (1 + ε)-spanners with small hop diameter for metrics of low doubling dimension.In this paper, we show that given any metric with constant doubling dimension k, and any 0 < ε < 1, one can find a (1 + ε)-spanner for the metric with nearly linear number of edges (i.e., only O(n log* n + nε^-O(k)) edges) and a constant hop diameter, and also a (1 + ε)-spanner with linear number of edges (i.e., only nε^-O(k) edges) which achieves a hop diameter that grows like the functional inverse of the Ackermann's function. Moreover, we prove that such tradeoffs between the number of edges and the hop diameter are asymptotically optimal.

References

[1]

Ittai Abraham and Dahlia Malkhi. Compact routing on euclidian metrics. In PODC, pages 141--149, 2004.

Digital Library

[2]

N. Alon and B. Schieber. Optimal preprocessing for answering on-line product queries. Tech. report 71/87, 1987.

[3]

Ingo Althöfer, Gautam Das, David Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discrete Comput. Geom., 9(1):81--100, 1993.

Digital Library

[4]

Sunil Arya, Gautam Das, David M. Mount, Jeffrey S. Salowe, and Michiel H. M. Smid. Euclidean spanners: short, thin, and lanky. STOC, pages 489--498, 1995.

Digital Library

[5]

Patrice Assouad. Plongements lipschitziens dans Rn. Bull. Soc. Math. France, 111(4):429--448, 1983.

[6]

Paul B. Callahan and S. Rao Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. Assoc. Comput. Mach., 42(1):67--90, 1995.

Digital Library

[7]

Hubert T-H. Chan, Anupam Gupta, Bruce M. Maggs, and Shuheng Zhou. On hierarchical routing in doubling metrics. Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 762--771, 2005.

Digital Library

[8]

Barun Chandra, Gautam Das, Giri Narasimhan, and José Soares. New sparseness results on graph spanners. Internat. J. Comput. Geom. Appl., 5(1-2):125--144, 1995. Eighth Annual ACM Symposium on Computational Geometry (Berlin, 1992).

Digital Library

[9]

Bernard Chazelle. Computing on a free tree via complexity-preserving mappings. In Algorithmica 2, pages 337--361, 1987.

[10]

K. L. Clarkson. Nearest neighbor queries in metric spaces. Discrete Comput. Geom., 22(1):63--93, 1999.

[11]

Anupam Gupta, Robert Krauthgamer, and James R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, pages 534--543, 2003.

Digital Library

[12]

Sariel Har-Peled and Manor Mendel. Fast construction of nets in low dimensional metrics, and their applications. Symposium on Computational Geometry, pages 150--158, 2005.

Digital Library

[13]

J. Heinonen. Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001.

[14]

David R. Karger and Matthias Ruhl. Finding nearest neighbors in growth-restricted metrics. In Proceedings of the 34th Annual ACM Symposium on the Theory of Computing, pages 63--66, 2002.

Digital Library

[15]

Robert Krauthgamer and James R. Lee. The black-box complexity of nearest neighbor search. In ICALP, pages 858--869, 2004.

[16]

Robert Krauthgamer and James R. Lee. Navigating nets: Simple algorithms for proximity search. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 791--801, 2004.

Digital Library

[17]

Vincent D. Park and M. Scott Corson. A highly adaptive distributed routing algorithm for mobile wireless networks. In INFOCOM, pages 1405--1413, 1997.

Digital Library

[18]

D. Peleg and A. Schäffer. Graph spanners. In J. Graph Theory 13, pages 99--116, 1989.

[19]

Charles E. Perkins and Elizabeth M. Belding-Royer. Ad-hoc on-demand distance vector routing. In WMCSA, pages 90--100, 1999.

Digital Library

[20]

Charles E. Perkins and Pravin Bhagwat. Highly dynamic destination-sequenced distance-vector routing (DSDV) for mobile computers. In SIGCOMM, pages 234--244, 1994.

Digital Library

[21]

J.S. Salowe. Constructing multidimensional spanner graphs. In Internat. J. Comput. Geom. Appl. 1, pages 99--107, 1991.

[22]

Kunal Talwar. Bypassing the embedding: Algorithms for low-dimensional metrics. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pages 281--290, 2004.

Digital Library

[23]

R. E. Tarjan. Efficiency of a good but not linear set union algorithm. Journal of ACM 22, pages 215--225, 1975.

Digital Library

[24]

P.M. Vaidya. A sparse graph almost as good as the complete graph on points in k dimensions. In Discrete Comput. Geom. 6, pages 369--381, 1991.

[25]

Andrew C. Yao. Space-time tradeoff for answering range queries. In Proceedings of the 14th Annual ACM Symposium on Theory of Computing, pages 128--136, 1982.

Digital Library

Cited By

Filtser ASolomon SGiakkoupis G(2016)The Greedy Spanner is Existentially OptimalProceedings of the 2016 ACM Symposium on Principles of Distributed Computing10.1145/2933057.2933114(9-17)Online publication date: 25-Jul-2016
https://dl.acm.org/doi/10.1145/2933057.2933114
Chan TGupta AMaggs BZhou S(2016)On Hierarchical Routing in Doubling MetricsACM Transactions on Algorithms10.1145/291518312:4(1-22)Online publication date: 16-Aug-2016
https://dl.acm.org/doi/10.1145/2915183
Elkin MSolomon S(2015)Optimal Euclidean SpannersJournal of the ACM10.1145/281900862:5(1-45)Online publication date: 2-Nov-2015
https://dl.acm.org/doi/10.1145/2819008
Show More Cited By

Index Terms

Small hop-diameter sparse spanners for doubling metrics
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms
    2. Graph theory
      1. Graph algorithms
      2. Paths and connectivity problems
2. Theory of computation
  1. Design and analysis of algorithms

Recommendations

Small Hop-diameter Sparse Spanners for Doubling Metrics

Given a metric M=(V,d), a graph G=(V,E) is a t-spanner for M if every pair of nodes in V has a "short" path (i.e., of length at most t times their actual distance) between them in the spanner. Furthermore, this spanner has a hop diameter bounded by D if ...
Sparse Euclidean Spanners with Tiny Diameter
Special Issue on SODA'11

In STOC’95, Arya et al. [1995] showed that for any set of n points in R^d, a (1 + ε)-spanner with diameter at most 2 (respectively, 3) and O(n log n) edges (respectively, O(n log log n) edges) can be built in O(n log n) time. Moreover, it was shown in ...
Sparse Fault-Tolerant Spanners for Doubling Metrics with Bounded Hop-Diameter or Degree

We study fault-tolerant spanners in doubling metrics. A subgraph H for a metric space X is called a k -vertex-fault-tolerant t -spanner (( k , t )-VFTS or simply k -VFTS), if for any subset S ⊆ X with | S |≤ k , it holds that d _{H S ...}

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

SODA '06: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm

January 2006

1261 pages

ISBN:0898716055

Sponsors

SIAM Activity Group on Discrete Mathematics
SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 22 January 2006

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

14
Total Citations
View Citations
201
Total Downloads

Downloads (Last 12 months)1
Downloads (Last 6 weeks)0

Reflects downloads up to 14 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Filtser ASolomon SGiakkoupis G(2016)The Greedy Spanner is Existentially OptimalProceedings of the 2016 ACM Symposium on Principles of Distributed Computing10.1145/2933057.2933114(9-17)Online publication date: 25-Jul-2016
https://dl.acm.org/doi/10.1145/2933057.2933114
Chan TGupta AMaggs BZhou S(2016)On Hierarchical Routing in Doubling MetricsACM Transactions on Algorithms10.1145/291518312:4(1-22)Online publication date: 16-Aug-2016
https://dl.acm.org/doi/10.1145/2915183
Elkin MSolomon S(2015)Optimal Euclidean SpannersJournal of the ACM10.1145/281900862:5(1-45)Online publication date: 2-Nov-2015
https://dl.acm.org/doi/10.1145/2819008
Solomon SShmoys D(2014)From hierarchical partitions to hierarchical coversProceedings of the forty-sixth annual ACM symposium on Theory of computing10.1145/2591796.2591864(363-372)Online publication date: 31-May-2014
https://dl.acm.org/doi/10.1145/2591796.2591864
Bartal YGottlieb LNeiman OCheng SDevillers O(2014)On the Impossibility of Dimension Reduction for Doubling Subsets of ℓpProceedings of the thirtieth annual symposium on Computational geometry10.1145/2582112.2582170(60-66)Online publication date: 8-Jun-2014
https://dl.acm.org/doi/10.1145/2582112.2582170
Gottlieb LSolomon SCheng SDevillers O(2014)Light spanners for Snowflake MetricsProceedings of the thirtieth annual symposium on Computational geometry10.1145/2582112.2582140(387-395)Online publication date: 8-Jun-2014
https://dl.acm.org/doi/10.1145/2582112.2582140
Elkin MSolomon SBoneh DRoughgarden TFeigenbaum J(2013)Optimal euclidean spannersProceedings of the forty-fifth annual ACM symposium on Theory of Computing10.1145/2488608.2488691(645-654)Online publication date: 1-Jun-2013
https://dl.acm.org/doi/10.1145/2488608.2488691
Solomon S(2013)Sparse Euclidean Spanners with Tiny DiameterACM Transactions on Algorithms10.1145/2483699.24837089:3(1-33)Online publication date: 1-Jun-2013
https://dl.acm.org/doi/10.1145/2483699.2483708
Solomon SRandall D(2011)An optimal-time construction of sparse Euclidean spanners with tiny diameterProceedings of the twenty-second annual ACM-SIAM symposium on Discrete algorithms10.5555/2133036.2133101(820-839)Online publication date: 23-Jan-2011
https://dl.acm.org/doi/10.5555/2133036.2133101
Dinitz YElkin MSolomon S(2010)Low-Light Trees, and Tight Lower Bounds for Euclidean SpannersDiscrete & Computational Geometry10.5555/3116254.311632343:4(736-783)Online publication date: 1-Jun-2010
https://dl.acm.org/doi/10.5555/3116254.3116323
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.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents