research-article

Fast computation of empirically tight bounds for the diameter of massive graphs

Authors:

Clémence Magnien,

Matthieu Latapy,

Michel HabibAuthors Info & Claims

Journal of Experimental Algorithmics (JEA), Volume 13

Article No.: 10, Pages 1.10 - 1.9

https://doi.org/10.1145/1412228.1455266

Published: 23 February 2009 Publication History

Abstract

The diameter of a graph is among its most basic parameters. Since a few years ago, it moreover became a key issue to compute it for massive graphs in the context of complex network analysis. However, known algorithms, including the ones producing approximate values, have too high a time and/or space complexity to be used in such cases. We propose here a new approach relying on very simple and fast algorithms that compute (upper and lower) bounds for the diameter. We show empirically that, on various real-world cases representative of complex networks studied in the literature, the obtained bounds are very tight (and even equal in some cases). This leads to rigorous and very accurate estimations of the actual diameter in cases which were previously untractable in practice.

References

[1]

Aingworth, Chekuri, and Motwani. 1996. Fast estimation of diameter and shortest paths (without matrix multiplication). In ACM-SIAM SODA.

Digital Library

[2]

Alon, N., Galil, Z., Margalit, O., and Naor, M. 1992. Witnesses for boolean matrix multiplication and for shortest paths. In IEEE FOCS.

Digital Library

[3]

Broder, A., Kumar, S., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomkins, A., and Wiener, J. L. 2000. Graph structure in the web. Computer Networks 33.

Digital Library

[4]

Chan, T. M. 2006. All-pairs shortest paths for unweighted undirected graphs in o(mn) time. In Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm. 514--523.

Digital Library

[5]

Chepoi, V. D. and Dragan, F. F. 1994. Linear-time algorithm for finding a central vertex of a chordal graph. In Proceedings of the second Annual European Symposium Algorithms, ESA'94, LNCS. Vol. 855.

Digital Library

[6]

Corneil, D., Dragan, F., Habib, M., and Paul, C. 2001. Diameter determination on restricted graph families. Discrete Applied Mathematics 113, 2-3.

Digital Library

[7]

Corneil, D., Dragan, F., and Köhler, E. 2003. On the power of bfs to determine a graph's diameter. Networks 42 (4).

[8]

Dor, D., Halperin, S., and Zwick, U. 1997. All pairs almost shortest paths. ECCC 4, 040.

[9]

Dragan, F., Nicolai, F., and Brandstädt, A. 1997. Lexbfs-orderings and powers of graphs. In Proceedings of the 22nd international workshop. Graph-Theoretic Concepts in Computer Science, WG'96, LNCS. Vol. 1197.

Digital Library

[10]

Dragan, F. F. 2005. Estimating all pairs shortest paths in restricted graph families: a unified approach. J. Algorithms 57, 1, 1--21.

Digital Library

[11]

Faloutsos, M., Faloutsos, P., and Faloutsos, C. 1999. On power-law relationships of the internet topology. In ACM SIGCOMM.

Digital Library

[12]

Feder, T. and Motwani, R. 1991. Clique partitions, graph compression, and speeding-up algorithms. In ACM STOC.

Digital Library

[13]

Handler, G. 1973. Minimax location of a facility in an undirected tree graph. Transportation Science 7 (3).

[14]

Latapy, M. and Magnien, C. 2008. Complex network measurements: Estimating the relevance of observed properties. To appear in the proceedings of ieee infocom.

[15]

Leskovec, J., Kleinberg, J., and Faloutsos, C. 2005. Graphs over time: Densification laws, shrinking diameters and possible explanations. In ACM SIGKDD.

Digital Library

[16]

Parnas, M. and Ron, D. 2002. Testing the diameter of graphs. Random Struct. Algorithms 20, 2, 165--183.

Digital Library

[17]

Seidel, R. 1992. On the all-pairs-shortest-path problem. In ACM STOC.

Digital Library

[18]

Watts, D. and Strogatz, S. 1998. Collective dynamics of small-world networks. Nature 393.

[19]

Zwick, U. 2001. Exact and approximate distances in graphs—A survey. In ESA. Vol. 2161.

Digital Library

Cited By

Bläsius TFischbeck P(2024)On the External Validity of Average-case Analyses of Graph AlgorithmsACM Transactions on Algorithms10.1145/363377820:1(1-42)Online publication date: 22-Jan-2024
https://dl.acm.org/doi/10.1145/3633778
Adriaens FGionis A(2022)Diameter Minimization by Shortcutting with Degree Constraints2022 IEEE International Conference on Data Mining (ICDM)10.1109/ICDM54844.2022.00095(843-848)Online publication date: Nov-2022
https://doi.org/10.1109/ICDM54844.2022.00095
Dalirrooyfard MLi RWilliams V(2022)Hardness of Approximate Diameter: Now for Undirected Graphs2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00102(1021-1032)Online publication date: Feb-2022
https://doi.org/10.1109/FOCS52979.2021.00102
Show More Cited By

Index Terms

Fast computation of empirically tight bounds for the diameter of massive graphs
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory

Recommendations

Tight conditional lower bounds for approximating diameter in directed graphs
STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

Among the most fundamental graph parameters is the Diameter, the largest distance between any pair of vertices in a graph. Computing the Diameter of a graph with m edges requires m^2−o(1) time under the Strong Exponential Time Hypothesis (SETH), which ...
A diameter-revealing proof of the Bondy-Lovász lemma
Highlights
- The Bondy-Lovász lemma is strengthened with an upper bound on the graph diameter.
Abstract
We present a strengthened version of a lemma due to Bondy and Lovász. This lemma establishes the connectivity of a certain graph whose nodes correspond to the spanning trees of a 2-vertex-connected graph, and implies the k = 2 case of ...
Bounds on the fault-diameter of graphs

Let G be a k+1-connected or k+1-edge-connected graph, where k∈ï ź. The k-fault-diameter and k-edge-fault-diameter of G is the largest diameter of the subgraphs obtained from G by removing up to k vertices and edges, respectively. In this paper we give ...

Comments

Information & Contributors

Information

Published In

cover image ACM Journal of Experimental Algorithmics

ACM Journal of Experimental Algorithmics Volume 13, Issue

2009

482 pages

ISSN:1084-6654

EISSN:1084-6654

DOI:10.1145/1412228

Issue’s Table of Contents

Copyright © 2009 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 23 February 2009

Published in JEA Volume 13

Author Tags

Qualifiers

Research-article
Research
Refereed

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

65
Total Citations
View Citations
506
Total Downloads

Downloads (Last 12 months)15
Downloads (Last 6 weeks)4

Reflects downloads up to 10 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Bläsius TFischbeck P(2024)On the External Validity of Average-case Analyses of Graph AlgorithmsACM Transactions on Algorithms10.1145/363377820:1(1-42)Online publication date: 22-Jan-2024
https://dl.acm.org/doi/10.1145/3633778
Adriaens FGionis A(2022)Diameter Minimization by Shortcutting with Degree Constraints2022 IEEE International Conference on Data Mining (ICDM)10.1109/ICDM54844.2022.00095(843-848)Online publication date: Nov-2022
https://doi.org/10.1109/ICDM54844.2022.00095
Dalirrooyfard MLi RWilliams V(2022)Hardness of Approximate Diameter: Now for Undirected Graphs2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00102(1021-1032)Online publication date: Feb-2022
https://doi.org/10.1109/FOCS52979.2021.00102
Pellizzoni PPietracaprina APucci G(2022)Adaptive k-center and diameter estimation in sliding windowsInternational Journal of Data Science and Analytics10.1007/s41060-022-00318-z14:2(155-173)Online publication date: 2-Apr-2022
https://doi.org/10.1007/s41060-022-00318-z
Crescenzi PMarino A(2022)Degrees of Separation and Diameter in Large GraphsEncyclopedia of Big Data Technologies10.1007/978-3-319-63962-8_59-2(1-7)Online publication date: 15-Jun-2022
https://doi.org/10.1007/978-3-319-63962-8_59-2
Dalirrooyfard MWein NKhuller SVassilevska Williams V(2021)Tight conditional lower bounds for approximating diameter in directed graphsProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451130(1697-1710)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3451130
Yang YEmer JSanchez DMartínez JDuato JJohn L(2021)SpZipProceedings of the 48th Annual International Symposium on Computer Architecture10.1109/ISCA52012.2021.00087(1069-1082)Online publication date: 14-Jun-2021
https://dl.acm.org/doi/10.1109/ISCA52012.2021.00087
Ceccarello MPietracaprina APucci GUpfal E(2020)Distributed Graph Diameter ApproximationAlgorithms10.3390/a1309021613:9(216)Online publication date: 1-Sep-2020
https://doi.org/10.3390/a13090216
Huang BLiu ZWu K(2020)Structure Preserved Graph Reordering for Fast Graph Processing Without the Pain2020 IEEE 22nd International Conference on High Performance Computing and Communications; IEEE 18th International Conference on Smart City; IEEE 6th International Conference on Data Science and Systems (HPCC/SmartCity/DSS)10.1109/HPCC-SmartCity-DSS50907.2020.00007(44-51)Online publication date: Dec-2020
https://doi.org/10.1109/HPCC-SmartCity-DSS50907.2020.00007
Faldu PDiamond JGrot B(2020)Domain-Specialized Cache Management for Graph Analytics2020 IEEE International Symposium on High Performance Computer Architecture (HPCA)10.1109/HPCA47549.2020.00028(234-248)Online publication date: Feb-2020
https://doi.org/10.1109/HPCA47549.2020.00028
Show More Cited By

View Options

Get Access

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 Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Issue’s Table of Contents