Article

A new approach to dynamic all pairs shortest paths

Authors:

Camil Demetrescu,

Giuseppe F. ItalianoAuthors Info & Claims

STOC '03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing

Pages 159 - 166

https://doi.org/10.1145/780542.780567

Published: 09 June 2003 Publication History

Abstract

We study novel combinatorial properties of graphs that allow us to devise a completely new approach to dynamic all pairs shortest paths problems. Our approach yields a fully dynamic algorithm for general directed graphs with non-negative real-valued edge weights that supports any sequence of operations in Õ(n²) amortized time per update and unit worst-case time per distance query, where n is the number of vertices. We can also report shortest paths in optimal worst-case time. These bounds improve substantially over previous results and solve a long-standing open problem. Our algorithm is deterministic and uses simple data structures.

References

[1]

G. Ausiello, G. Italiano, A. Marchetti-Spaccamela, and U. Nanni. Incremental algorithms for minimal length paths. Journal of Algorithms, 12(4):615--38, 1991.

Digital Library

[2]

S. Baswana, R. Hariharan, and S. Sen. Improved decremental algorithms for transitive closure and all-pairs shortest paths. In Proc. 34th ACM Symposium on Theory of Computing (STOC'02), pages 117--123, 2002.

Digital Library

[3]

C. Demetrescu and G. Italiano. Fully dynamic all pairs shortest paths with real edge weights. In Proc. of the 42nd IEEE Annual Symposium on Foundations of Computer Science (FOCS'01), Las Vegas, Nevada, pages 260--267, 2001.

Digital Library

[4]

C. Demetrescu and G. Italiano. Improved bounds and new trade-offs for dynamic all pairs shortest paths. In Proc. of the 29-th International Colloquium on Automata, Languages, and Programming (ICALP'02), Malaga, Spain, 2002.

Digital Library

[5]

E. Dijkstra. A note on two problems in connection with graphs. Numerische Mathematik, 1:269--271, 1959.

Digital Library

[6]

S. Even and H. Gazit. Updating distances in dynamic graphs. Methods of Operations Research, 49:371--387, 1985.

[7]

J. Fakcharoemphol and S. Rao. Planar graphs, negative weight edges, shortest paths, and near linear time. In Proc. of the 42nd IEEE Annual Symposium on Foundations of Computer Science (FOCS'01), Las Vegas, Nevada, pages 232--241, 2001.

Digital Library

[8]

M. Fredman and R. Tarjan. Fibonacci heaps and their use in improved network optimization algorithms. Journal of the ACM, 34:596--615, 1987.

Digital Library

[9]

D. Frigioni, A. Marchetti-Spaccamela, and U. Nanni. Semi-dynamic algorithms for maintaining single source shortest paths trees. Algorithmica, 22(3):250--274, 1998.

[10]

D. Frigioni, A. Marchetti-Spaccamela, and U. Nanni. Fully dynamic algorithms for maintaining shortest paths trees. Journal of Algorithms, 34:351--381, 2000.

Digital Library

[11]

D. Greene and D. Knuth. Mathematics for the analysis of algorithms. Birkhäuser, 1982.

Digital Library

[12]

M. Henzinger, P. Klein, S. Rao, and S. Subramanian. Faster shortest-path algorithms for planar graphs. Journal of Computer and System Sciences, 55(1):3--23, Aug. 1997.

Digital Library

[13]

V. King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Proc. 40th IEEE Symposium on Foundations of Computer Science (FOCS'99), pages 81--99, 1999.

Digital Library

[14]

P. Loubal. A network evaluation procedure. Highway Research Record 205, pages 96--109, 1967.

[15]

J. Murchland. The effect of increasing or decreasing the length of a single arc on all shortest distances in a graph. Technical report, LBS-TNT-26, London Business School, Transport Network Theory Unit, London, UK, 1967.

[16]

G. Ramalingam and T. Reps. An incremental algorithm for a generalization of the shortest path problem. Journal of Algorithms, 21:267--305, 1996.

Digital Library

[17]

G. Ramalingam and T. Reps. On the computational complexity of dynamic graph problems. Theoretical Computer Science, 158:233--277, 1996.

Digital Library

[18]

V. Rodionov. The parametric problem of shortest distances. U.S.S.R. Computational Math. and Math. Phys., 8(5):336--343, 1968.

[19]

H. Rohnert. A dynamization of the all-pairs least cost problem. In Proc. 2nd Annual Symposium on Theoretical Aspects of Computer Science, (STACS'85), LNCS 182, pages 279--286, 1985.

Digital Library

Cited By

Mao XMohar BShinkar IO'Donnell R(2024)Fully Dynamic All-Pairs Shortest Paths: Likely Optimal Worst-Case Update TimeProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649695(1141-1152)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649695
Pitchanathan ACohen AZinenko OGrosser T(2024)Strided Difference Bound MatricesComputer Aided Verification10.1007/978-3-031-65627-9_14(279-302)Online publication date: 24-Jul-2024
https://dl.acm.org/doi/10.1007/978-3-031-65627-9_14
Sunita Garg D(2018)Dynamizing Dijkstra: A solution to dynamic shortest path problem through retroactive priority queueJournal of King Saud University - Computer and Information Sciences10.1016/j.jksuci.2018.03.003Online publication date: Mar-2018
https://doi.org/10.1016/j.jksuci.2018.03.003
Show More Cited By

Index Terms

A new approach to dynamic all pairs shortest paths
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
2. Theory of computation
  1. Randomness, geometry and discrete structures

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Published In

cover image ACM Conferences

STOC '03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing

June 2003

740 pages

ISBN:1581136749

DOI:10.1145/780542

Conference Chair:
Lawrence L. Larmore
University of Nevada Las Vegas, Las Vegas, NV
,
Program Chair:
Michel X. Goemans
Massachusetts Institute of Technology, Cambridge, MA

Copyright © 2003 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]

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Published: 09 June 2003

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STOC03: The 35th Annual ACM Symposium on Theory of Computing

June 9 - 11, 2003

CA, San Diego, USA

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STOC '03 Paper Acceptance Rate 80 of 270 submissions, 30%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Cited By

Mao XMohar BShinkar IO'Donnell R(2024)Fully Dynamic All-Pairs Shortest Paths: Likely Optimal Worst-Case Update TimeProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649695(1141-1152)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649695
Pitchanathan ACohen AZinenko OGrosser T(2024)Strided Difference Bound MatricesComputer Aided Verification10.1007/978-3-031-65627-9_14(279-302)Online publication date: 24-Jul-2024
https://dl.acm.org/doi/10.1007/978-3-031-65627-9_14
Sunita Garg D(2018)Dynamizing Dijkstra: A solution to dynamic shortest path problem through retroactive priority queueJournal of King Saud University - Computer and Information Sciences10.1016/j.jksuci.2018.03.003Online publication date: Mar-2018
https://doi.org/10.1016/j.jksuci.2018.03.003
Bhattacharya SHenzinger MNanongkai DKlein P(2017)Fully dynamic approximate maximum matching and minimum vertex cover in O(log3 n) worst case update timeProceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3039686.3039716(470-489)Online publication date: 16-Jan-2017
https://dl.acm.org/doi/10.5555/3039686.3039716
Nanongkai DSaranurak TWulff-Nilsen C(2017)Dynamic Minimum Spanning Forest with Subpolynomial Worst-Case Update Time2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2017.92(950-961)Online publication date: Oct-2017
https://doi.org/10.1109/FOCS.2017.92
Iwata YOka K(2014)Fast Dynamic Graph Algorithms for Parameterized ProblemsAlgorithm Theory – SWAT 201410.1007/978-3-319-08404-6_21(241-252)Online publication date: 2014
https://doi.org/10.1007/978-3-319-08404-6_21
Colin JCheikh ANakechbandi M(2013)Route planning in a weakly dynamic undirected graph2013 International Conference on Advanced Logistics and Transport10.1109/ICAdLT.2013.6568493(406-410)Online publication date: May-2013
https://doi.org/10.1109/ICAdLT.2013.6568493
Castronovo SKunz BMüller C(2013)Combining Uniform and Heuristic Search: Solving DSSSP with Restricted Knowledge of Graph TopologyAgents and Artificial Intelligence10.1007/978-3-642-36907-0_12(173-187)Online publication date: 2013
https://doi.org/10.1007/978-3-642-36907-0_12
Khopkar SNagi RNikolaev A(2012)An Efficient Map-Reduce Algorithm for the Incremental Computation of All-Pairs Shortest Paths in Social NetworksProceedings of the 2012 International Conference on Advances in Social Networks Analysis and Mining (ASONAM 2012)10.1109/ASONAM.2012.197(1144-1148)Online publication date: 26-Aug-2012
https://dl.acm.org/doi/10.1109/ASONAM.2012.197
Madry AMitzenmacher MSchulman L(2010)Faster approximation schemes for fractional multicommodity flow problems via dynamic graph algorithmsProceedings of the forty-second ACM symposium on Theory of computing10.1145/1806689.1806708(121-130)Online publication date: 5-Jun-2010
https://dl.acm.org/doi/10.1145/1806689.1806708
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