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Computing almost shortest paths

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Michael ElkinAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 1, Issue 2

Pages 283 - 323

https://doi.org/10.1145/1103963.1103968

Published: 01 October 2005 Publication History

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Abstract

We study the s-sources almost shortest paths (abbreviated s-ASP) problem. Given an unweighted graph G = (V,E), and a subset S &sube; V of s nodes, the goal is to compute almost shortest paths between all the pairs of nodes S × V. We devise an algorithm with running time O(&mid;E&mid;nρ + s · n1 + ζ) for this problem that computes the paths Pu,w for all pairs (u,w) ∈ S × V such that the length of Pu,w is at most (1 + &epsi;) dG(u,w) + β(ζ,ρ,&epsi;), and β(ζ,ρ,&epsi;) is constant when ζ, ρ, and &epsi; are arbitrarily small constants.

We also devise a distributed protocol for the s-ASP problem that computes the paths P<inf>u,w</inf> as above, and has time and communication complexities of O(s · Diam(G) + n1 + ζ/2) (respectively, O(s · Diam(G) log3 n + n1 + ζ/2 log n)) and O(&mid;E&mid; nρ + s · n1 + ζ) (respectively, O(&mid;E&mid; nρ + s · n1 + ζ + n1 + ρ + ζ(ρ − ζ/2)/2)) in the synchronous (respectively asynchronous) setting.

Our sequential algorithm, as well as the distributed protocol, is based on a novel algorithm for constructing (1 + &epsi;, β(ζ,ρ, &epsi;))-spanners of size O(n1 + ζ), developed in this article. This algorithm has running time of O(&mid;E&mid; nρ), which is significantly faster than the previously known algorithm given in Elkin and Peleg [2001], whose running time is Õ(n2 + ρ). We also develop the first distributed protocol for constructing (1 + &epsi;,β)-spanners. The communication complexity of this protocol is near optimal.

References

[1]

Awerbuch, B., Berger, B., Cowen, L., and Peleg, D. 1998. Near-linear time construction of sparse neighborhood covers. SIAM J. Comput. 28, 1, 263--277.

Digital Library

Google Scholar

[2]

Awerbuch, B., Baratz, A., and Peleg, D. 1991. Efficient broadcast and light-weight spanners. Unpublished manuscript.

Google Scholar

[3]

Aingworth, D., Chekuri, C., Indyk, P., and Motwani, R. 1996. Fast estimation of diameter and shortest paths (without matrix multiplication). In Proceedings of the 7th ACM-SIAM Symposium on Discrete Algorithms (Atlanta, GA). 547--553.

Digital Library

Google Scholar

[4]

Althöfer, I., Das, G., Dobkin, D., and Joseph, D. 1990. Generating sparse spanners for weighted graphs. In Proceedings of the 2nd Scandinavian Workshop on Algorithm Theory. Lecture Notes in Computer Science, vol. 447. Springer-Verlag, New York, NY/Berlin, Germany, 26--37.

Digital Library

Google Scholar

[5]

Awerbuch, B., and Gallagher, G. 1987. A new distributed algorithm to find breadth first search trees. IEEE Trans. Inform. Theory. IT-33, 3, 315--322.

Digital Library

Google Scholar

[6]

Alon, N., Galil, Z., and Margalit, O. 1997. On the exponent of the all pairs shortest paths problem. J. Comput Syst. Sci. 54, 255--262.

Digital Library

Google Scholar

[7]

Alon, N., Galil, Z., Margalit, O., and Naor, M. 1992. Witnesses for Boolean matrix multiplication and for shortest paths. In Proceeedings of the 33rd IEEE Symposium on Foundations of Computer Science (Pittsburgh, PA). 417--426.

Google Scholar

[8]

Awerbuch, B., and Peleg, D. 1990. Network synchronization with polylogarithmic overhead. In Proceedings of the 31st Symposium on Foundations of Computer Science. 514--522.

Google Scholar

[9]

Afek, Y., and Ricklin, M. 1992. A paradigm for running distributed algorithms. In Proceedings of the 6th Workshop on Distributed Algorithms. Lecture Notes in Computer Science, vol. 647. Springer, New York, NY/Berlin, Germany, 1--10.

Digital Library

Google Scholar

[10]

Awerbuch, B. 1985a. Complexity of network synchronization. J. ACM 4, 804--823.

Digital Library

Google Scholar

[11]

Awerbuch, B. 1985b. Reducing complexities of the distributed max-flow and the breadth-first-search algorithms by means of network synchronization. Networks 15, 425--437.

Crossref

Google Scholar

[12]

Awerbuch, B. 1989. Distributed shortest paths algorithms. In Proceedings of the 21st ACM Symposium on Theory of Computing. 230--240.

Digital Library

Google Scholar

[13]

Bollobas, B., Coppersmith, D., and Elkin, M. L. 2003. Sparse distance preservers and additive spanners. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA'03, Baltimore, MD, January.)

Digital Library

Google Scholar

[14]

Baswana, S., and Sen, S. 2003. A simple linear time algorithm for computing a (2k − 1)-spanner of O(n1+1/k) size in weighted graphs. In Proceedings of the International Colloquium on Automata, Languages and Computing. 284--296.

Digital Library

Google Scholar

[15]

Chandra, B., Das, G., Narasimhan, G., and Soares, J. 1992. New sparseness results on graph spanners. In Proceedings of the 8th ACM Symposium on Computational Geometry. 192--201.

Digital Library

Google Scholar

[16]

Chew, L. P. 1986. There is a planar graph almost as good as complete graph. In Proceedings of the 2nd Symposium on Computational Geometry. 169--177.

Digital Library

Google Scholar

[17]

Cohen, E. 1993. Fast algorithms for constructing t-spanners and paths of stretch t. In Proceedings of the 34th IEEE Symposium on Foundations of Computer Science. IEEE Press, Piscataway, NJ, 648--658.

Digital Library

Google Scholar

[18]

Cohen, E. 1994. Polylog-time and near-linear work approximation scheme for undirected shortest paths. In Proceedings of the 26th ACM Symposium on Theory of Computation. 16--26.

Digital Library

Google Scholar

[19]

Dor, D., Halperin, S., and Zwick, U. 2000. All pairs almost shortest paths. SIAM J. Comput. 29, 1740--1759.

Digital Library

Google Scholar

[20]

Dobkin, D. P., Friedman, S. J., and Supowit, K. J. 1987. Delaunay graphs are almost as good as complete graphs. In Proceedings of the 28th IEEE Symposium on Foundations of Computer Science. 20--26.

Google Scholar

[21]

Elkin, M. 2001. Computing almost shortest paths, In Proceedings of the 20th ACM Symposium on Principles of Distributed Computing (Newport, RI, August). 53--63.

Digital Library

Google Scholar

[22]

Elkin, M., and Peleg, D. 2001. (1 + &epsi;,β)-spanner constructions for general graphs. In Proceedings of the 33rd ACM Symposium on Theory of Computing (Crete, Greece, July). Full version is to appear in SIAM Journal of Computing.

Digital Library

Google Scholar

[23]

Elkin, M., and Zhang, J. 2004. Efficient algorithms for constructing (1 + &epsi;,β)-spanners in the distributed and streaming models. In Proceedings of the 23rd ACM Symposium on Principles of Distributed Computing (St. John's, Newfoundland, Canada, July).

Digital Library

Google Scholar

[24]

Frederickson, G. N. 1985. Trade-offs for selection in distributed networks. In Proceedings of the 2nd Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 182. Springer, Berlin, Germany, 143--150.

Digital Library

Google Scholar

[25]

Gallagher, R. G. 1982. Distributed minimum hop algorithms. Tech. rep. LIDS-P-175. Laboratory for Information and Decision Systems, MIT, Cambridge, MA.

Google Scholar

[26]

Galil, Z., and Margalit, O. 1993. Witnesses for Boolean matrix multiplication. J. Complex. 9, 201--221.

Digital Library

Google Scholar

[27]

Galil, Z., and Margalit, O. 1997. All pairs shortest paths for graphs with small integer length edges. J. Comput. Syst. Sci. 54, 243--254.

Digital Library

Google Scholar

[28]

Kortsarz, G. 2001. On the hardness of approximating spanners. Algorithmica 30, 3, 432--450.

Crossref

Google Scholar

[29]

Peleg, D. 2000. Distributed Computing: A Locality-Sensitive Approach. SIAM, Press, Philadelphia, PA.

Crossref

Google Scholar

[30]

Peleg, D., and Schäffer, A. 1989. Graph spanners, J. Graph Theor. 13, 99--116.

Crossref

Google Scholar

[31]

Peleg, D., and Ullman, J. D. 1989. An optimal synchronizer for the hypercube. SIAM J. Comput. 18 740--747.

Digital Library

Google Scholar

[32]

Roditty, L., and Zwick, U. 2004. Dynamic approximate all-pairs shortest paths in undirected graphs. In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science (Rome, Italy, October).

Digital Library

Google Scholar

[33]

Roditty, L., Thorup, M., and Zwick, U. 2002. Roundtrip spanners and roundtrip routing in directed graphs. Proceedings of the Symposium on Discrete Algorithms. 844--851.

Digital Library

Google Scholar

[34]

Seidel, R. 1995. On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Syst. Sci. 51, 400--403.

Digital Library

Google Scholar

[35]

Segall, A. 1983. Distributed network protocols. IEEE Trans. Inform. Theor. IT-29, 1 (Jan.), 23--34.

Crossref

Google Scholar

[36]

Thorup, M. 2001. Compact oracles for reachability and approximate distances in planar digraphs. Proceedings of the Symposium on Foundations of Computer Science. 242--251.

Digital Library

Google Scholar

[37]

Thorup, M., and Zwick, U. 2001a. Compact routing schemes. In Proceedings of the Symposium on Parallel Algorithms and Architecture. 1--10.

Digital Library

Google Scholar

[38]

Thorup, M., and Zwick, U. 2001b. Approximate distance oracles. In Proceedings of the Symposium on Theory of Computing. 183--192.

Digital Library

Google Scholar

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Roditty LSaha BServedio R(2023)New Algorithms for All Pairs Approximate Shortest PathsProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585197(309-320)Online publication date: 2-Jun-2023
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Bezdrighin MElkin MGhaffari MGrunau CHaeupler BIlchi SRozhoň VAgrawal KLee I(2022)Deterministic Distributed Sparse and Ultra-Sparse Spanners and Connectivity CertificatesProceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3490148.3538565(1-10)Online publication date: 11-Jul-2022
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Filtser ANeiman O(2022)Light Spanners for High Dimensional Norms via Stochastic DecompositionsAlgorithmica10.1007/s00453-022-00994-084:10(2987-3007)Online publication date: 1-Oct-2022
https://dl.acm.org/doi/10.1007/s00453-022-00994-0
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Index Terms

Computing almost shortest paths
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
    2. Graph theory
2. Theory of computation
  1. Design and analysis of algorithms

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Reviews

Reviewer: Burkhard Englert

Computing the shortest paths between all vertices in a graph is a very important problem. It has many applications and can be used to solve problems like network routing, where the goal is to find the shortest path for data packets to take through a switching network. It is also used in more general search algorithms for a variety of problems, ranging from automated circuit layout to speech recognition. Algorithms that solve this problem generally involve nontrivial matrix multiplication, and as a result are impractical since their time complexities involve huge constants. Moreover, the currently best-known time complexity of algorithms of this type is O(n ^ 2.376), where n is the number of vertices in the graph. Often, approximations are cheaper to compute than exact values. In the s-sources almost shortest path (s-ASP) problem, given an unweighted, undirected graph G=(V, E) and a subset S of V of s sources, the goal is to compute paths between all the pairs of nodes (u, w) in S x V. The length of the computed path must be close to the actual distance d(u,w) between the nodes u and w in G, such that the length returned by the algorithm is at most α · d(u,w) + β. Here, α is called the multiplicative term of the error, while β is the additive term. This paper provides an important, new, and more efficient solution to this problem. It begins with a description of the problem, followed by a comparison of the new solution with previous results and a summary. The second and third sections contain the actual technical contributions of the paper. Both begin with overviews of the subsequent constructions. This significantly helps the reader in understanding the protocols. In the second section, a solution of the sequential s-ASP and the all pairs almost shortest path (APASP) problem is presented. The new algorithm has a running time of O(|E| · n ^ __?__ + s · n ^(1+ ζ)) with a multiplicative error of 1+ ε and an additive error of β(ζ,__?__,ε), where β(ζ,__?__,ε) is a constant whenever 0 < ζ,__?__,ε < 1 for arbitrarily small constants ζ,__?__ and ε. Then, in the third part, the first distributed protocol for constructing (1+ε, β) spanners is presented. This protocol has a near optimal communication complexity, and is used to provide a distributed protocol for computing almost shortest paths. Both the second and third parts conclude with an extension of the respective results to weighted graphs. This is by no means an introductory paper about the construction of almost shortest paths. The author uses a multitude of appropriate notations that require the reader's full attention. In the end, however, the reader's patience will be rewarded with a very insightful presentation. Online Computing Reviews Service

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cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 1, Issue 2

October 2005

190 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/1103963

Issue’s Table of Contents

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Publication History

Published: 01 October 2005

Published in TALG Volume 1, Issue 2

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Cited By

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Roditty LSaha BServedio R(2023)New Algorithms for All Pairs Approximate Shortest PathsProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585197(309-320)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585197
Bezdrighin MElkin MGhaffari MGrunau CHaeupler BIlchi SRozhoň VAgrawal KLee I(2022)Deterministic Distributed Sparse and Ultra-Sparse Spanners and Connectivity CertificatesProceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3490148.3538565(1-10)Online publication date: 11-Jul-2022
https://dl.acm.org/doi/10.1145/3490148.3538565
Filtser ANeiman O(2022)Light Spanners for High Dimensional Norms via Stochastic DecompositionsAlgorithmica10.1007/s00453-022-00994-084:10(2987-3007)Online publication date: 1-Oct-2022
https://dl.acm.org/doi/10.1007/s00453-022-00994-0
Dory MFischer OKhoury SLeitersdorf DMiller ACensor-Hillel K(2021)Constant-Round Spanners and Shortest Paths in Congested Clique and MPCProceedings of the 2021 ACM Symposium on Principles of Distributed Computing10.1145/3465084.3467928(223-233)Online publication date: 21-Jul-2021
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Censor-Hillel KDory M(2021)Distributed Spanner ApproximationSIAM Journal on Computing10.1137/20M131263050:3(1103-1147)Online publication date: 22-Jun-2021
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Filtser ASolomon S(2020)The Greedy Spanner Is Existentially OptimalSIAM Journal on Computing10.1137/18M121067849:2(429-447)Online publication date: 9-Apr-2020
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Censor-Hillel KPaz ARavid N(2020)The sparsest additive spanner via multiple weighted BFS treesTheoretical Computer Science10.1016/j.tcs.2020.05.035840(33-44)Online publication date: Nov-2020
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Ahmed RBodwin GSahneh FHamm KLatifi Jebelli MKobourov SSpence R(2020)Graph spanners: A tutorial reviewComputer Science Review10.1016/j.cosrev.2020.10025337(100253)Online publication date: Aug-2020
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Ahmed RBodwin GDarabi Sahneh FKobourov SSpence R(2020)Weighted Additive SpannersGraph-Theoretic Concepts in Computer Science10.1007/978-3-030-60440-0_32(401-413)Online publication date: 24-Jun-2020
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