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Work-Efficient Batch-Incremental Minimum Spanning Trees with Applications to the Sliding-Window Model

Authors:

Daniel Anderson,

Guy E. Blelloch,

Kanat TangwongsanAuthors Info & Claims

SPAA '20: Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures

Pages 51 - 61

https://doi.org/10.1145/3350755.3400241

Published: 09 July 2020 Publication History

Abstract

Algorithms for dynamically maintaining minimum spanning trees (MSTs) have received much attention in both the parallel and sequential settings. While previous work has given optimal algorithms for dense graphs, all existing parallel batch-dynamic algorithms perform polynomial work per update in the worst case for sparse graphs. In this paper, we present the first work-efficient parallel batch-dynamic algorithm for incremental MST, which can insert l edges in O(l log(1+n/l) work in expectation and O(polylog(n)) span w.h.p. The key ingredient of our algorithm is an algorithm for constructing a compressed path tree of an edge-weighted tree, which is a smaller tree that contains all pairwise heaviest edges between a given set of marked vertices. Using our batch-incremental MST algorithm, we demonstrate a range of applications that become efficiently solvable in parallel in the sliding-window model, such as graph connectivity, approximate MSTs, testing bipartiteness, k-certificates, cycle-freeness, and maintaining sparsifiers.

References

[1]

Umut A Acar, Daniel Anderson, Guy E Blelloch, and Laxman Dhulipala. 2019. Parallel Batch-Dynamic Graph Connectivity. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA).

[2]

Umut A Acar, Daniel Anderson, Guy E Blelloch, Laxman Dhulipala, and Sam Westrick. 2020. Parallel Batch-dynamic Trees via Change Propagation. arXiv preprint arXiv:2002.05129 [cs.DS] (2020).

[3]

Umut A Acar, Guy E Blelloch, and Jorge L Vittes. 2005. An experimental analysis of change propagation in dynamic trees. In Algorithm Engineering and Experiments (ALENEX).

[4]

Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. 2012. Analyzing graph structure via linear measurements. In ACM-SIAM Symposium on Discrete Algorithms(SODA).

[5]

Stephen Alstrup, Jacob Holm, Kristian De Lichtenberg, and Mikkel Thorup. 2005. Maintaining Information in Fully Dynamic Trees with Top Trees. ACM Trans. Algorithms 1, 2 (2005), 243--264.

Digital Library

[6]

András A. Benczúr and David R. Karger. 1996. Approximating s-t Minimum Cuts in Õ(n2) Time. In ACM Symposium on Theory of Computing (STOC).

[7]

Guy E. Blelloch. 1996. Programming Parallel Algorithms. Commun. ACM 39, 3(1996), 85--97.

Digital Library

[8]

Guy E. Blelloch, Daniel Ferizovic, and Yihan Sun. 2016. Just Join for Parallel Ordered Sets. In ACM Symposium on Parallelism in Algorithms and Architectures(SPAA).

[9]

Guy E. Blelloch and Margaret Reid-Miller. 1998. Fast Set Operations Using Treaps. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA).

[10]

Otakar Boruvka. 1926. O jistém problému minimálním. Práce Mor. Prirodved. Spol. v Brne (Acta Societ. Scienc. Natur. Moravicae)3, 3 (1926), 37--58.

[11]

Bernard Chazelle, Ronitt Rubinfeld, and Luca Trevisan. 2005. Approximating the Minimum Spanning Tree Weight in Sublinear Time. SIAM J. Comput.34, 6(2005), 1370--1379.

Digital Library

[12]

Richard Cole, Philip N. Klein, and Robert E. Tarjan. 1996. Finding Minimum Spanning Forests in Logarithmic Time and Linear Work Using Random Sampling. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA).

[13]

Michael S Crouch, Andrew McGregor, and Daniel Stubbs. 2013. Dynamic graphs in the sliding-window model. In European Symposium on Algorithms (ESA).

[14]

Sajal K Das and Paolo Ferragina. 1994. An o(n) work EREW parallel algorithm for updating MST. In European Symposium on Algorithms (ESA).

[15]

Sajal K Das and Paolo Ferragina. 1995. Parallel Dynamic Algorithms for Minimum Spanning Trees. Citeseer. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.24.136&rep=rep1&type=pdf.

[16]

Sajal K Das and Paolo Ferragina. 1999. An EREW PRAM algorithm for updating minimum spanning trees.Parallel Process. Lett. 9, 01 (1999), 111--122.

[17]

Mayur. Datar, Aristides. Gionis, Piotr. Indyk, and Rajeev. Motwani. 2002. Maintaining Stream Statistics over Sliding Windows. SIAM J. on Computing 31, 6(2002), 1794--1813.

Digital Library

[18]

Edsger W Dijkstra et al.1959. A note on two problems in connexion with graphs. Numerische mathematik 1, 1 (1959), 269--271.

[19]

David Eppstein, Zvi Galil, Giuseppe F Italiano, and Amnon Nissenzweig. 1997. Sparsification-a technique for speeding up dynamic graph algorithms. J. ACM 44, 5 (1997), 669--696.

Digital Library

[20]

Paolo Ferragina. 1995. An EREW PRAM fully-dynamic algorithm for MST. In International Parallel Processing Symposium (IPPS).

[21]

Paolo Ferragina. 1995. A technique to speed up parallel fully dynamic algorithms for MST. J. Parallel Distrib. Comput. 31, 2 (1995), 181--189.

Digital Library

[22]

Paolo Ferragina and Fabrizio Luccio. 1994. Batch dynamic algorithms for two graph problems. In International Conference on Parallel Architectures and Languages Europe (PARLE).

[23]

Paolo Ferragina and Fabrizio Luccio. 1996. Three techniques for parallel maintenance of a minimum spanning tree under batch of updates. Parallel Process. Lett. 6, 02 (1996), 213--222.

[24]

Greg N Frederickson. 1985. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. on Computing 14, 4 (1985), 781--798.

Digital Library

[25]

Wai Shing Fung, Ramesh Hariharan, Nicholas J. A. Harvey, and Debmalya Panigrahi. 2019. A General Framework for Graph Sparsification. SIAM J. Comput. 48, 4 (2019), 1196--1223.

Digital Library

[26]

Hillel Gazit. 1991. An Optimal Randomized Parallel Algorithm for Finding Connected Components in a Graph.SIAM J. Comput. 20, 6 (1991), 1046--1067.

Digital Library

[27]

Barbara Geissmann and Lukas Gianinazzi. 2018. Parallel Minimum Cuts in Near-linear Work and Low Depth. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA).

[28]

Mohsen Ghaffari, Krzysztof Nowicki, and Mikkel Thorup. 2020. Faster Algorithms for Edge Connectivity via Random 2-Out Contractions. In ACM-SIAM Symposium on Discrete Algorithms (SODA).

[29]

Ashish Goel, Michael Kapralov, and Ian Post. 2012. Single pass sparsification in the streaming model with edge deletions.arXiv preprint arXiv:1203.4900 [cs.DS](2012).

[30]

Gramoz Goranci, Monika Henzinger, and Mikkel Thorup. 2018. Incremental Exact Min-Cut in Polylogarithmic Amortized Update Time. ACM Trans. Algorithms14,2 (2018), 17:1--17:21.

[31]

Yan Gu, Julian Shun, Yihan Sun, and Guy E. Blelloch. 2015. A Top-Down Parallel Semisort. In ACM Symposium on Parallelism in Algorithms and Architectures(SPAA).

[32]

Jacob Holm, Kristian de Lichtenberg, and Mikkel Thorup. 2001. Poly-Logarithmic Deterministic Fully-Dynamic Algorithms for Connectivity, Minimum Spanning Tree, 2-Edge, and Biconnectivity. J. ACM 48, 4 (2001), 723--760.

Digital Library

[33]

Jacob Holm, Eva Rotenberg, and Christian Wulff-Nilsen. 2015. Faster Fully-Dynamic Minimum Spanning Forest. In European Symposium on Algorithms(ESA).

[34]

Joseph JáJá. 1992. An introduction to parallel algorithms. Vol. 17. Addison-Wesley Reading.

Digital Library

[35]

Vojtech Jarník. 1930.O jistém problému minimálním. Práca Moravské Prírodovedecké Spolecnosti 6 (1930), 57--63.

[36]

Donald B Johnson and Panagiotis Metaxas. 1992. Optimal algorithms for the vertex updating problem of a minimum spanning tree. In International Parallel Processing Symposium (IPPS).

Digital Library

[37]

Hermann Jung and Kurt Mehlhorn. 1988. Parallel algorithms for computing maximal independent sets in trees and for updating minimum spanning trees.Inform. Process. Lett. 27, 5 (1988), 227--236.

Digital Library

[38]

David R. Karger, Philip N. Klein, and Robert E. Tarjan. 1995. A Randomized Linear-Time Algorithm to Find Minimum Spanning Trees. J. ACM 42, 2 (1995),321--328.

Digital Library

[39]

Tsvi Kopelowitz, Ely Porat, and Yair Rosenmutter. 2018. Improved Worst-Case Deterministic Parallel Dynamic Minimum Spanning Forest. In ACM Symposiumon Parallelism in Algorithms and Architectures (SPAA).

[40]

Joseph B Kruskal. 1956. On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical society 7, 1 (1956), 48--50.

[41]

Gary L. Miller and John H. Reif. 1989. Parallel Tree Contraction Part 1: Fundamentals. In Randomness and Computation, Vol. 5. 47--72.

[42]

Shaunak Pawagi. 1989. A parallel algorithm for multiple updates of minimum spanning trees. In International Conference on Parallel Processing (ICPP).

[43]

Shaunak Pawagi and Owen Kaser. 1993. Optimal parallel algorithms for multiple updates of minimum spanning trees. Algorithmica 9, 4 (1993), 357--381.

[44]

Shaunak Pawagi and IV Ramakrishnan. 1986. An O(lgn) algorithm for parallel update of minimum spanning trees. Inform. Process. Lett.22, 5 (1986), 223--229.

Digital Library

[45]

Robert Clay Prim. 1957. Shortest connection networks and some generalizations. The Bell System Technical Journal 36, 6 (1957), 1389--1401.

[46]

Xiaojun Shen and Weifa Liang. 1993. A parallel algorithm for multiple edge up-dates of minimum spanning trees. In International Parallel Processing Symposium(IPPS).

[47]

Natcha Simsiri, Kanat Tangwongsan, Srikanta Tirthapura, and Kun-Lung Wu. 2016. Work-efficient parallel union-find with applications to incremental graph connectivity. In European Conference on Parallel Processing (Euro-Par).

Digital Library

[48]

Daniel D Sleator and Robert Endre Tarjan. 1983. A data structure for dynamic trees. J. of computer and system sciences 26, 3 (1983), 362--391.

[49]

Philip M. Spira and A Pan. 1975. On finding and updating spanning trees and shortest paths.SIAM J. on Computing4, 3 (1975), 375--380.

[50]

Xiaoming Sun and David P Woodruff. 2015. Tight bounds for graph problems in insertion streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM).

[51]

Robert Endre Tarjan. 1983. Data structures and network algorithms. SIAM.

[52]

Yung Hyang Tsin. 1988. On handling vertex deletion in updating spanning trees. Inform. Process. Lett. 27, 4 (1988), 167--168.

Digital Library

[53]

Peter Varman and Kshitij Doshi. 1986. A parallel vertex insertion algorithm for minimum spanning trees. In Intl. Colloq. on Automata, Languages and Programming (ICALP).

Digital Library

[54]

Peter Varman and Kshitij Doshi. 1988. An efficient parallel algorithm for updating minimum spanning trees. Theoretical Computer Science (TCS)58, 1--3 (1988), 379--397.

Cited By

Czumaj AMishra GMukherjee AKuznetsov PGelles ROlivetti D(2024)Streaming Graph Algorithms in the Massively Parallel Computation ModelProceedings of the 43rd ACM Symposium on Principles of Distributed Computing10.1145/3662158.3662770(496-507)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3662158.3662770
Anderson DBlelloch GAgrawal KPetrank E(2024)Deterministic and Low-Span Work-Efficient Parallel Batch-Dynamic TreesProceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3626183.3659976(247-258)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3626183.3659976
Blelloch GGu YSun Y(2024)Teaching Parallel Algorithms Using the Binary-Forking Model2024 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW)10.1109/IPDPSW63119.2024.00080(346-351)Online publication date: 27-May-2024
https://doi.org/10.1109/IPDPSW63119.2024.00080
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Index Terms

Work-Efficient Batch-Incremental Minimum Spanning Trees with Applications to the Sliding-Window Model
1. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
      1. Dynamic graph algorithms
    2. Parallel algorithms
      1. Shared memory algorithms

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cover image ACM Conferences

SPAA '20: Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures

July 2020

601 pages

ISBN:9781450369350

DOI:10.1145/3350755

General Chair:
Christian Scheideler
Institut fuer Informatik Universitaet Paderborn Fuerstenallee 11
,
Program Chair:
Michael Spear

Copyright © 2020 Owner/Author.

Permission to make digital or hard copies of part or all 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 third-party components of this work must be honored. For all other uses, contact the Owner/Author.

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SIGACT: ACM Special Interest Group on Algorithms and Computation Theory
SIGARCH: ACM Special Interest Group on Computer Architecture
EATCS: European Association for Theoretical Computer Science

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Association for Computing Machinery

New York, NY, United States

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Published: 09 July 2020

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SPAA '20

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SPAA '20: 32nd ACM Symposium on Parallelism in Algorithms and Architectures

July 15 - 17, 2020

Virtual Event, USA

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Cited By

Czumaj AMishra GMukherjee AKuznetsov PGelles ROlivetti D(2024)Streaming Graph Algorithms in the Massively Parallel Computation ModelProceedings of the 43rd ACM Symposium on Principles of Distributed Computing10.1145/3662158.3662770(496-507)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3662158.3662770
Anderson DBlelloch GAgrawal KPetrank E(2024)Deterministic and Low-Span Work-Efficient Parallel Batch-Dynamic TreesProceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3626183.3659976(247-258)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3626183.3659976
Blelloch GGu YSun Y(2024)Teaching Parallel Algorithms Using the Binary-Forking Model2024 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW)10.1109/IPDPSW63119.2024.00080(346-351)Online publication date: 27-May-2024
https://doi.org/10.1109/IPDPSW63119.2024.00080
Fedorov AHashemi DNadiradze GAlistarh DAgrawal KShun J(2023)Provably-Efficient and Internally-Deterministic Parallel Union-FindProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591082(261-271)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591082
Dong XWu YWang ZDhulipala LGu YSun YAgrawal KShun J(2023)High-Performance and Flexible Parallel Algorithms for Semisort and Related ProblemsProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591071(341-353)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591071
Anderson DBlelloch G(2022)Parallel Minimum Cuts in O(m log2 n) Work and Low DepthACM Transactions on Parallel Computing10.1145/3565557Online publication date: 16-Dec-2022
https://dl.acm.org/doi/10.1145/3565557
Tseng TDhulipala LShun JAgrawal KLee I(2022)Parallel Batch-Dynamic Minimum Spanning Forest and the Efficiency of Dynamic Agglomerative Graph ClusteringProceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3490148.3538584(233-245)Online publication date: 11-Jul-2022
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Anderson DBlelloch GAgrawal KAzar Y(2021)Parallel Minimum Cuts in O(m log2n) Work and Low DepthProceedings of the 33rd ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3409964.3461797(71-82)Online publication date: 6-Jul-2021
https://dl.acm.org/doi/10.1145/3409964.3461797
Dong XGu YSun YZhang YAgrawal KAzar Y(2021)Efficient Stepping Algorithms and Implementations for Parallel Shortest PathsProceedings of the 33rd ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3409964.3461782(184-197)Online publication date: 6-Jul-2021
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