Cited By
View all- Parter M(2023)Secure Computation Meets Distributed Universal Optimality2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00144(2336-2368)Online publication date: 6-Nov-2023
Low-Congestion Shortcuts in Constant Diameter Graphs
Pages 203 - 211
Abstract
Low congestion shortcuts, introduced by Ghaffari and Haeupler (SODA 2016), provide a unified framework for global optimization problems in the CONGEST model of distributed computing. Roughly speaking, for a given graph G and a collection of vertex-disjoint connected subsets S1,…,Sℓ ⊆V(G), (c,d) low-congestion shortcuts augment each subgraph G[Si] with a subgraph Hi ⊆G such that: (i) each edge appears on at most c subgraphs (congestion bound), and (ii) the diameter of each subgraph G[Si] ∪ Hi is bounded by d (dilation bound). It is desirable to compute shortcuts of small congestion and dilation as these quantities capture the round complexity of many global optimization problems in the CONGEST model. For n-vertex graphs with constant diameter D=O(1), Elkin (STOC 2004) presented an (implicit) shortcuts lower bound with1 c + d + Ωe (n (D-2)/(2D-2)). A nearly matching upper bound, however, was only recently obtained for D ∈ {3,4} by Kitamura et al. (DISC 2019).
In this work, we resolve the long-standing complexity gap of shortcuts in constant diameter graphs, originally posed by Lotker et al. (PODC 2001). We present new shortcut constructions which match, up to poly-logarithmic terms, the lower bounds of Elkin. As a result, we provide improved and existentially optimal algorithms for several network optimization tasks in constant diameter graphs, including MST, (1+ε)-approximate minimum cuts and more.
Supplementary Material
Presentation video
- Download
- 61.58 MB
References
[1]
Réka Albert, Hawoong Jeong, and Albert-László Barabási. 1999. Diameter of the world-wide web. nature 401, 6749 (1999), 130--131.
[2]
Julia Chuzhoy, Merav Parter, and Zihan Tan. 2020. On Packing Low- Diameter Spanning Trees. In 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference). 33:1--33:18.
[3]
Michal Dory and Mohsen Ghaffari. 2019. Improved distributed approximations for minimum-weight two-edge-connected spanning subgraph. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing. 521--530.
[4]
Michael Elkin. 2004. Unconditional lower bounds on the time approximation tradeoffs for the distributed minimum spanning tree problem. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004. 331--340. https://doi.org/10.1145/1007352.1007407
[5]
Mohsen Ghaffari. 2015. Near-optimal scheduling of distributed algorithms. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing. ACM, 3--12.
[6]
Mohsen Ghaffari. 2017. Improved Distributed Algorithms for Fundamental Graph Problems. Ph.D. Dissertation. MIT, USA. https://groups.csail.mit.edu/tds/papers/Ghaffari/PhDThesis-Ghaffari.pdf
[7]
Mohsen Ghaffari and Bernhard Haeupler. 2016. Distributed Algorithms for Planar Networks II: Low-Congestion Shortcuts, MST, and Min-Cut. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, Robert Krauthgamer (Ed.). SIAM, 202--219.
[8]
Mohsen Ghaffari and Bernhard Haeupler. 2020. Low-Congestion Shortcuts for Graphs Excluding Dense Minors. CoRR abs/2008.03091 (2020). arXiv:2008.03091 https://arxiv.org/abs/2008.03091
[9]
Mohsen Ghaffari, Fabian Kuhn, and Hsin-Hao Su. 2017. Distributed MST and Routing in Almost Mixing Time. In Proceedings of the ACM Symposium on Principles of Distributed Computing, PODC 2017, Washington, DC, USA, July 25-27, 2017. 131--140. https://doi.org/10.1145/3087801.3087827
[10]
Mohsen Ghaffari and Merav Parter. 2016. MST in Log-Star Rounds of Congested Clique. In Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, Chicago, IL, USA, July 25-28, 2016. 19--28.
[11]
Mohsen Ghaffari and Merav Parter. 2017. Near-Optimal Distributed DFS in Planar Graphs. In 31st International Symposium on Distributed Computing, DISC 2017, October 16--20, 2017, Vienna, Austria. 21:1--21:16.
[12]
Bernhard Haeupler, D. Ellis Hershkowitz, and David Wajc. 2018. Roundand Message-Optimal Distributed Graph Algorithms. In Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, PODC 2018, Egham, United Kingdom, July 23-27, 2018. 119--128.
[13]
Bernhard Haeupler, Taisuke Izumi, and Goran Zuzic. 2016. Low congestion shortcuts without embedding. In Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing. ACM, 451--460.
[14]
Bernhard Haeupler and Jason Li. 2018. Faster Distributed Shortest Path Approximations via Shortcuts. In 32nd International Symposium on Distributed Computing, DISC 2018, New Orleans, LA, USA, October 15-19, 2018. 33:1--33:14.
[15]
Bernhard Haeupler, Jason Li, and Goran Zuzic. 2018. Minor Excluded Network Families Admit Fast Distributed Algorithms. In Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, PODC 2018, Egham, United Kingdom, July 23-27, 2018. 465--474.
[16]
James W. Hegeman, Gopal Pandurangan, Sriram V. Pemmaraju, Vivek B. Sardeshmukh, and Michele Scquizzato. 2015. Toward Optimal Bounds in the Congested Clique: Graph Connectivity and MST. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, PODC 2015, Donostia-San Sebastián, Spain, July 21 - 23, 2015. 91--100.
[17]
Tomasz Jurdzinski and Krzysztof Nowicki. 2018. MST in O(1) Rounds of Congested Clique. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018. 2620--2632.
[18]
Naoki Kitamura, Hirotaka Kitagawa, Yota Otachi, and Taisuke Izumi. 2019. Low-Congestion Shortcut and Graph Parameters. In 33rd International Symposium on Distributed Computing, DISC 2019, October 14-18, 2019, Budapest, Hungary (LIPIcs, Vol. 146), Jukka Suomela (Ed.). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 25:1--25:17.
[19]
Tom Leighton, Bruce Maggs, and Andrea W Richa. 1999. Fast algorithms for finding O(congestion+ dilation) packet routing schedules. Combinatorica 19, 3 (1999), 375--401.
[20]
Jason Li and Merav Parter. 2019. Planar diameter via metric compression. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019. 152--163.
[21]
Zvi Lotker, Boaz Patt-Shamir, and David Peleg. 2001. Distributed MST for constant diameter graphs. In Proceedings of the Twentieth Annual ACM Symposium on Principles of Distributed Computing, PODC 2001, Newport, Rhode Island, USA, August 26-29, 2001. 63--71.
[22]
Zvi Lotker, Boaz Patt-Shamir, and David Peleg. 2006. Distributed MST for constant diameter graphs. Distributed Comput. 18, 6 (2006), 453--460.
[23]
Zvi Lotker, Elan Pavlov, Boaz Patt-Shamir, and David Peleg. 2003. MST construction in O(log logn) communication rounds. In the Proceedings of the Symposium on Parallel Algorithms and Architectures. ACM, 94--100.
[24]
Krzysztof Nowicki. 2019. A Deterministic Algorithm for the MST Problem in Constant Rounds of Congested Clique. CoRR abs/1912.04239 (2019). arXiv:1912.04239 http://arxiv.org/abs/1912.04239
[25]
David Peleg. 2000. Distributed Computing: A Locality-sensitive Approach. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.
[26]
Atish Das Sarma, Stephan Holzer, Liah Kor, Amos Korman, Danupon Nanongkai, Gopal Pandurangan, David Peleg, and Roger Wattenhofer. 2011. Distributed verification and hardness of distributed approximation. In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011. 363--372.
[27]
Atish Das Sarma, Stephan Holzer, Liah Kor, Amos Korman, Danupon Nanongkai, Gopal Pandurangan, David Peleg, and Roger Wattenhofer. 2012. Distributed verification and hardness of distributed approximation. SIAM J. Comput. 41, 5 (2012), 1235--1265.
Index Terms
- Low-Congestion Shortcuts in Constant Diameter Graphs
Recommendations
Low-Congestion Shortcuts for Graphs Excluding Dense Minors
PODC'21: Proceedings of the 2021 ACM Symposium on Principles of Distributed ComputingWe prove that any n-node graph G with diameter D admits shortcuts with congestion O(δ D log n) and dilation O(δ D), where δ is the maximum edge-density of any minor of G. Our proof is simple and constructive with a tildeΘ (δ D)-round1 distributed ...
Low-Congestion Shortcuts without Embedding
PODC '16: Proceedings of the 2016 ACM Symposium on Principles of Distributed ComputingDistributed optimization algorithms are frequently faced with solving sub-problems on disjoint connected parts of a network. Unfortunately, the diameter of these parts can be significantly larger than the diameter of the underlying network, leading to ...
Sublinear-Time Quantum Computation of the Diameter in CONGEST Networks
PODC '18: Proceedings of the 2018 ACM Symposium on Principles of Distributed ComputingThe computation of the diameter is one of the most central problems in distributed computation. In the standard CONGEST model, in which two adjacent nodes can exchange O(log n) bits per round (here n denotes the number of nodes of the network), it is ...
Comments
Information & Contributors
Information
Published In
July 2021
590 pages
Copyright © 2021 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]
Sponsors
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 23 July 2021
Check for updates
Author Tags
Qualifiers
- Research-article
Funding Sources
- European Research Council (ERC) Grant
Conference
PODC '21: ACM Symposium on Principles of Distributed Computing
July 26 - 30, 2021
Virtual Event, Italy
Acceptance Rates
Overall Acceptance Rate 740 of 2,477 submissions, 30%
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- View Citations1Total Citations
- 121Total Downloads
- Downloads (Last 12 months)15
- Downloads (Last 6 weeks)2
Reflects downloads up to 03 Oct 2024
Other Metrics
Citations
Cited By
View all- Parter M(2023)Secure Computation Meets Distributed Universal Optimality2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00144(2336-2368)Online publication date: 6-Nov-2023
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.
Sign in