Broadcast and minimum spanning tree with o(m) messages in the asynchronous CONGEST model
We provide the first asynchronous distributed algorithms to compute broadcast and minimum spanning tree with o ( m ) bits of communication, in a sufficiently dense graph with n nodes and m edges. For decades, it was believed that Ω ( m ) bits of communication are required for any algorithm that cons...
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| Veröffentlicht in: | Distributed computing Jg. 34; H. 4; S. 283 - 299 |
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01.08.2021
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| Abstract | We provide the first asynchronous distributed algorithms to compute broadcast and minimum spanning tree with
o
(
m
) bits of communication, in a sufficiently dense graph with
n
nodes and
m
edges. For decades, it was believed that
Ω
(
m
)
bits of communication are required for any algorithm that constructs a broadcast tree. In 2015, King, Kutten and Thorup showed that in the KT1 model where nodes have initial knowledge of their neighbours’ identities it is possible to construct MST in
O
~
(
n
)
messages in the synchronous CONGEST model. In the CONGEST model messages are of size
O
(
log
n
)
. However, no algorithm with
o
(
m
) messages was known for the asynchronous case. Here, we provide an algorithm that uses
O
(
n
3
/
2
log
3
/
2
n
)
messages to find MST in the asynchronous CONGEST model. Our algorithm is randomized Monte Carlo and outputs MST with high probability. We will provide an algorithm for computing a spanning tree with
O
(
n
3
/
2
log
3
/
2
n
)
messages. Given a spanning tree, we can compute MST with
O
~
(
n
)
messages. |
|---|---|
| AbstractList | We provide the first asynchronous distributed algorithms to compute broadcast and minimum spanning tree with o(m) bits of communication, in a sufficiently dense graph with n nodes and m edges. For decades, it was believed that Ω(m) bits of communication are required for any algorithm that constructs a broadcast tree. In 2015, King, Kutten and Thorup showed that in the KT1 model where nodes have initial knowledge of their neighbours’ identities it is possible to construct MST in O~(n) messages in the synchronous CONGEST model. In the CONGEST model messages are of size O(logn). However, no algorithm with o(m) messages was known for the asynchronous case. Here, we provide an algorithm that uses O(n3/2log3/2n) messages to find MST in the asynchronous CONGEST model. Our algorithm is randomized Monte Carlo and outputs MST with high probability. We will provide an algorithm for computing a spanning tree with O(n3/2log3/2n) messages. Given a spanning tree, we can compute MST with O~(n) messages. We provide the first asynchronous distributed algorithms to compute broadcast and minimum spanning tree with o ( m ) bits of communication, in a sufficiently dense graph with n nodes and m edges. For decades, it was believed that Ω ( m ) bits of communication are required for any algorithm that constructs a broadcast tree. In 2015, King, Kutten and Thorup showed that in the KT1 model where nodes have initial knowledge of their neighbours’ identities it is possible to construct MST in O ~ ( n ) messages in the synchronous CONGEST model. In the CONGEST model messages are of size O ( log n ) . However, no algorithm with o ( m ) messages was known for the asynchronous case. Here, we provide an algorithm that uses O ( n 3 / 2 log 3 / 2 n ) messages to find MST in the asynchronous CONGEST model. Our algorithm is randomized Monte Carlo and outputs MST with high probability. We will provide an algorithm for computing a spanning tree with O ( n 3 / 2 log 3 / 2 n ) messages. Given a spanning tree, we can compute MST with O ~ ( n ) messages. |
| Author | Mashreghi, Ali King, Valerie |
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| Cites_doi | 10.1145/4221.4227 10.1007/978-0-387-30162-4_417 10.1145/77600.77618 10.4230/LIPIcs.DISC.2018.32 10.1145/323596.323612 10.1145/3007748.3007775 10.1137/11085178X 10.1109/TDSC.2007.1007 10.1145/2699440 10.1007/BF02242717 10.1145/224964.224990 10.1145/3055399.3055449 10.1137/S0097539794261118 10.1145/3087801.3087823 10.1137/S0097539704441058 10.1145/41840.41847 10.1145/3055399.3055452 10.1145/2213556.2213560 10.1145/2767386.2767405 10.1137/S0097539700369740 10.1137/1.9781611973105.81 10.1145/224964.225474 10.1145/357195.357200 10.1109/SFCS.1985.7 10.1145/1835698.1835742 10.4230/LIPIcs.DISC.2018.30 10.1145/28395.28421 |
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| References | Emek, Y., Korman, A.: Efficient threshold detection in a distributed environment. In: Proceedings of the 29th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing, pp. 183–191. ACM (2010) GallagerRGHumbletPASpiraPMA distributed algorithm for minimum-weight spanning treesACM Trans. Program. Lang. Syst.198351667710.1145/357195.357200 Kapron, B.M., King, V., Mountjoy, B.: Dynamic graph connectivity in polylogarithmic worst case time. In: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1131–1142. Society for Industrial and Applied Mathematics (2013) AwerbuchBComplexity of network synchronizationJ ACM198532480482381033810.1145/4221.4227 Peleg, D., Ullman, J.D.: An optimal synchronizer for the hypercube. In: Proceedings of the Sixth Annual ACM Symposium on Principles of Distributed Computing, pp. 77–85. ACM (1987) Mashreghi, A., King, V.: Time-communication trade-offs for minimum spanning tree construction. In: Proceedings of the 18th International Conference on Distributed Computing and Networking, p. 8. ACM (2017) Awerbuch, B.: Optimal distributed algorithms for minimum weight spanning tree, counting, leader election, and related problems. In: Proceedings of the Nineteenth Annual ACM Symposium on Theory of computing, pp. 230–240. ACM (1987) PelegDRubinovichVA near-tight lower bound on the time complexity of distributed minimum-weight spanning tree constructionSIAM J. Comput.200030514271442181395710.1137/S00975397003697400973.05074 Elkin, M.: A faster distributed protocol for constructing a minimum spanning tree. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 359–368. Society for Industrial and Applied Mathematics (2004) Gmyr, R., Pandurangan, G.: Time-Message Trade-Offs in Distributed Algorithms. In: Schmid, U., Widder , J. (eds.) 32nd International Symposium on Distributed Computing (DISC 2018), Leibniz International Proceedings in Informatics (LIPIcs), vol. 121, pp. 32:1–32:18. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2018). https://doi.org/10.4230/LIPIcs.DISC.2018.32. http://drops.dagstuhl.de/opus/volltexte/2018/9821 ElkinMAn unconditional lower bound on the time-approximation trade-off for the distributed minimum spanning tree problemSIAM J. Comput.2006362433456224773510.1137/S0097539704441058 Ghaffari, M., Kuhn, F.: Distributed MST and broadcast with fewer messages, and faster gossiping. In: Schmid, U., Widder, J. (eds.) 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), vol. 121, pp. 30:1–30:12. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2018). https://doi.org/10.4230/LIPIcs.DISC.2018.30. http://drops.dagstuhl.de/opus/volltexte/2018/9819 KuttenSPanduranganGPelegDRobinsonPTrehanAOn the complexity of leader electionJ. ACM20156217332377310.1145/2699440 Kutten, S., Peleg, D.: Fast distributed construction of k-dominating sets and applications. In: Proceedings of the Fourteenth Annual ACM Symposium on Principles of Distributed Computing, pp. 238–251. ACM (1995) GarayJAKuttenSPelegDA sublinear time distributed algorithm for minimum-weight spanning treesSIAM J. Comput.1998271302316161483310.1137/S0097539794261118 Awerbuch, B., Peleg, D.: Network synchronization with polylogarithmic overhead. In: 31st Annual Symposium on Foundations of Computer Science, 1990. Proceedings., pp. 514–522. IEEE (1990) Elkin, M.: Synchronizers, spanners. In: Kao, Ming-Yang (ed) Encyclopedia of Algorithms, pp. 1–99. Springer (2008) Pandurangan, G., Robinson, P., Scquizzato, M.: A time-and message-optimal distributed algorithm for minimum spanning trees. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pp. 743–756. ACM (2017) AwerbuchBGoldreichOVainishRPelegDA trade-off between information and communication in broadcast protocolsJ ACM1990372238256107225810.1145/77600.77618 Faloutsos, M., Molle, M.: Optimal distributed algorithm for minimum spanning trees revisited. In: Proceedings of the Fourteenth Annual ACM Symposium on Principles of DistributCd computing, pp. 231–237. ACM (1995) Elkin, M.: Distributed exact shortest paths in sublinear time. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pp. 757–770. ACM, New York, NY, USA (2017). https://doi.org/10.1145/3055399.3055452 SinghGBernsteinAJA highly asynchronous minimum spanning tree protocolDistrib. Comput.19958315116110.1007/BF02242717 AwerbuchBGoldreichOVainishROn the Message Complexity of Broadcast: A Basic Lower Bound1987CambridgeLaboratory for Computer Science, Massachusetts Institute of Technology King, V., Kutten, S., Thorup, M.: Construction and impromptu repair of an mst in a distributed network with o (m) communication. In: Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, pp. 71–80. ACM (2015) SarmaADHolzerSKorLKormanANanongkaiDPanduranganGPelegDWattenhoferRDistributed verification and hardness of distributed approximationSIAM J. Comput.201241512351265302379210.1137/11085178X Gafni, E.: Improvements in the time complexity of two message-optimal election algorithms. In: Proceedings of the Fourth Annual ACM Symposium on Principles of Distributed Computing, pp. 175–185. ACM (1985) Ahn, K.J., Guha, S., McGregor, A.: Graph sketches: sparsification, spanners, and subgraphs. In: Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pp. 5–14. ACM (2012) Chin, F., Ting, H.: An almost linear time and o (nlogn+ e) messages distributed algorithm for minimum-weight spanning trees. In: 26th Annual Symposium on Foundations of Computer Science, pp. 257–266. IEEE (1985) Mashreghi, A., King, V.: Faster asynchronous mst and low diameter tree construction with sublinear communication. arXiv preprint arXiv:1907.12152 (2019) AwerbuchBKuttenSMansourYPatt-ShamirBVargheseGA time-optimal self-stabilizing synchronizer using a phase clockIEEE Trans Dependable Secure Comput20074318019010.1109/TDSC.2007.1007 Elkin, M.: A simple deterministic distributed mst algorithm, with near-optimal time and message complexities. arXiv preprint arXiv:1703.02411 (2017) Mashreghi, A., King, V.: Brief announcement: faster asynchronous mst and low diameter tree construction with sublinear communication. In: 33rd International Symposium on Distributed Computing (DISC 2019). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2019) JA Garay (387_CR18) 1998; 27 B Awerbuch (387_CR2) 1985; 32 B Awerbuch (387_CR5) 1990; 37 387_CR27 387_CR26 G Singh (387_CR32) 1995; 8 387_CR28 387_CR12 387_CR11 387_CR14 387_CR13 387_CR30 AD Sarma (387_CR31) 2012; 41 B Awerbuch (387_CR4) 1987 D Peleg (387_CR29) 2000; 30 387_CR19 RG Gallager (387_CR17) 1983; 5 M Elkin (387_CR10) 2006; 36 387_CR16 387_CR15 B Awerbuch (387_CR6) 2007; 4 387_CR22 387_CR25 387_CR24 387_CR1 387_CR21 387_CR3 387_CR20 S Kutten (387_CR23) 2015; 62 387_CR7 387_CR8 387_CR9 |
| References_xml | – reference: SinghGBernsteinAJA highly asynchronous minimum spanning tree protocolDistrib. Comput.19958315116110.1007/BF02242717 – reference: Elkin, M.: A faster distributed protocol for constructing a minimum spanning tree. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 359–368. Society for Industrial and Applied Mathematics (2004) – reference: AwerbuchBKuttenSMansourYPatt-ShamirBVargheseGA time-optimal self-stabilizing synchronizer using a phase clockIEEE Trans Dependable Secure Comput20074318019010.1109/TDSC.2007.1007 – reference: Faloutsos, M., Molle, M.: Optimal distributed algorithm for minimum spanning trees revisited. In: Proceedings of the Fourteenth Annual ACM Symposium on Principles of DistributCd computing, pp. 231–237. ACM (1995) – reference: Elkin, M.: A simple deterministic distributed mst algorithm, with near-optimal time and message complexities. arXiv preprint arXiv:1703.02411 (2017) – reference: Kapron, B.M., King, V., Mountjoy, B.: Dynamic graph connectivity in polylogarithmic worst case time. In: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1131–1142. Society for Industrial and Applied Mathematics (2013) – reference: Emek, Y., Korman, A.: Efficient threshold detection in a distributed environment. In: Proceedings of the 29th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing, pp. 183–191. ACM (2010) – reference: AwerbuchBComplexity of network synchronizationJ ACM198532480482381033810.1145/4221.4227 – reference: GarayJAKuttenSPelegDA sublinear time distributed algorithm for minimum-weight spanning treesSIAM J. Comput.1998271302316161483310.1137/S0097539794261118 – reference: Mashreghi, A., King, V.: Brief announcement: faster asynchronous mst and low diameter tree construction with sublinear communication. In: 33rd International Symposium on Distributed Computing (DISC 2019). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2019) – reference: Ahn, K.J., Guha, S., McGregor, A.: Graph sketches: sparsification, spanners, and subgraphs. In: Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pp. 5–14. ACM (2012) – reference: KuttenSPanduranganGPelegDRobinsonPTrehanAOn the complexity of leader electionJ. ACM20156217332377310.1145/2699440 – reference: Elkin, M.: Synchronizers, spanners. In: Kao, Ming-Yang (ed) Encyclopedia of Algorithms, pp. 1–99. Springer (2008) – reference: Awerbuch, B.: Optimal distributed algorithms for minimum weight spanning tree, counting, leader election, and related problems. In: Proceedings of the Nineteenth Annual ACM Symposium on Theory of computing, pp. 230–240. ACM (1987) – reference: GallagerRGHumbletPASpiraPMA distributed algorithm for minimum-weight spanning treesACM Trans. Program. Lang. Syst.198351667710.1145/357195.357200 – reference: Awerbuch, B., Peleg, D.: Network synchronization with polylogarithmic overhead. In: 31st Annual Symposium on Foundations of Computer Science, 1990. Proceedings., pp. 514–522. IEEE (1990) – reference: Mashreghi, A., King, V.: Time-communication trade-offs for minimum spanning tree construction. In: Proceedings of the 18th International Conference on Distributed Computing and Networking, p. 8. ACM (2017) – reference: AwerbuchBGoldreichOVainishROn the Message Complexity of Broadcast: A Basic Lower Bound1987CambridgeLaboratory for Computer Science, Massachusetts Institute of Technology – reference: Pandurangan, G., Robinson, P., Scquizzato, M.: A time-and message-optimal distributed algorithm for minimum spanning trees. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pp. 743–756. ACM (2017) – reference: Ghaffari, M., Kuhn, F.: Distributed MST and broadcast with fewer messages, and faster gossiping. In: Schmid, U., Widder, J. (eds.) 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), vol. 121, pp. 30:1–30:12. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2018). https://doi.org/10.4230/LIPIcs.DISC.2018.30. http://drops.dagstuhl.de/opus/volltexte/2018/9819 – reference: Peleg, D., Ullman, J.D.: An optimal synchronizer for the hypercube. In: Proceedings of the Sixth Annual ACM Symposium on Principles of Distributed Computing, pp. 77–85. ACM (1987) – reference: Mashreghi, A., King, V.: Faster asynchronous mst and low diameter tree construction with sublinear communication. arXiv preprint arXiv:1907.12152 (2019) – reference: King, V., Kutten, S., Thorup, M.: Construction and impromptu repair of an mst in a distributed network with o (m) communication. In: Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, pp. 71–80. ACM (2015) – reference: Chin, F., Ting, H.: An almost linear time and o (nlogn+ e) messages distributed algorithm for minimum-weight spanning trees. In: 26th Annual Symposium on Foundations of Computer Science, pp. 257–266. IEEE (1985) – reference: Elkin, M.: Distributed exact shortest paths in sublinear time. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pp. 757–770. 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| Title | Broadcast and minimum spanning tree with o(m) messages in the asynchronous CONGEST model |
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