More Web Proxy on the site http://driver.im/

research-article

Local Computation: Lower and Upper Bounds

Authors:

Thomas Moscibroda,

Roger WattenhoferAuthors Info & Claims

Journal of the ACM (JACM), Volume 63, Issue 2

Article No.: 17, Pages 1 - 44

https://doi.org/10.1145/2742012

Published: 31 March 2016 Publication History

Abstract

The question of what can be computed, and how efficiently, is at the core of computer science. Not surprisingly, in distributed systems and networking research, an equally fundamental question is what can be computed in a distributed fashion. More precisely, if nodes of a network must base their decision on information in their local neighborhood only, how well can they compute or approximate a global (optimization) problem? In this paper we give the first polylogarithmic lower bound on such local computation for (optimization) problems including minimum vertex cover, minimum (connected) dominating set, maximum matching, maximal independent set, and maximal matching. In addition, we present a new distributed algorithm for solving general covering and packing linear programs. For some problems this algorithm is tight with the lower bounds, whereas for others it is a distributed approximation scheme. Together, our lower and upper bounds establish the local computability and approximability of a large class of problems, characterizing how much local information is required to solve these tasks.

References

[1]

Yehuda Afek, Shay Kutten, and Moti Yung. 1990. Memory-efficient self stabilizing protocols for general networks. In WDAG (Lecture Notes in Computer Science), Jan van Leeuwen and Nicola Santoro (Eds.), Vol. 486. Springer, Berlin, 15--28.

[2]

J. Akiyama, H. Era, and F. Harary. 1983. Regular graphs containing a given graph. Elem. Math. 83 83 (1983), 15--17.

[3]

Noga Alon, Laszlo Babai, and Alon Itai. 1986. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithm. 7, 4 (1986), 567--583.

Digital Library

[4]

D. Angluin and A. Gardiner. 1981. Finite common coverings of pairs of regular graphs. J. Comb. Theor. Ser. B 30, 2 (1981), 184--187.

[5]

Baruch Awerbuch and Michael Sipser. 1988. Dynamic networks are as fast as static networks. In Proc. Symp. Foundations of Computer Science (FOCS). 206--219.

Digital Library

[6]

Baruch Awerbuch and George Varghese. 1991. Distributed program checking: A paradigm for building self-stabilizing distributed protocols. In Proc. Symp. on Foundations of Computer Science (FOCS). 258--267.

Digital Library

[7]

R. Bar-Yehuda, K. Censor-Hillel, and G. Schwartzman. 2016. A distributed (2 + &epsi;)-approximation for vertex cover in O(log Δ/&epsi;log log Δ) rounds. CoRR abs/1602.03713v2.

[8]

L. Barenboim and M. Elkin. 2013. Distributed Graph Coloring: Fundamentals and Recent Developments. Morgan & Claypool Publishers.

[9]

Yair Bartal, John W. Byers, and Danny Raz. 1997. Global optimization using local information with applications to flow control. In Proc. Symp. on Foundations of Computer Science (FOCS). 303--312.

[10]

B. Bollobas. 1978. Extremal Graph Theory. Academic Press.

[11]

R. Cole and U. Vishkin. 1986. Deterministic coin tossing with applications to optimal parallel list ranking. Inform. Control 70, 1 (1986), 32--53.

Digital Library

[12]

Andrzej Czygrinow, Michał Hańćkowiak, and Wojciech Wawrzyniak. 2008. Fast distributed approximations in planar graphs. In Proc. 22nd Symp. on Distributed Computing (DISC). 78--92.

Digital Library

[13]

Erik D. Demaine, Uriel Feige, Mohammad Taghi Hajiaghayi, and Mohammad R. Salavatipour. 2006. Combination can be hard: Approximability of the unique coverage problem. In Proc. of the 17^th ACM-SIAM Symposium on Discrete Algorithm (SODA). 162--171.

[14]

Edsger W. Dijkstra. 1974. Self-stabilizing systems in spite of distributed control. Communications of the ACM 17, 11 (1974), 643--644.

Digital Library

[15]

D. Dubhashi, A. Mei, A. Panconesi, J. Radhakrishnan, and A. Srinivasan. 2003. Fast distributed algorithms for (weakly) connected dominating sets and linear-size skeletons. In Proc. of the ACM-SIAM Symposium on Discrete Algorithms (SODA). 717--724.

[16]

Michael Elkin. 2004a. Distributed approximation—A survey. ACM SIGACT News—Distributed Computing Column 35, 4 (2004).

[17]

Michael Elkin. 2004b. An unconditional lower bound on the hardness of approximation of distributed minimum spanning tree problem. In Proc. of the 36^th ACM Symposium on Theory of Computing (STOC). 331--340.

[18]

P. Erdős and H. Sachs. 1963. Reguläre graphen gegebener taillenweite mit minimaler knotenzahl. Wiss. Z. Martin-Luther-U. Halle Math.-Nat. 12 (1963), 251--257.

[19]

Uriel Feige, Magnús M. Halldórsson, Guy Kortsarz, and Aravind Srinivasan. 2003. Approximating the domatic number. SIAM J. Comput. 32, 1 (2003), 172--195.

Digital Library

[20]

F. Fich and E. Ruppert. 2003. Hundreds of impossibility results for distributed computing. Distrib. Comput. 16, 2--3 (2003), 121--163.

Digital Library

[21]

Michael J. Fischer, Nancy A. Lynch, and Michael S. Paterson. 1985. Impossibility of distributed consensus with one faulty process. J. ACM 32, 2 (1985), 374--382.

Digital Library

[22]

L. Fleischer. 2000. Approximating fractional multicommodity flow independent of the number of commodities. SIAM J. Discr. Math. 13, 4 (2000), 505--520.

Digital Library

[23]

P. Fraigniaud, A. Korman, and D. Peleg. 2013. Towards a complexity theory for local distributed computing. J. ACM 60, 5 (2013), 35.

Digital Library

[24]

Seth Copen Goldstein, Jason D. Campbell, and Todd C. Mowry. 2005. Programmable matter. Computer 38, 6 (2005), 99--101.

Digital Library

[25]

A. Israeli and A. Itai. 1986. A fast and simple randomized parallel algorithm for maximal matching. Inform. Process. Lett. 22 (1986), 77--80.

[26]

L. Jia, R. Rajaraman, and R. Suel. 2001. An efficient distributed algorithm for constructing small dominating sets. In Proc. of the 20^th ACM Symposium on Principles of Distributed Computing (PODC). 33--42.

[27]

Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. 2004. What cannot be computed locally&excl;. In Proc. of the 23^rd ACM Symposium on the Principles of Distributed Computing (PODC). 300--309.

[28]

Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. 2006. The price of being near-sighted. In Proc. of the 17^th ACM-SIAM Symposium on Discrete Algorithms (SODA).

[29]

F. Kuhn and R. Wattenhofer. 2003. Constant-time distributed dominating set approximation. In Proc. of the 22^nd Annual ACM Symp. on Principles of Distributed Computing (PODC). 25--32.

[30]

Leslie Lamport, Robert Shostak, and Marshall Pease. 1982. The byzantine generals problem. ACM Trans. Program. Lang. Syst. 4, 3 (1982), 382--401.

Digital Library

[31]

F. Lazebnik and V. A. Ustimenko. 1995. Explicit construction of graphs with an arbitrary large girth and of large size. Discr. Appl. Math. 60, 1--3 (1995), 275--284.

Digital Library

[32]

Christoph Lenzen, Yvonne Anne Oswald, and Roger Wattenhofer. 2008. What can be approximated locally? In 20th ACM Symposium on Parallelism in Algorithms and Architecture (SPAA), Munich, Germany.

Digital Library

[33]

Christoph Lenzen, Jukka Suomela, and Roger Wattenhofer. 2009. Local algorithms: Self-stabilization on speed. In 11th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS). 17--34.

Digital Library

[34]

Christoph Lenzen and Roger Wattenhofer. 2008. Leveraging linial’s locality limit. In Proc. 22nd Symp. on Distributed Computing (DISC). 394--407.

Digital Library

[35]

Nathan Linial. 1992. Locality in distributed graph algorithms. SIAM J. Comput. 21, 1 (1992), 193--201.

Digital Library

[36]

N. Linial and M. Saks. 1993. Low diameter graph decompositions. Combinatorica 13, 4 (1993), 441--454.

[37]

M. Luby. 1986. A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15 (1986), 1036--1053.

Digital Library

[38]

Grzegorz Malewicz, Matthew H. Austern, Aart J. C. Bik, James C. Dehnert, Ilan Horn, Naty Leiser, and Grzegorz Czajkowski. 2010. Pregel: A system for large-scale graph processing. In Proceedings of the International Conference on Management of Data (SIGMOD).

Digital Library

[39]

Moni Naor and Larry Stockmeyer. 1995. What can be computed locally? SIAM J. Comput. 24, 6 (1995), 1259--1277.

Digital Library

[40]

Huy N. Nguyen and Krzysztof Onak. 2008. Constant-time approximation algorithms via local improvements. In Proc. of the 49^th Symposium on Foundations of Computer Science (FOCS). 327--336.

[41]

C. H. Papadimitriou and M. Yannakakis. 1991. On the value of information in distributed decision making. In Proc. of the 10^th ACM Symposium on Principles of Distributed Computing (PODC). 61--64.

[42]

C. H. Papadimitriou and M. Yannakakis. 1993. Linear programming without the matrix. In Proc. of the 25^th ACM Symposium on Theory of Computing (STOC). 121--129.

[43]

Michal Parnas and Dana Ron. 2007. Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms. Theor. Comput. Sci. 381, 1-3 (2007), 183--196.

Digital Library

[44]

David Peleg. 2000. Distributed Computing: A Locality-Sensitive Approach. SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelpia, PA.

Digital Library

[45]

S. Plotkin, D. Shmoys, and E. Tardos. 1995. Fast approximation algorithms for fractional packing and covering problems. Math. Operat. Res. 20 (1995), 257--301.

Digital Library

[46]

S. Rajagopalan and V. Vazirani. 1998. Primal-dual RNC approximation algorithms for set cover and covering integer programs. SIAM J. Comput. 28 (1998), 525--540.

Digital Library

[47]

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. Computing 41, 5 (2012), 1235--1265.

[48]

Johannes Schneider and Roger Wattenhofer. 2008. A log-star distributed maximal independent set algorithm for growth-bounded graphs. In 27th ACM Symposium on Principles of Distributed Computing (PODC), Toronto, Canada. ACM, New York, NY, 35--44.

Digital Library

[49]

Johannes Schneider and Roger Wattenhofer. 2011. Bounds on contention management algorithms. Theoretical Computer Science 412, 32 (2011), 4151--4160.

Digital Library

[50]

A. Srinivasan. 1995. Improved approximations of packing and covering problems. In Proc. of the 27th ACM Symposium on Theory of Computing (STOC). ACM, New York, NY, 268--276.

Digital Library

[51]

Aaron Sterling. 2009. Memory consistency conditions for self-assembly programming. In 1st Innovations in Computer Science (ICS). Beijing, China, 490--500.

[52]

J. Suomela. 2011. Survey of local algorithms. ACM Computing Surveys 45, 2 (2011), 24:1--24:40.

[53]

Mirjam Wattenhofer and Roger Wattenhofer. 2004. Distributed weighted matching. In Proc. of the 18^th Conference on Distributed Computing (DISC). 335--348.

[54]

N. E. Young. 2001. Sequential and parallel algorithms for mixed packing and covering. In Proc. of the 42^nd Symposium on Foundations of Computer Science (FOCS). 538--546.

Cited By

Heydt OKublenz SOssona de Mendez PSiebertz SVigny A(2025)Distributed domination on sparse graph classesEuropean Journal of Combinatorics10.1016/j.ejc.2023.103773123(103773)Online publication date: Jan-2025
https://doi.org/10.1016/j.ejc.2023.103773
Giliberti JParsaeian ZKuznetsov PGelles ROlivetti D(2024)Brief Announcement: Massively Parallel Ruling Set Made DeterministicProceedings of the 43rd ACM Symposium on Principles of Distributed Computing10.1145/3662158.3662816(523-526)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3662158.3662816
Assadi SKonrad CNaidu KSundaresan JMohar BShinkar IO'Donnell R(2024)O(log log n) Passes Is Optimal for Semi-streaming Maximal Independent SetProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649763(847-858)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649763
Show More Cited By

Index Terms

Local Computation: Lower and Upper Bounds
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
      2. Network flows
2. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
      1. Network flows
  2. Randomness, geometry and discrete structures

Recommendations

Lower Bounds for Maximal Matchings and Maximal Independent Sets
There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O(Δ + log^* n) communication rounds; here, n is the number of nodes and Δ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the ...
What cannot be computed locally!
PODC '04: Proceedings of the twenty-third annual ACM symposium on Principles of distributed computing

We give time lower bounds for the distributed approximation of minimum vertex cover (MVC) and related problems such as minimum dominating set (MDS). In k communication rounds, MVC and MDS can only be approximated by factors Ω(n^c/k2/k) and Ω(Δ>^1/k/k) for ...
Distributed maximal matching: greedy is optimal
PODC '12: Proceedings of the 2012 ACM symposium on Principles of distributed computing

We study distributed algorithms that find a maximal matching in an anonymous, edge-coloured graph. If the edges are properly coloured with k colours, there is a trivial greedy algorithm that finds a maximal matching in k-1 synchronous communication ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Journal of the ACM

Journal of the ACM Volume 63, Issue 2

May 2016

249 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/2906142

Editor:
Éva Tardos
Cornell University

Issue’s Table of Contents

Copyright © 2016 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 the author(s) 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].

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 31 March 2016

Accepted: 01 January 2015

Revised: 01 December 2014

Received: 01 November 2011

Published in JACM Volume 63, Issue 2

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

98
Total Citations
View Citations
922
Total Downloads

Downloads (Last 12 months)61
Downloads (Last 6 weeks)12

Reflects downloads up to 11 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Heydt OKublenz SOssona de Mendez PSiebertz SVigny A(2025)Distributed domination on sparse graph classesEuropean Journal of Combinatorics10.1016/j.ejc.2023.103773123(103773)Online publication date: Jan-2025
https://doi.org/10.1016/j.ejc.2023.103773
Giliberti JParsaeian ZKuznetsov PGelles ROlivetti D(2024)Brief Announcement: Massively Parallel Ruling Set Made DeterministicProceedings of the 43rd ACM Symposium on Principles of Distributed Computing10.1145/3662158.3662816(523-526)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3662158.3662816
Assadi SKonrad CNaidu KSundaresan JMohar BShinkar IO'Donnell R(2024)O(log log n) Passes Is Optimal for Semi-streaming Maximal Independent SetProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649763(847-858)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649763
Assadi SKol GZhang Z(2024)Rounds vs. Communication Tradeoffs for Maximal Independent SetsSIAM Journal on Computing10.1137/22M1536972(FOCS22-20-FOCS22-59)Online publication date: 25-Mar-2024
https://doi.org/10.1137/22M1536972
Ghaffari MGrunau C(2024)Near-Optimal Deterministic Network Decomposition and Ruling Set, and Improved MIS2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00007(2148-2179)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00007
Chang Y(2024)The energy complexity of diameter and minimum cut computation in bounded-genus networksTheoretical Computer Science10.1016/j.tcs.2023.114279982(114279)Online publication date: Jan-2024
https://doi.org/10.1016/j.tcs.2023.114279
Halldórsson MRawitz D(2024)Distributed Fractional Local Ratio and Independent Set ApproximationInformation and Computation10.1016/j.ic.2024.105238(105238)Online publication date: Nov-2024
https://doi.org/10.1016/j.ic.2024.105238
Lenzen CWenning S(2024)Tight Bounds for Constant-Round Domination on Graphs of High Girth and Low ExpansionStabilization, Safety, and Security of Distributed Systems10.1007/978-3-031-74498-3_20(277-291)Online publication date: 20-Oct-2024
https://doi.org/10.1007/978-3-031-74498-3_20
Halldórsson MRawitz D(2024)Distributed Fractional Local Ratio and Independent Set ApproximationStructural Information and Communication Complexity10.1007/978-3-031-60603-8_16(281-299)Online publication date: 23-May-2024
https://doi.org/10.1007/978-3-031-60603-8_16
Maus Y(2023)Distributed Graph Coloring Made EasyACM Transactions on Parallel Computing10.1145/360589610:4(1-21)Online publication date: 17-Aug-2023
https://dl.acm.org/doi/10.1145/3605896
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Issue’s Table of Contents