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Theory of Computing Systems

, Volume 47, Issue 4, pp 920–933 | Cite as

Local MST Computation with Short Advice

  • Pierre Fraigniaud
  • Amos Korman
  • Emmanuelle LebharEmail author
Article

Abstract

We use the recently introduced advising scheme framework for measuring the difficulty of locally distributively computing a Minimum Spanning Tree (MST). An (m,t)-advising scheme for a distributed problem ℘ is a way, for every possible input I of ℘, to provide an “advice” (i.e., a bit string) about I to each node so that: (1) the maximum size of the advices is at most m bits, and (2) the problem ℘ can be solved distributively in at most t rounds using the advices as inputs. In case of MST, the output returned by each node of a weighted graph G is the edge leading to its parent in some rooted MST T of G. Clearly, there is a trivial (⌈log n⌉,0)-advising scheme for MST (each node is given the local port number of the edge leading to the root of some MST T), and it is known that any (0,t)-advising scheme satisfies \(t\geq\tilde{\Omega}(\sqrt{n})\). Our main result is the construction of an (O(1),O(log n))-advising scheme for MST. That is, by only giving a constant number of bits of advice to each node, one can decrease exponentially the distributed computation time of MST in arbitrary graph, compared to algorithms dealing with the problem in absence of any a priori information. We also consider the average size of the advices. On the one hand, we show that any (m,0)-advising scheme for MST gives advices of average size Ω(log n). On the other hand we design an (m,1)-advising scheme for MST with advices of constant average size, that is one round is enough to decrease the average size of the advices from log n to constant.

Keywords

Minimum spanning tree Distributed algorithm Local computation 

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References

  1. 1.
    Chin, F., Ting, H.F.: An almost linear time and O(nlog n+e) messages distributed algorithm for minimum-weight spanning trees. In: Proc. 26th IEEE Symp. on Foundations of Computer Science (FOCS), pp. 257–266 (1985) Google Scholar
  2. 2.
    Cohen, R., Fraigniaud, P., Ilcinkas, D., Korman, A., Peleg, D.: Labeling schemes for tree representation. In: Proc. 7th International Workshop on Distributed Computing (IWDC), pp. 13–24 (2005) Google Scholar
  3. 3.
    Cohen, R., Fraigniaud, P., Ilcinkas, D., Korman, A., Peleg, D.: Label-guided graph exploration by a finite automaton. In: 32nd Int. Colloquium on Automata, Languages and Programming (ICALP). LNCS, vol. 3580, pp. 335–346. Springer, Berlin (2005) CrossRefGoogle Scholar
  4. 4.
    Cormen, T.H., Leiserson, T., Rivest, R.L.: Introduction to Algorithms. MIT Press/McGraw-Hill, New York (1990) zbMATHGoogle Scholar
  5. 5.
    Elkin, M.: A faster distributed protocol for constructing a minimum spanning tree. In: Proc. ACM-SIAM on Discrete Algorithms (SODA), pp. 352–361 (2004) Google Scholar
  6. 6.
    Elkin, M.: An unconditional lower bound on the hardness of approximation of distributed minimum spanning tree problem. In: Proc. 36th Annual ACM Symp. on Theory of Computing (STOC), pp. 331–340 (2004) Google Scholar
  7. 7.
    Fraigniaud, P., Ilcinkas, D., Pelc, A.: Oracle size: a new measure of difficulty for communication tasks. In: Proc. 25th Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 179–187 (2006) Google Scholar
  8. 8.
    Fraigniaud, P., Ilcinkas, D., Pelc, A.: Tree exploration with an oracle. In: 31st Int. Symp. on Mathematical Foundations of Computer Science (MFCS). LNCS, vol. 4162, pp. 24–37. Springer, Berlin (2006) CrossRefGoogle Scholar
  9. 9.
    Gafni, E.: Improvements in the time complexity of two message-optimal election algorithms. In: Proc 4th ACM Symp. on Principles of Distributed Computing (PODC), pp. 175–185 (1985) Google Scholar
  10. 10.
    Gallager, R.G., Humblet, P.A., Spira, P.M.: A distributed algorithm for minimum-weight spanning trees. ACM Trans. Program. Lang. Syst. 5, 66–67 (1983) zbMATHCrossRefGoogle Scholar
  11. 11.
    Korman, A., Kutten, S.: Distributed verification of minimum spanning trees. In: Proc. 25th Annual Symposium on Principles of Distributed Computing (PODC), pp. 26–34 (2006) Google Scholar
  12. 12.
    Korman, A., Kutten, S., Peleg, D.: Proof labeling schemes. In: Proc. 24th Annual Symposium on Principles of Distributed Computing (PODC), pp. 9–18 (2005) Google Scholar
  13. 13.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: What cannot be compute locally! In: Proc. 23th ACM Symp. on Principles of Distributed Computing (PODC), pp. 300–309 (2004) Google Scholar
  14. 14.
    Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21(1), 193–201 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lotker, Z., Patt-Shamir, B., Peleg, D.: Distributed MST for constant diameter graphs. In: Proc. 20th ACM Symp. on Principles of Distributed Computing (PODC), pp. 63–72 (2001) Google Scholar
  16. 16.
    Lotker, Z., Pavlov, E., Patt-Shamir, B., Peleg, D.: MST construction in O(log  log n) communication rounds. In: SPAA ’03: Proceedings of the Fifteenth Annual ACM Symposium on Parallel Algorithms and Architectures, pp. 94–100 (2003) Google Scholar
  17. 17.
    Naor, M., Stockmeyer, L.: What can be computed locally? In: 25th ACM Symposium on Theory of Computing (STOC), pp. 184–193 (1993) Google Scholar
  18. 18.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM Monographs on Discrete Mathematics. SIAM, Philadelphia (2000) zbMATHGoogle Scholar
  19. 19.
    Peleg, D., Rubinovich, R.: A near-tight lower bound on the time complexity of distributed MST construction. In: 40th IEEE Symp. on Foundations of Computer Science (FOCS), pp. 253–261 (1999) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Pierre Fraigniaud
    • 1
  • Amos Korman
    • 1
  • Emmanuelle Lebhar
    • 1
    Email author
  1. 1.CNRS and Univ. Paris 7ParisFrance

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