The complexity of broadcasting in planar and decomposable graphs

  • Andreas Jakoby
  • Rüdiger Reischuk
  • Christian Schindelhauer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 903)


Broadcasting in processor networks means disseminating a single piece of information, which is originally known only at some nodes, to all members of the network. The goal is to inform everybody using as few rounds as possible, that is to minimize the broadcasting time.

Given a graph and a subset of nodes, the sources, the problem to determine its specific broadcast time, or more general to find a broadcast schedule of minimal length has been shown to be NP-complete. In contrast to other optimization problems for graphs, like vertex cover or traveling salesman, little was known about restricted graph classes for which polynomial time algorithms exist, for example for graphs of bounded treewidth. The broadcasting problem is harder in this respect because it does not have the finite state property. Here, we will investigate this problem in detail and prove that it remains hard even if one restricts to planar graphs of bounded degree or constant broadcasting time. A simple consequence is that the minimal broadcasting time cannot even be approximated with an error less than 1/8, unless P=NP.

On the other hand, we will investigate for which classes of graphs this problem can be solved efficiently and show that broadcasting and even a more general version of this problem becomes easy for graphs with good decomposition properties. The solution strategy can efficiently be parallelized, too. Combining the negative and the positive results reveals the parameters that make broadcasting difficult. Depending on simple graph properties the complexity jumps from NC or P to NP.


graph algorithms graph decomposition computational complexity 


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  1. [ALS91]
    S. Arnborg, J. Lagergren, and D. Seese, Easy Problems for Tree-Decomposable Graphs J. Algorithms 12, 1991, 308–340.Google Scholar
  2. [BHLP92]
    J.-C. Bermond, P. Hell, A. Liestman, and J. Peters, Broadcasting in Bounded Degree Graphs, SIAM J. Disc. Math. 5, 1992, 10–24.Google Scholar
  3. [BK91]
    H. Bodlaender, T. Kloks, Better Algorithms for the Pathwidth and Tree-width of Graphs, Proc. 18 ICALP, 1991, 544–555.Google Scholar
  4. [DF86]
    M. Dyer and A. Frieze, Planar 3DM Is NP-Complete, J. Algorithms 7, 1986, 174–184.Google Scholar
  5. [GJ79]
    M. Garey and D. Johnson, Computers and Intractability, A Guide To the Theory of NP-Completeness, Freeman 1979.Google Scholar
  6. [H90]
    W. Hohberg, The Decomposition of Graphs into k-Connected Components for Arbitraryk, Technical Report, TH Darmstadt, 1990.Google Scholar
  7. [HHL88]
    S. Hedetniemi, S. Hedetniemi, and A. Liestman, A Survey of Gossiping and Broadcasting in Communication Networks, Networks 18, 1988, 319–349.Google Scholar
  8. [HJM90]
    J. Hromkovič, C.-D. Jeschke, and B. Moinien, Optimal Algorithms for Dissemination of Information in Some Interconnection Networks, Proc. 15. MFCS, 1990, 337–346.Google Scholar
  9. [HR89]
    W. Hohberg, R. Reischuk, Decomposition of Graphs — A Uniform Approach for the Design of Fast Sequential and Parallel Algorithms on Graphs, Technical Report, TH Darmstadt, 1989.Google Scholar
  10. [JRS93]
    A. Jakoby, R. Reischuk, C. Schindelhauer, The Complexity of Broadcasting in Planar and Decomposable Graphs, Technical Report, TH Darmstadt, 1993.Google Scholar
  11. [L90]
    J. Lagergren, Efficient Parallel Algorithms for Tree-Decomposition and Related Problems, Proc. 31. FoCS, 1990, 173–182.Google Scholar
  12. [LP88]
    A. Liestman and J. Peters, Broadcast Networks of Bounded Degree, SIAM J. Disc. Math. 4, 1988, 531–540.Google Scholar
  13. [M93]
    M. Middendorf, Minimum Broadcast Time is NP-complete for 3-regular planar graphs and deadline 2, IPL 46, 1993, 281–287.Google Scholar
  14. [R91a]
    R. Reischuk, An Algebraic Divide-and-Conquer Approach to Design Highly Parallel Solution Strategies for Optimization Problems on Graphs, Technical Report, TH Darmstadt, 1991.Google Scholar
  15. [R91b]
    R. Reischuk, Graph Theoretical Methods for the Design of Parallel Algorithms, Proc. 8. FCT, 1991, 61–67.Google Scholar
  16. [RS86]
    N. Robertson, P. Seymour, Graph Minors II. Algorithmic Aspects of Tree-Width, J. Alg. 7, 1986, 309–322.Google Scholar
  17. [SCH81]
    P. Slater, E. Cockayne, and S. Hedetniemi, Information Dissemination in Trees, SIAM J. Comput. 10, 1981, 692–701.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Andreas Jakoby
    • 1
  • Rüdiger Reischuk
    • 1
  • Christian Schindelhauer
    • 1
  1. 1.Med. Universität zu LübeckGermany

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