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A Parametrized Analysis of Algorithms on Hierarchical Graphs

  • Rachel FaranEmail author
  • Orna Kupferman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10316)

Abstract

Hierarchical graphs are used in order to describe systems with a sequential composition of sub-systems. A hierarchical graph consists of a vector of subgraphs. Vertices in a subgraph may “call” other subgraphs. The reuse of subgraphs, possibly in a nested way, causes hierarchical graphs to be exponentially more succinct than equivalent flat graphs. Early research on hierarchical graphs and the computational price of their succinctness suggests that there is no strong correlation between the complexity of problems when applied to flat graphs and their complexity in the hierarchical setting. That is, the complexity in the hierarchical setting is higher, but all “jumps” in complexity up to an exponential one are exhibited, including no jumps in some problems.

We continue the study of the complexity of algorithms for hierarchical graphs, with the following contributions: (1) In many applications, the subgraphs have a small, often a constant, number of exit vertices, namely vertices from which control returns to the calling subgraph. We offer a parameterized analysis of the complexity and point to problems where the complexity becomes lower when the number of exit vertices is bounded by a constant. (2) We describe a general methodology for algorithms on hierarchical graphs. The methodology is based on an iterative compression of subgraphs in a way that maintains the solution to the problems and results in subgraphs whose size depends only on the number of exit vertices, and (3) We handle labeled hierarchical graphs, where edges are labeled by letters from some alphabet, and the problems refer to the languages of the graphs.

Keywords

Flow Network Hamiltonian Path Graph Reachability Hierarchical Setting Hierarchical Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Alur, R., Kannan, S., Yannakakis, M.: Communicating hierarchical state machines. In: Wiedermann, J., Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 169–178. Springer, Heidelberg (1999). doi: 10.1007/3-540-48523-6_14 CrossRefGoogle Scholar
  2. 2.
    Alur, R., Yannakakis, M.: Model checking of hierarchical state machines. ACM TOPLAS 23(3), 273–303 (2001)CrossRefGoogle Scholar
  3. 3.
    Aminof, B., Kupferman, O., Murano, A.: Improved model checking of hierarchical systems. Inf. Comput. 210, 68–86 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barrett, C., Jacob, R., Marathe, M.: Formal-language-constrained path problems. SIAM J. Comput. 30(3), 809–837 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Clarke, E.M., Grumberg, O., Peled, D.: Model Checking. MIT Press, Cambridge (1999)Google Scholar
  6. 6.
    de Roever, W.-P.: The need for compositional proof systems: a survey. In: de Roever, W.-P., Langmaack, H., Pnueli, A. (eds.) COMPOS 1997. LNCS, vol. 1536, pp. 1–22. Springer, Heidelberg (1998). doi: 10.1007/3-540-49213-5_1 CrossRefGoogle Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, Heidelberg (2013)CrossRefzbMATHGoogle Scholar
  8. 8.
    Drusinsky, D., Harel, D.: On the power of bounded concurrency I: finite automata. J. ACM 41(3), 517–539 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Can. J. Math. 8(3), 399–404 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Galperin, H., Wigderson, A.: Succinct representations of graphs. Inf. Control 56(3), 183–198 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Harel, D., Kupferman, O., Vardi, M.Y.: On the complexity of verifying concurrent transition systems. Inf. Comput. 173, 1–19 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Immerman, N.: Length of predicate calculus formulas as a new complexity measure. In: Proceedings of 20th FOCS, pp. 337–347 (1979)Google Scholar
  13. 13.
    Kupferman, O., Tamir, T.: Hierarchical network formation games. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10205, pp. 229–246. Springer, Heidelberg (2017). doi: 10.1007/978-3-662-54577-5_13 CrossRefGoogle Scholar
  14. 14.
    Kupferman, O., Vardi, M.Y., Wolper, P.: An automata-theoretic approach to branching-time model checking. J. ACM 47(2), 312–360 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lengauer, T.: The complexity of compacting hierarchically specified layouts of integrated circuits. In: Proceedings of 23rd FOCS, pp. 358–368 (1982)Google Scholar
  16. 16.
    Lengauer, T., Wagner, K.W.: The correlation between the complexities of the nonhierarchical and hierarchical versions of graph problems. JCSS 44, 63–93 (1990)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Lengauer, T., Wanke, E.: Efficient solutions of connectivity problems on hierarchically defined graphs. SIAM J. Comput. 17(6), 1063–1081 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Megiddo, N.: Optimal flows in networks with multiple sources and sinks. Math. Program. 7(1), 97–107 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mendelzon, A.O., Wood, P.T.: Finding regular simple paths in graph databases. SIAM J. Comput. 24(6), 1235–1258 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rothvoß, T.: The matching polytope has exponential extension complexity. In: Proceedings of 46th STOC, pp. 263–272 (2014)Google Scholar

Copyright information

© IFIP International Federation for Information Processing 2017

Authors and Affiliations

  1. 1.School of Engineering and Computer ScienceHebrew UniversityJerusalemIsrael

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