A linear-time algorithm for edge-disjoint paths in planar graphs

  • Dorothea Wagner
  • Karsten Weihe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 726)


In this paper we discuss the problem of finding edge-disjoint paths in a planar, undirected graph s.t. each path connects two specified vertices on the outer face boundary. We will focus on the “classical” case where an instance must additionally fulfill the so-called evenness-condition. The fastest algorithm for this problem known from the literature requires \(\mathcal{O}\left( {n^{{5 \mathord{\left/{\vphantom {5 3}} \right.\kern-\nulldelimiterspace} 3}} \left( {\log \log n} \right)^{{1 \mathord{\left/{\vphantom {1 3}} \right.\kern-\nulldelimiterspace} 3}} } \right)\) time, where n denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which yields an \(\mathcal{O}\left( n \right)\) algorithm.


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  1. [BM]
    M. Becker and K. Mehlhorn (1986): Algorithms for Routing in Planar Graphs. Acta Informatica 23, 163–176.Google Scholar
  2. [Fr]
    G.N. Frederickson (1987): Fast Algorithms for Shortest Paths in Planar Graphs, with Applications. SIAM J. Comput. 16, 1004–1022.Google Scholar
  3. [GT]
    H.N. Gabow and R.E. Tarjan (1985): A Linear-Time Algorithm for a Special Case of Disjoint Set Union. J. Comp. System Sciences 30, 209–221.Google Scholar
  4. [Ha]
    R. Hassin (1984): On Multicommodity Flows in Planar Graphs. Networks 14, 225–235.Google Scholar
  5. [Ka]
    M. Kaufmann (1990): A Linear-Time Algorithm for Routing in a Convex Grid. IEEE Transact. Computer-Aided Design 9, 180–184.Google Scholar
  6. [KK]
    M. Kaufmann and G. Klär (1991): A Faster Algorithm for Edge-Disjoint Paths in Planar Graphs. Proc. Int. Symp. on Algorithms (ISA '91), LNCS 557, 336–348.Google Scholar
  7. [KM]
    M. Kaufmann and K. Mehlhorn (1986): Generalized Switchbox Routing. J. Algorithms 7, 510–531.Google Scholar
  8. [KL]
    M.R. Kramer and J. van Leeuwen (1984): The complexity of Wire-Routing and Finding Minimum Area Layouts for Arbitrary VLSI-Circuits. Advances Comp. Res. 2, 129–146.Google Scholar
  9. [MNS]
    K. Matsumuto, T. Nishizeki and N. Saito (1985): An Efficient Algorithm for Finding Multicommodity Flows in Planar Networks. SIAM J. Comp. 14, 289–302.Google Scholar
  10. [NSS]
    T. Nishizeki, N. Saito and K. Suzuki (1985): A Linear-Time Routing Algorithm for Convex Grids. IEEE Transact. Computer-Aided Design CAD-4, 68–76.Google Scholar
  11. [SAN]
    H. Suzuki, T. Akama and T. Nishizeki (1990): Finding Steiner Forests in Planar Graphs. First Proc. ACM-SIAM Symp. Discrete Algorithms, 444–453.Google Scholar
  12. [OS]
    H. Okamura and P.D. Seymour (1981): Multicommodity Flows in Planar Graphs. J. Combinatorial Th. B 31, 75–81.Google Scholar
  13. [Ta]
    R.E. Tarjan (1979): A Class of Algorithms Which Require Non-Linear Time to Maintain Disjoint Sets. J. Comp. System Sciences 18, 110–127.Google Scholar
  14. [WW]
    D. Wagner and K. Weihe (1993): A Linear-Time Algorithm for Edge-Disjoint Paths in Planar Graphs. Report No. 344 TU BerlinGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Dorothea Wagner
    • 1
  • Karsten Weihe
    • 1
  1. 1.Fachbereich MathematikTechnische Universität BerlinBerlinGermany

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