Directing Road Networks by Listing Strong Orientations

  • Alessio Conte
  • Roberto Grossi
  • Andrea Marino
  • Romeo Rizzi
  • Luca Versari
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9843)

Abstract

A connected road network with N nodes and L edges has \(K \le L\) edges identified as one-way roads. In a feasible direction, these one-way roads are assigned a direction each, so that every node can reach any other [Robbins ’39]. Using O(L) preprocessing time and space usage, it is shown that all feasible directions can be found in O(K) amortized time each. To do so, we give a new algorithm that lists all the strong orientations of an undirected connected graph with m edges in O(m) amortized time each, using O(m) space. The cost can be deamortized to obtain O(m) delay with \(O(m^2)\) preprocessing time and space.

References

  1. 1.
    Arkin, E.M., Hassin, R.: A note on orientations of mixed graphs. Discrete Appl. Math. 116(3), 271–278 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ben-Ameur, W., Glorieux, A., Neto, J.: On the most imbalanced orientation of a graph. In: Xu, D., Du, D., Du, D. (eds.) COCOON 2015. LNCS, vol. 9198, pp. 16–29. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  3. 3.
    Boesch, F., Tindell, R.: Robbins’ theorem for mixed multigraphs. Am. Math. Monthly 87(9), 716–719 (1980)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bollobás, B.: Modern Graph Theory. Springer-Verlag, New York (1998)CrossRefMATHGoogle Scholar
  5. 5.
    Rainer, E., Feldbacher, K., Klinz, B., Woeginger, G.J.: Minimum-cost strong network orientation problems: classification, complexity, and algorithms. Networks 33(1), 57–70 (1999)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chartrand, G., Harary, F., Schultz, M., Wall, C.E.: Forced orientation number of a graph. Congressus Numerantium, pp. 183–192 (1994)Google Scholar
  7. 7.
    Chung, F.R.K., Garey, M.R., Tarjan, R.E.: Strongly connected orientations of mixed multigraphs. Networks 15(4), 477–484 (1985)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chvátal, V., Thomassen, C.: Distances in orientations of graphs. J. Comb. Theory Ser. B 24(1), 61–75 (1978)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Conte, A., Grossi, R., Marino, A., Rizzi, R.: Enumerating cyclic orientations of a graph. In: Lipták, Z., Smyth, W.F. (eds.) IWOCA 2015. LNCS, vol. 9538, pp. 88–99. Springer, Heidelberg (2016)CrossRefGoogle Scholar
  10. 10.
    Conte, A., Grossi, R., Marino, A., Rizzi, R.: Listing acyclic orientations of graphs with single, multiple sources. In: Proceedings LATIN (2011) Observation of strains: Theoretical Informatics - 12th Latin American Symposium, Ensenada, 11-15 April 2016, pp. 319–333 (2016)Google Scholar
  11. 11.
    Dankelmann, P., Oellermann, O.R., Jian-Liang, W.: Minimum average distance of strong orientations of graphs. Discrete Appl. Math. 143(1–3), 204–212 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fomin, E.V., Matamala, M., Prisner, E., Rapaport, I.: At-free graphs: linear bounds for the oriented diameter. Discrete Appl. Math. 141(1), 135–148 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fomin, F.V., Matamala, M.: Complexity of approximating the oriented diameter of chordal graphs. J. Graph Theory 45(4), 255–269 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fukuda, K., Prodon, A., Sakuma, T.: Combinatorics and computer science notes on acyclic orientations and the shelling lemma. Theoret. Comput. Sci. 263(1), 9–16 (2001)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gutin, G.: Minimizing and maximizing the diameter in orientations of graphs. Graphs Comb. 10(2), 225–230 (1994)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hassin, R., Megiddo, N.: On orientations, shortest paths. Linear Algebra Appl. 114, 115, 589–602 (1989). Special Issue Dedicated to Alan J. HoffmanMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Italiano, G.F., Laura, L., Santaroni, F.: Finding strong bridges and strong articulation points in linear time. Theoret. Comput. Sci. 447, 74–84 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Koh, K.M., Tay, E.G.: Optimal orientations of graphs and digraphs: a survey. Graphs Comb. 18(4), 745–756 (2002)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kurz, S., Lätsch, M.: Bounds for the minimum oriented diameter. Discrete Math. Theoret. Comput. Sci. 14(1), 109–140 (2012)MathSciNetMATHGoogle Scholar
  20. 20.
    Plesník, J.: On the sum of all distances in a graph or digraph. J. Graph Theory 8(1), 1–21 (1984)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Robbins, H.E.: A theorem on graphs, with an application to a problem of traffic control. The American Mathematical Monthly 46(5), 281–283 (1939)CrossRefMATHGoogle Scholar
  22. 22.
    Roberts, F.S.: Graph theory and its applications to problems of society. NSF-CBSM Monograph No. 29. SIAM Publications (1978)Google Scholar
  23. 23.
    Roberts, F.S., Xu, Y.: On the optimal strongly connected orientations of city street graphs. II: two East-West avenues or North-South streets. Networks 19(2), 221–233 (1989)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Roberts, F.S., Xu, Y.: On the optimal strongly connected orientations of city street graphs I: large grids. SIAM J. Discrete Math. 1(2), 199–222 (1988)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Roberts, F.S., Xu, Y.: On the optimal strongly connected orientations of city street graphs. III: three East-West avenues or North-South streets. Networks 22(2), 109–143 (1992)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Roberts, F.S., Xu, Y.: On the optimal strongly connected orientations of city street graphs IV: four East-West avenues or North-South streets. Discrete Appl. Math. 49(1), 331–356 (1994)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Uno, T.: Two general methods to reduce delay and change of enumeration algorithms: NII Technical report NII-2003-004E, Tokyo (2003)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Alessio Conte
    • 1
  • Roberto Grossi
    • 1
  • Andrea Marino
    • 1
  • Romeo Rizzi
    • 2
  • Luca Versari
    • 3
  1. 1.Università di PisaPisaItaly
  2. 2.Università di VeronaVeronaItaly
  3. 3.Scuola Normale SuperiorePisaItaly

Personalised recommendations