Computing and Listing st-Paths in Public Transportation Networks

  • Kateřina Böhmová
  • Matúš Mihalák
  • Tobias PrögerEmail author
  • Gustavo Sacomoto
  • Marie-France Sagot
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)


Given a set of directed paths (called lines) L, a public transportation network is a directed graph \(G_L=(V_L,A_L)\) which contains exactly the vertices and arcs of every line \(l\in L\). An st-route is a pair \((\pi ,\gamma )\) where \(\gamma =\langle l_1,\ldots ,l_h \rangle \) is a line sequence and \(\pi \) is an st-path in \(G_L\) which is the concatenation of subpaths of the lines \(l_1,\ldots ,l_h\), in this order. Given a threshold \(\beta \), we present an algorithm for listing all st-paths \(\pi \) for which a route \((\pi ,\gamma )\) with \(|\gamma | \le \beta \) exists, and we show that the running time of this algorithm is polynomial with respect to the input and the output size. We also present an algorithm for listing all line sequences \(\gamma \) with \(|\gamma |\le \beta \) for which a route \((\pi ,\gamma )\) exists, and show how to speed it up using preprocessing. Moreover, we show that for the problem of finding an st-route \((\pi ,\gamma )\) that minimizes the number of different lines in \(\gamma \), even computing an \(o(\log |V|)\)-approximation is NP-hard.


Directed Graph Short Path Problem Recursive Call Listing Problem Transit Network 
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.



We thank the anonymous reviewers for pointing out how the running times of our listing algorithms can be improved by a factor of \(\varTheta (\log M)\). Furthermore we thank Peter Widmayer for many helpful discussions. This work has been partially supported by the Swiss National Science Foundation (SNF) under the grant number 200021 138117/1, and by the EU FP7/2007-2013 (DG CONNECT.H5-Smart Cities and Sustainability), under grant agreement no. 288094 (project eCOMPASS). Kateřina Böhmová is a recipient of a Google Europe Fellowship in Optimization Algorithms, and this research is supported in part by this Google Fellowship. Gustavo Sacomoto is a recipient of a grant from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement \(\text {n}^\circ \) [247073]10 SISYPHE.


  1. 1.
    Bast, H., Delling, D., Goldberg, A.V., Müller-Hannemann, M., Pajor, T., Sanders, P., Wagner, D., Werneck, R.F.: Route planning in transportation networks. CoRR abs/1504.05140 (2015)Google Scholar
  2. 2.
    Bezem, G., Leeuwen, J.V.: Enumeration in graphs. Technical report RUU-CS-87-07, Utrecht University (1987)Google Scholar
  3. 3.
    Birmelé, E., Ferreira, R.A., Grossi, R., Marino, A., Pisanti, N., Rizzi, R., Sacomoto, G.: Optimal listing of cycles and st-paths in undirected graphs. In: SODA 2013, pp. 1884–1896 (2013)Google Scholar
  4. 4.
    Böhmová, K., Mihalák, M., Neubert, P., Pröger, T., Widmayer, P.: Robust routing in urban public transportation: evaluating strategies that learn from the past. In: ATMOS 2015, pp. 68–81 (2015)Google Scholar
  5. 5.
    Böhmová, K., Mihalák, M., Pröger, T., Šrámek, R., Widmayer, P.: Robust routing in urban public transportation: how to find reliable journeys based on past observations. In: ATMOS 2013, pp. 27–41 (2013)Google Scholar
  6. 6.
    Brodal, G.S., Jacob, R.: Time-dependent networks as models to achieve fast exact time-table queries. Electr. Notes Theor. Comput. Sci. 92, 3–15 (2004)CrossRefzbMATHGoogle Scholar
  7. 7.
    Dial, R.B.: Algorithm 360: shortest-path forest with topological ordering. Commun. ACM 12(11), 632–633 (1969)CrossRefGoogle Scholar
  8. 8.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1(1), 269–271 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dinur, I., Steurer, D.: Analytical approach to parallel repetition. In: STOC 2014, pp. 624–633 (2014)Google Scholar
  10. 10.
    Eppstein, D.: Finding the k shortest paths. SIAM J. Comput. 28(2), 652–673 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34(3), 596–615 (1987)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Johnson, D.B.: Finding all the elementary circuits of a directed graph. SIAM J. Comput. 4(1), 77–84 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Katoh, N., Ibaraki, T., Mine, H.: An efficient algorithm for \(k\) shortest simple paths. Networks 12(4), 411–427 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lawler, E.L.: A procedure for computing the \(k\) best solutions to discrete optimization problems and its application to the shortest path problem. Mgmt. Sci. 18, 401–405 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Müller-Hannemann, M., Schulz, F., Wagner, D., Zaroliagis, C.D.: Timetable information: models and algorithms. In: Geraets, F., Kroon, L., et al. (eds.) ATMOS 2004, Part I. LNCS, vol. 4359, pp. 67–90. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Rizzi, R., Sacomoto, G., Sagot, M.-F.: Efficiently listing bounded length st-paths. In: Jan, K., Miller, M., Froncek, D. (eds.) IWOCA 2014. LNCS, vol. 8986, pp. 318–329. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  18. 18.
    Schulz, F., Wagner, D., Zaroliagis, C.D.: Using multi-level graphs for timetable information in railway systems. In: Mount, D.M., Stein, C. (eds.) ALENEX 2002. LNCS, vol. 2409, pp. 43–59. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  19. 19.
    Yen, J.Y.: Finding the \(k\) shortest loopless paths in a network. Mgmt. Sci. 17, 712–716 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Yuan, S., Varma, S., Jue, J.P.: Minimum-color path problems for reliability in mesh networks. In: INFOCOM 2005, pp. 2658–2669 (2005)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Kateřina Böhmová
    • 1
  • Matúš Mihalák
    • 2
  • Tobias Pröger
    • 1
    Email author
  • Gustavo Sacomoto
    • 3
    • 4
  • Marie-France Sagot
    • 3
    • 4
  1. 1.Institut für Theoretische InformatikETH ZürichZürichSwitzerland
  2. 2.Department of Knowledge EngineeringMaastricht UniversityMaastrichtThe Netherlands
  3. 3.INRIA Grenoble Rhône-AlpesMontbonnot-Saint-MartinFrance
  4. 4.UMR CNRS 5558 – LBBEUniversité Lyon 1LyonFrance

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