On Balanced Separators in Road Networks

  • Aaron Schild
  • Christian Sommer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9125)


The following algorithm partitions road networks surprisingly well: (i) sort the vertices by longitude (or latitude, or some linear combination) and (ii) compute the maximum flow from the first \(k\) nodes (forming the source) to the last \(k\) nodes (forming the sink). Return the corresponding minimum cut as an edge separator (or recurse until the resulting subgraphs are sufficiently small).


Road Network Planar Graph Graph Partitioning Inertial Flow Distance Oracle 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AK07]
    Arora, S., Kale, S.: A combinatorial, primal-dual approach to semidefinite programs. In: 39th ACM Symposium on Theory of Computing (STOC), pp. 227–236 (2007)Google Scholar
  2. [AL08]
    Andersen, R., Lang, K.J.: An algorithm for improving graph partitions. In: 19th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 651–660 (2008)Google Scholar
  3. [And86]
    Andreae, T.: On a pursuit game played on graphs for which a minor is excluded. Journal of Combinatorial Theory, Series B 41(1), 37–47 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  4. [AST90]
    Alon, N., Seymour, P.D., Thomas, R.: A separator theorem for nonplanar graphs. Journal of the American Mathematical Society 3(4), 801–808 (1990). Announced at STOC 1990zbMATHMathSciNetCrossRefGoogle Scholar
  5. [BCLS87]
    Bui, T.N., Chaudhuri, S., Leighton, F.T., Sipser, M.: Graph bisection algorithms with good average case behavior. Combinatorica 7(2), 171–191 (1987). Announced at FOCS 1984MathSciNetCrossRefGoogle Scholar
  6. [BCRW13]
    Bauer, R., Columbus, T., Rutter, I., Wagner, D.: Search-space size in contraction hierarchies. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 93–104. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  7. [BK04]
    Boykov, Y., Kolmogorov, V.: An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. In: Figueiredo, M., Zerubia, J., Jain, A.K. (eds.) EMMCVPR 2001. LNCS, vol. 2134, pp. 359–374. Springer, Heidelberg (2001) CrossRefGoogle Scholar
  8. [BMS+13]
    Buluç, A., Meyerhenke, H., Safro, I., Sanders, P., Schulz, C.: Recent advances in graph partitioning (2013). arXiv, abs/1311.3144Google Scholar
  9. [BMS+14]
    Bader, D.A., Meyerhenke, H., Sanders, P., Schulz, C., Kappes, A., Wagner, D.: Benchmarking for graph clustering and partitioning. In: Encyclopedia of Social Network Analysis and Mining, pp. 73–82 (2014)Google Scholar
  10. [BMSW13]
    Bader, D.A., Meyerhenke, H., Sanders, P., Wagner, D. (eds.): Graph Partitioning and Graph Clustering. In: 10th DIMACS Implementation Challenge Workshop of Contemporary Mathematics, vol. 588 (2013)Google Scholar
  11. [BVZ01]
    Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence 23(11), 1222–1239 (2001). Announced at ICCV 1999CrossRefGoogle Scholar
  12. [CKK+06]
    Chawla, S., Krauthgamer, R., Kumar, R., Rabani, Y., Sivakumar, D.: On the hardness of approximating multicut and sparsest-cut. Computational Complexity 15(2), 94–114 (2006). Announced at CCC 2005zbMATHMathSciNetCrossRefGoogle Scholar
  13. [DFG+14]
    Delling, D., Fleischman, D., Goldberg, A.V., Razenshteyn, I., Werneck, R.F.: An exact combinatorial algorithm for minimum graph bisection. Mathematical Programming Series A (2014)Google Scholar
  14. [DGJ08]
    Demetrescu, C., Goldberg, A.V., Johnson, D.S.: Implementation challenge for shortest paths. In: Encyclopedia of Algorithms (2008)Google Scholar
  15. [DGPW13]
    Delling, D., Goldberg, A.V., Pajor, T., Werneck, R.F.: Customizable route planning. In: Pardalos, P.M., Rebennack, S. (eds.) SEA 2011. LNCS, vol. 6630, pp. 376–387. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  16. [DGRW11]
    Delling, D., Goldberg, A.V., Razenshteyn, I., Werneck, R.F.F.: Graph partitioning with natural cuts. In: 25th IEEE International Symposium on Parallel and Distributed Processing (IPDPS), pp. 1135–1146 (2011)Google Scholar
  17. [DHM+09]
    Delling, D., Holzer, M., Müller, K., Schulz, F., Wagner, D.: High-performance multi-level routing. In: The Shortest Path Problem: 9th DIMACS Implementation Challenge, vol. 74, pp. 73–92 (2009)Google Scholar
  18. [Din70]
    Dinic, E.A.: Algorithm for solution of a problem of maximum flow in a network with power estimation. Doklady Akademii Nauk SSSR; Translation in Soviet Mathematics Doklady 11(5), 1277–1280 (1970)Google Scholar
  19. [Dji85]
    Djidjev, H.N.: A linear algorithm for partitioning graphs of fixed genus. Serdica. Bulgariacae mathematicae publicationes 11(4), 369–387 (1985)zbMATHMathSciNetGoogle Scholar
  20. [Dji96]
    Djidjev, H.N.: Efficient algorithms for shortest path queries in planar digraphs. In: D’Amore, F., Marchetti-Spaccamela, A., Franciosa, P.G. (eds.) WG 1996. LNCS, vol. 1197, pp. 151–165. Springer, Heidelberg (1997) CrossRefGoogle Scholar
  21. [DSW14]
    Dibbelt, J., Strasser, B., Wagner, D.: Customizable contraction hierarchies. In: Gudmundsson, J., Katajainen, J. (eds.) SEA 2014. LNCS, vol. 8504, pp. 271–282. Springer, Heidelberg (2014) Google Scholar
  22. [FR06]
    Fakcharoenphol, J., Rao, S.: Planar graphs, negative weight edges, shortest paths, and near linear time. Journal of Computer and System Sciences 72(5), 868–889 (2006). Announced at FOCS 2001zbMATHMathSciNetCrossRefGoogle Scholar
  23. [Fre87]
    Frederickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM Journal on Computing 16(6), 1004–1022 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  24. [GHK+11]
    Goldberg, A.V., Hed, S., Kaplan, H., Tarjan, R.E., Werneck, R.F.: Maximum flows by incremental breadth-first search. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 457–468. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  25. [GHT84]
    Gilbert, J.R., Hutchinson, J.P., Tarjan, R.E.: A separator theorem for graphs of bounded genus. Journal of Algorithms 5(3), 391–407 (1984). Announced as TR82-506 in 1982zbMATHMathSciNetCrossRefGoogle Scholar
  26. [GR98]
    Goldberg, A.V., Rao, S.: Beyond the flow decomposition barrier. Journal of the ACM 45(5), 783–797 (1998). Announced at FOCS 1997zbMATHMathSciNetCrossRefGoogle Scholar
  27. [HKRS97]
    Henzinger, M.R., Klein, P.N., Rao, S., Subramanian, S.: Faster shortest-path algorithms for planar graphs. Journal of Computer and System Sciences 55(1), 3–23 (1997). Announced at STOC 1994zbMATHMathSciNetCrossRefGoogle Scholar
  28. [HSW08]
    Holzer, M., Schulz, F., Wagner, D.: Engineering multilevel overlay graphs for shortest-path queries. ACM Journal of Experimental Algorithmics 13(2008). Announced at ALENEX 2006Google Scholar
  29. [KKS11]
    Kawarabayashi, K., Klein, P.N., Sommer, C.: Linear-space approximate distance oracles for planar, bounded-genus and minor-free graphs. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 135–146. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  30. [KLSV10]
    Kieritz, T., Luxen, D., Sanders, P., Vetter, C.: Distributed time-dependent contraction hierarchies. In: Festa, P. (ed.) SEA 2010. LNCS, vol. 6049, pp. 83–93. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  31. [KMS13]
    Klein, P.N., Mozes, S., Sommer, C.: Structured recursive separator decompositions for planar graphs in linear time. In: 45th ACM Symposium on Theory of Computing (STOC), pp. 505–514 (2013)Google Scholar
  32. [KRV09]
    Khandekar, R., Rao, S., Vazirani, U.V.: Graph partitioning using single commodity flows. Journal of the ACM 56(4) (2009). Announced at STOC 2006Google Scholar
  33. [KV05]
    Khot, S., Vishnoi, N.K.: The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into \(\ell _1\). In: 46th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 53–62 (2005)Google Scholar
  34. [LR04]
    Lang, K., Rao, S.: A flow-based method for improving the expansion or conductance of graph cuts. In: Bienstock, D., Nemhauser, G. (eds.) IPCO 2004. LNCS, vol. 3064, pp. 325–337. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  35. [LT79]
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM Journal on Applied Mathematics 36(2), 177–189 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  36. [Mad13]
    Madry, A.: Navigating central path with electrical flows: from flows to matchings, and back. In: 54th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 253–262 (2013)Google Scholar
  37. [MS12]
    Mozes, S., Sommer, C.: Exact distance oracles for planar graphs. In: 23rd ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 209–222 (2012)Google Scholar
  38. [OSVV08]
    Orecchia, L., Schulman, L.J., Vazirani, U.V., Vishnoi, N.K.: On partitioning graphs via single commodity flows. In: 40th ACM Symposium on Theory of Computing (STOC), pp. 461–470 (2008)Google Scholar
  39. [She09]
    Sherman, J.: Breaking the multicommodity flow barrier for \(O(\sqrt{\log n})\)-approximations to sparsest cut. In: 50th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 363–372 (2009)Google Scholar
  40. [Som14]
    Sommer, C.: Shortest-path queries in static networks. ACM Computing Surveys 46, 45:1–45:31 (2014)CrossRefGoogle Scholar
  41. [SS11]
    Sanders, P., Schulz, C.: Engineering multilevel graph partitioning algorithms. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 469–480. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  42. [SS12]
    Sanders, P., Schulz, C.: Distributed evolutionary graph partitioning. In: 14th Meeting on Algorithm Engineering & Experiments, (ALENEX), pp. 16–29 (2012)Google Scholar
  43. [SS13]
    Sanders, P., Schulz, C.: Think locally, act globally: highly balanced graph partitioning. In: Demetrescu, C., Marchetti-Spaccamela, A., Bonifaci, V. (eds.) SEA 2013. LNCS, vol. 7933, pp. 164–175. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  44. [ST97]
    Simon, H.D., Teng, S.-H.: How good is recursive bisection? SIAM Journal on Scientific Computing 18, 1436–1445 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  45. [Ung51]
    Ungar, P.: A theorem on planar graphs. Journal of the London Mathematical Society s1–26(4), 256–262 (1951)MathSciNetCrossRefGoogle Scholar
  46. [vWZA13]
    van Walderveen, F., Zeh, N., Arge, L.: Multiway simple cycle separators and I/O-efficient algorithms for planar graphs. In: 24th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 901–918 (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.UC BerkeleyBerkeleyUSA
  2. 2.Apple Inc.CupertinoUSA

Personalised recommendations