Reactive Proximity Data Structures for Graphs

  • David Eppstein
  • Michael T. Goodrich
  • Nil Mamano
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


We consider data structures for graphs where we maintain a subset of the nodes called sites, and allow proximity queries, such as asking for the closest site to a query node, and update operations that enable or disable nodes as sites. We refer to a data structure that can efficiently react to such updates as reactive. We present novel reactive proximity data structures for graphs of polynomial expansion, i.e., the class of graphs with small separators, such as planar graphs and road networks. Our data structures can be used directly in several logistical problems and geographic information systems dealing with real-time data, such as emergency dispatching. We experimentally compare our data structure to Dijkstra’s algorithm in a system emulating random queries in a real road network.


  1. 1.
    Agarwal, P.K., Eppstein, D., Matoušek, J.: Dynamic half-space reporting, geometric optimization, and minimum spanning trees. In: 33rd Symposium Foundations of Computer Science (FOCS), pp. 80–89 (1992)Google Scholar
  2. 2.
    Arkin, E.M., Bae, S.W., Efrat, A., Okamoto, K., Mitchell, J.S., Polishchuk, V.: Geometric stable roommates. Inf. Process. Lett. 109(4), 219–224 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aurenhammer, F.: Voronoi diagrams–a survey of a fundamental geometric data structure. ACM Comput. Surv. 23(3), 345–405 (1991)CrossRefGoogle Scholar
  4. 4.
    Buriol, L.S., Resende, M.G.C., Thorup, M.: Speeding up dynamic shortest-path algorithms. INFORMS J. Comput. 20(2), 191–204 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chan, E.P.F., Yang, Y.: Shortest path tree computation in dynamic graphs. IEEE Trans. Comput. 58(4), 541–557 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chan, T.M.: A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries. J. ACM 57(3), 16:1–16:15 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chung, F.R.K.: Separator theorems and their applications. In: Paths, flows, and VLSI-layout (Bonn, 1988), Algorithms Combin. vol. 9, pp. 17–34. Springer, Berlin (1990)Google Scholar
  8. 8.
    Clarkson, K.L.: Nearest-neighbor searching and metric space dimensions. In: Shakhnarovich, G., Darrell, T., Indyk, P. (eds.) Nearest-Neighbor Methods in Learning and Vision: Theory and Practice, pp. 15–59. MIT Press (2006). Chapter 2Google Scholar
  9. 9.
    Cormen, T.H., Stein, C., Rivest, R.L., Leiserson, C.E.: Introduction to Algorithms, 2nd edn. McGraw-Hill, New York City (2001)zbMATHGoogle Scholar
  10. 10.
    De Berg, M., Cheong, O., Van Kreveld, M., Overmars, M.: Computational Geometry: Introduction. Springer, Heidelberg (2008). CrossRefzbMATHGoogle Scholar
  11. 11.
    Demetrescu, C., Goldberg, A.V., Johnson, D.S.: 9th DIMACS implementation challenge: shortest paths (2006).
  12. 12.
    Demetrescu, C., Italiano, G.F.: A new approach to dynamic all pairs shortest paths. J. ACM 51(6), 968–992 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Djidjev, H.N., Pantziou, G.E., Zaroliagis, C.D.: On-line and dynamic algorithms for shortest path problems. In: Mayr, E.W., Puech, C. (eds.) STACS 1995. LNCS, vol. 900, pp. 193–204. Springer, Heidelberg (1995). CrossRefGoogle Scholar
  14. 14.
    Dujmović, V., Eppstein, D., Wood, D.R.: Structure of graphs with locally restricted crossings. SIAM J. Discrete Math. 31(2), 805–824 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dvořák, Z., Norin, S.: Strongly sublinear separators and polynomial expansion. SIAM J. Discrete Math. 30(2), 1095–1101 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    van Emde Boas, P.: Preserving order in a forest in less than logarithmic time. In: 16th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 75–84 (1975)Google Scholar
  17. 17.
    Eppstein, D.: Dynamic Euclidean minimum spanning trees and extrema of binary functions. Discrete Comput. Geom. 13(1), 111–122 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Eppstein, D.: Fast hierarchical clustering and other applications of dynamic closest pairs. J. Exp. Algorithmics 5 (2000)Google Scholar
  19. 19.
    Eppstein, D.: All maximal independent sets and dynamic dominance for sparse graphs. ACM Trans. Algorithms 5(4), Article No. 38 (2009)Google Scholar
  20. 20.
    Eppstein, D.: Treetopes and their graphs. In: 27th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 969–984 (2016).
  21. 21.
    Eppstein, D., Galil, Z., Italiano, G.F.: Dynamic graph algorithms. In: Atallah, M.J. (ed.) Algorithms and Theory of Computation Handbook, pp. 9.1–9.28, 2nd edn. CRC Press(2010).
  22. 22.
    Eppstein, D., Goodrich, M.T., Korkmaz, D., Mamano, N.: Defining equitable geographic districts in road networks via stable matching. In: 25th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (2017)Google Scholar
  23. 23.
    Eppstein, D., Goodrich, M.T., Sun, J.Z.: Skip quadtrees: dynamic data structures for multidimensional point sets. Int. J. Comput. Geom. Appl. 18(1–2), 131–160 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Eppstein, D., Goodrich, M.T., Trott, L.: Going off-road: transversal complexity in road networks. In: 17th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pp. 23–32 (2009)Google Scholar
  25. 25.
    Eppstein, D., Gupta, S.: Crossing patterns in nonplanar road networks. In: 25th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, September 2017Google Scholar
  26. 26.
    Erwig, M.: The graph Voronoi diagram with applications. Networks 36(3), 156–163 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Frieze, A.M., Miller, G.L., Teng, S.H.: Separator based parallel divide and conquer in computational geometry. In: 4th ACM Symposium on Parallel Algorithms and Architectures (SPAA), pp. 420–429 (1992).
  28. 28.
    Frigioni, D., Italiano, G.F.: Dynamically switching vertices in planar graphs. Algorithmica 28(1), 76–103 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Gilbert, J.R., Hutchinson, J.P., Tarjan, R.E.: A separator theorem for graphs of bounded genus. J. Algorithms 5(3), 391–407 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Goodrich, M.T.: Planar separators and parallel polygon triangulation. J. Comput. Syst. Sci. 51(3), 374–389 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Goodrich, M.T., Tamassia, R.: Algorithm Design and Applications, 1st edn. Wiley, Hoboken (2014)zbMATHGoogle Scholar
  32. 32.
    Henzinger, M.R., Klein, P., Rao, S., Subramanian, S.: Faster shortest-path algorithms for planar graphs. J. Comput. Syst. Sci. 55(1), 3–23 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kaplan, H., Mulzer, W., Roditty, L., Seiferth, P., Sharir, M.: Dynamic planar Voronoi diagrams for general distance functions and their algorithmic applications. In: 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2495–2504 (2017)Google Scholar
  34. 34.
    Kawarabayashi, K., Reed, B.: A separator theorem in minor-closed classes. In: 51st IEEE Symposium on Foundations of Computer Science (FOCS), pp. 153–162 (2010)Google Scholar
  35. 35.
    King, V.: Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In: 40th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 81–89 (1999)Google Scholar
  36. 36.
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36(2), 177–189 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Roditty, L., Zwick, U.: On dynamic shortest paths problems. Algorithmica 61(2), 389–401 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Samet, H.: The design and analysis of spatial data structures. Addison-Wesley Series in Computer Science. Addison-Wesley, Reading (1990)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • David Eppstein
    • 1
  • Michael T. Goodrich
    • 1
  • Nil Mamano
    • 1
  1. 1.Department of Computer ScienceUniversity of CaliforniaIrvineUSA

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