Approximate Distance Queries in Disk Graphs

  • Martin Fürer
  • Shiva Prasad Kasiviswanathan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4368)


We present efficient algorithms for approximately answering distance queries in disk graphs. Let G be a disk graph with n vertices and m edges. For any fixed ε> 0, we show that G can be preprocessed in \(O(m\sqrt{n}\epsilon^{-1}+m\epsilon^{-2}\log S)\) time, constructing a data structure of size O(n 3/2 ε − 1+ − 2logS), such that any subsequent distance query can be answered approximately in \(O(\sqrt{n}\epsilon^{-1}+\epsilon^{-2}\log S)\) time. Here S is the ratio between the largest and smallest radius. The estimate produced is within an additive error which is only ε times the longest edge on some shortest path.

The algorithm uses an efficient subdivision of the plane to construct a sparse graph having many of the same distance properties as the input disk graph. Additionally, the sparse graph has a small separator decomposition, which is then used to answer distance queries. The algorithm extends naturally to the higher dimensional ball graphs.


Short Path Sparse Graph Cluster Graph Unit Disk Graph Vertical Line Segment 
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.


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  1. 1.
    Mead, C., Conway, L.: Introduction to VLSI System. Addison-Wesley, Reading (1980)Google Scholar
  2. 2.
    Krumke, S.O., Marathe, M.V., Ravi, S.S.: Models and approximation algorithms for channel assignment in radio networks. Wireless Networks 7(6), 575–584 (2001)MATHCrossRefGoogle Scholar
  3. 3.
    Li, X.Y.: Algorithmic, geometric and graphs issues in wireless networks. Wireless Communications and Mobile Computing 3(2), 119–140 (2003)CrossRefGoogle Scholar
  4. 4.
    Li, X.Y., Wan, P.J., Frieder, O.: Coverage in wireless ad hoc sensor networks. IEEE Transactions on Computers 52(6), 753–763 (2003)CrossRefGoogle Scholar
  5. 5.
    Srinivas, A., Modiano, E.: Minimum energy disjoint path routing in wireless ad-hoc networks. In: MOBICOM 2003, pp. 122–133. ACM, New York (2003)CrossRefGoogle Scholar
  6. 6.
    Gao, J., Zhang, L.: Well-separated pair decomposition for the unit-disk graph metric and its applications. SIAM Journal on Computing 35(1), 151–169 (2005)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Thorup, M., Zwick, U.: Approximate distance oracles. Journal of ACM 52(1), 1–24 (2005)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Zwick, U.: Exact and approximate distances in graphs - A survey. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 33–48. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to K-nearest-neighbors and N-body potential fields. Journal of ACM 42(1), 67–90 (1995)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Fürer, M., Kasiviswanathan, S.P.: Spanners for geometric intersection graphs (2006), Available at
  11. 11.
    Miller, G.L., Teng, S.H., Vavasis, S.A.: A unified geometric approach to graph separators. In: FOCS 2001, pp. 538–547. IEEE, Los Alamitos (1991)Google Scholar
  12. 12.
    Arikati, S.R., Chen, D.Z., Chew, L.P., Das, G., Smid, M.H.M., Zaroliagis, C.D.: Planar spanners and approximate shortest path queries among obstacles in the plane. In: Díaz, J. (ed.) ESA 1996. LNCS, vol. 1136, pp. 514–528. Springer, Heidelberg (1996)Google Scholar
  13. 13.
    Schieber, B., Vishkin, U.: On finding lowest common ancestors: Simplification and parallelization. SIAM Journal on Computing 17(6), 1253–1262 (1988)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Balakrishnan, H., Barrett, C.L., Kumar, V.S.A., Marathe, M.V., Thite, S.: The distance-2 matching problem and its relationship to the mac-layer capacity of ad hoc wireless networks. IEEE Journal on Selected Areas in Communications 22, 1069–1079 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Martin Fürer
    • 1
  • Shiva Prasad Kasiviswanathan
    • 1
  1. 1.Computer Science and EngineeringPennsylvania State University 

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