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Abstract

Inspired by Feige (36th STOC, 2004), we initiate a study of sublinear randomized algorithms for approximating average parameters of a graph. Specifically, we consider the average degree of a graph and the average distance between pairs of vertices in a graph. Since our focus is on sublinear algorithms, these algorithms access the input graph via queries to an adequate oracle.

We consider two types of queries. The first type is standard neighborhood queries (i.e., what is the i th neighbor of vertex v ?), whereas the second type are queries regarding the quantities that we need to find the average of (i.e., what is the degree of vertex v ? and what is the distance between u and v ?, respectively).

Loosely speaking, our results indicate a difference between the two problems: For approximating the average degree, the standard neighbor queries suffice and in fact are preferable to degree queries. In contrast, for approximating average distances, the standard neighbor queries are of little help whereas distance queries are crucial.

Keywords

Average Distance Average Degree Average Parameter Input Graph Neighbor Query 
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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References

  1. 1.
    Barhum, K.: MSc. thesis, Weizmann Institute of Science (in preparation)Google Scholar
  2. 2.
    Bădoiu, M., Czumaj, A., Indyk, P., Sohler, C.: Facility Location in Sublinear Time. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, Springer, Heidelberg (2005)Google Scholar
  3. 3.
    Chazelle, B., Rubinfeld, R., Trevisan, L.: Approximating the Minimum Spanning Tree Weight in Sublinear Time. SIAM Journal on Computing 34, 1370–1379 (2005)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Coppersmith, D., Winograd, S.: Matrix Multiplication via Arithmetic Progression. Journal of Symbolic Computation 9, 251–280 (1990)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dor, D., Halperin, S., Zwick, U.: All Pairs Almost Shortest Paths. SIAM Journal on Computing 29, 1740–1759 (2000)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Elkin, M.L.: Computing Almost Shortest Paths. Technical Report MCS01–03, Faculty of Mathematics and Computer Science, Weizmann Institute of Science (2001)Google Scholar
  7. 7.
    Even, S.: Graph Algorithms. Computer Science Press (1979)Google Scholar
  8. 8.
    Feige, U.: On sums of independent random variables with unbounded variance, and estimating the average degree in a graph. In: Proc. of the 36th STOC, pp. 594–603 (2004)Google Scholar
  9. 9.
    Galil, Z., Margalit, O.: All Pairs Shortest Paths for Graphs with Small Integer Length Edges. Information and Computation 54, 243–254 (1997)MATHMathSciNetGoogle Scholar
  10. 10.
    Goldreich, O., Ron, D.: Property Testing in Bounded Degree Graphs. Algorithmica 32(2), 302–343 (2002)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Goldreich, O., Ron, D.: A Sublinear Bipartitness Tester for Bounded Degree Graphs. Combinatorica 19(3), 335–373 (1999)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Goldreich, O., Ron, D.: Approximating Average Parameters of Graphs. ECCC, TR05-073Google Scholar
  13. 13.
    Indyk, P.: Sublinear Time Algorithms for Metric Space Problems. In: Proc. of the 31st STOC, pp. 428–434 (1999)Google Scholar
  14. 14.
    Kaufman, T., Krivelevich, M., Ron, D.: Tight Bounds for Testing Bipartiteness in General Graphs. SIAM Journal on Computing 33, 1441–1483 (2004)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Parnas, M., Ron, D.: Testing the diameter of graphs. Random Structures and Algorithms 20(2), 165–183 (2002)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Siedel, R.: On the All-Pairs-Shortest-Path Problem in Unweighted Undirected Graphs. Journal of Computer and System Sciences 51, 400–403 (1995)CrossRefMathSciNetGoogle Scholar
  17. 17.
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Oded Goldreich
    • 1
  • Dana Ron
    • 2
  1. 1.Department of Computer ScienceWeizmann Institute of ScienceRehovotIsrael
  2. 2.Department of Electrical Engineering-SystemsTel Aviv UniversityTel AvivIsrael

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