Approximating the Minimum Spanning Tree Weight in Sublinear Time

  • Bernard Chazelle
  • Ronitt Rubinfeld
  • Luca Trevisan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2076)


We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of maximum degree d, with edge weights in the set 1,..., w, and given a parameter 0 < ε < 1/2, estimates in time O(dw-2 log w/∈ ) the weight of the minimum spanning tree of G with a relative error of at most ε. Note that the running time does not depend on the number of vertices in G. We also prove a nearly matching lower bound of Ω(dw-2) on the probe and time complexity of any approximation algorithm for MST weight.

The essential component of our algorithm is a procedure for estimating in time O(dε-2 log ε-1) the number of connected components of an unweighted graph to within an additive error of εn. The time bound is shown to be tight up to within the log ε-1 factor. Our connected- components algorithm picks O(1/∈2) vertices in the graph and then grows “local spanning trees” whose sizes are specified by a stochastic process. From the local information collected in this way, the algorithm is able to infer, with high confidence, an estimate of the number of connected components. We then show how estimates on the number of components in various subgraphs of G can be used to estimate the weight of its MST.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Alon, N., Dar, S., Parnas, M., Ron, D., Testing of clustering, Proc. FOCS, 2000.Google Scholar
  2. [2]
    Chazelle, B., A minimum spanning tree algorithm with inverse-Ackermann type complexity, J. ACM, 47 (2000), 1028–1047.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Chazelle, B., The Discrepancy Method: Randomness and Complexity, Cambridge University Press, 2000.Google Scholar
  4. [4]
    Eppstein, D., Representing all minimum spanning trees with applications to counting and generation, Tech. Rep. 95-50, ICS, UCI, 1995.Google Scholar
  5. [5]
    Fredman, M.L., Willard, D.E. Trans-dichotomous algorithms for minimum spanning trees and shortest paths, J. Comput. and System Sci., 48 (1993), 424–436.CrossRefMathSciNetGoogle Scholar
  6. [6]
    Frieze, A., Kannan, R. Quick approximation to matrices and applications, Combinatorica, 19 (1999)Google Scholar
  7. [7]
    Frieze, A., Kannan, R., Vempala, S., Fast monte-carlo algorithms for finding low-rank approximations, Proc. 39th FOCS (1998).Google Scholar
  8. [8]
    Goldreich, O., Goldwasser, S., Ron, D., Property testing and its connection to learning and approximation, Proc. 37th FOCS (1996), 339–348.Google Scholar
  9. [9]
    Goldreich, O., Ron, D., Property testing in bounded degree graphs, Proc. 29th STOC (1997), 406–415.Google Scholar
  10. [10]
    Graham, R.L., Hell, P. On the history of the minimum spanning tree problem, Ann. Hist. Comput. 7 (1985), 43–57.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Karger, D.R., Klein, P.N, Tarjan, R.E., A randomized linear-time algorithm to find minimum spanning trees, J. ACM, 42 (1995), 321–328.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Nešetřil, J. A few remarks on the history of MST-problem, Archivum Mathematicum, Brno 33 (1997), 15–22. Prelim. version in KAM Series, Charles University, Prague, No. 97-338, 1997.MATHGoogle Scholar
  13. [13]
    Pettie, S., Ramachandran, V. An optimal minimum spanning tree algorithm, Proc. 27th ICALP (2000).Google Scholar
  14. [14]
    Ron, D., Property testing (a tutorial), to appear in “Handbook on Randomization.”Google Scholar
  15. [15]
    Rubinfeld, R., Sudan, M., Robust characterizations of polynomials with applications to program testing, SIAM J. Comput. 25 (1996), 252–271.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Bernard Chazelle
    • 1
  • Ronitt Rubinfeld
    • 2
  • Luca Trevisan
    • 3
  1. 1.Princeton University and NEC Research InstitutePrinceton
  2. 2.NEC Research InstitutePrinceton
  3. 3.U.C. BerkeleyBerkeley

Personalised recommendations