Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Alon, N., Dar, S., Parnas, M., Ron, D., Testing of clustering, Proc. FOCS, 2000.
Chazelle, B., A minimum spanning tree algorithm with inverse-Ackermann type complexity, J. ACM, 47 (2000), 1028–1047.
Chazelle, B., The Discrepancy Method: Randomness and Complexity, Cambridge University Press, 2000.
Eppstein, D., Representing all minimum spanning trees with applications to counting and generation, Tech. Rep. 95-50, ICS, UCI, 1995.
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.
Frieze, A., Kannan, R. Quick approximation to matrices and applications, Combinatorica, 19 (1999)
Frieze, A., Kannan, R., Vempala, S., Fast monte-carlo algorithms for finding low-rank approximations, Proc. 39th FOCS (1998).
Goldreich, O., Goldwasser, S., Ron, D., Property testing and its connection to learning and approximation, Proc. 37th FOCS (1996), 339–348.
Goldreich, O., Ron, D., Property testing in bounded degree graphs, Proc. 29th STOC (1997), 406–415.
Graham, R.L., Hell, P. On the history of the minimum spanning tree problem, Ann. Hist. Comput. 7 (1985), 43–57.
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.
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.
Pettie, S., Ramachandran, V. An optimal minimum spanning tree algorithm, Proc. 27th ICALP (2000).
Ron, D., Property testing (a tutorial), to appear in “Handbook on Randomization.”
Rubinfeld, R., Sudan, M., Robust characterizations of polynomials with applications to program testing, SIAM J. Comput. 25 (1996), 252–271.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chazelle, B., Rubinfeld, R., Trevisan, L. (2001). Approximating the Minimum Spanning Tree Weight in Sublinear Time. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds) Automata, Languages and Programming. ICALP 2001. Lecture Notes in Computer Science, vol 2076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48224-5_16
Download citation
DOI: https://doi.org/10.1007/3-540-48224-5_16
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42287-7
Online ISBN: 978-3-540-48224-6
eBook Packages: Springer Book Archive