Approximating the Minimum Spanning Tree Weight in Sublinear Time

We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set{1,...,w}, and given a parameter $0 < \eps < 1/2$, estimates in time $O( dw \varepsilon^{-2} \log{\frac{dw}\varepsilon})$ the weight of the min...

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Bibliographic Details
Published in:SIAM journal on computing Vol. 34; no. 6; pp. 1370 - 1379
Main Authors: Chazelle, Bernard, Rubinfeld, Ronitt, Trevisan, Luca
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2005
Subjects:
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Approximation
Combinatorics
Combinatorics. Ordered structures
Computer science
Computer science; control theory; systems
Exact sciences and technology
Graph theory
Graphs
Information retrieval. Graph
Mathematics
Sciences and techniques of general use
Theoretical computing
Error estimation
minimum spanning tree
Connected graph
sublinear time algorithms
68W25
Approximation algorithm
Stochastic process
approximation algorithms 68W20
Sublinear time algorithm
Spanning tree
randomized algorithms
Relative error
68R10
Time complexity
ISSN:0097-5397, 1095-7111
Online Access:Get full text
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Summary:We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set{1,...,w}, and given a parameter $0 < \eps < 1/2$, estimates in time $O( dw \varepsilon^{-2} \log{\frac{dw}\varepsilon})$ the weight of the minimum spanning tree (MST) of G with a relative error of at most $\eps$. Note that the running time does not depend on the number of vertices in G. We also prove a nearly matching lower bound of $\Omega( dw \varepsilon^{-2} )$ 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 $O(d\eps^{-2}\log \frac{d}\varepsilon)$ the number of connected components of an unweighted graph to within an additive error of $\varepsilon n$. (This becomes $O(\eps^{-2}\log \frac{1}\varepsilon)$ for $d=O(1)$.) The time bound is shown to be tight up to within the $\log \frac{d}\varepsilon$ factor. Our connected-components algorithm picks $O(1/\varepsilon^2)$ 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.
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ISSN:0097-5397
1095-7111
DOI:10.1137/S0097539702403244