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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Veröffentlicht in:	SIAM journal on computing Jg. 34; H. 6; S. 1370 - 1379
Hauptverfasser:	Chazelle, Bernard, Rubinfeld, Ronitt, Trevisan, Luca
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2005
Schlagworte:	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
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Abstract	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.
AbstractList	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.
Author	Trevisan, Luca Chazelle, Bernard Rubinfeld, Ronitt
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Keywords	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
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SubjectTerms	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
Title	Approximating the Minimum Spanning Tree Weight in Sublinear Time
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