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 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.