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 < ε < 1/2, estimates in time O(dwε −2 log dw ε ) the weight of the minimum spanning tree (MST) 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(dwε −2 ) on the probe and time complexity of any approximation algorithm for MST weight. The factor. Our connected-components algorithm picks O(1/ε 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.
Sublinear time algorithms for estimating the number of connected component and the weight of MST were first given by Chazelle et al. REF .
Approximating the Minimum Spanning Tree Weight in Sublinear Time
SIAM J. Comput.
Mathematics Computer Science