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A Query Algorithm for Learning a Spanning Forest in Weighted Undirected Graphs
Proceedings of The 34th International Conference on Algorithmic Learning Theory, PMLR 201:259-274, 2023.
Abstract
We consider the problem of finding a spanning forest in an unknown {\em weighted} undirected graph when the access to the graph is via CUT queries, that is, one can query a subset S⊆V of vertices and get the cut-value ∑e∈∂Sw(e) as the response. It is not too hard to solve this problem using O(nlogn) queries mimicking a Prim-style algorithm using a binary-search style idea. In this paper we use the power of CUT queries to obtain a Monte-Carlo algorithm that makes O(nloglogn(logloglogn)2) CUT queries. At the core of our contribution is a generalization of a result in [Apers et al., 2022] which studies the same problem on unweighted graphs, but to handle weights, we need to combine their ideas with ideas for {\em support estimation} of weighted vectors, as in [Stockmeyer, 1983], and {\em weighted graph reconstruction} algorithms, as in [Bshouty and Mazzawi, 2012].