A Query Algorithm for Learning a Spanning Forest in Weighted Undirected Graphs

Hang Liao, Deeparnab Chakrabarty

Abstract:

We consider the problem of finding a spanning forest in an unknown 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 P e∈∂S w(e) as the response. It is not too hard to solve this problem using O(n log n) 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(n log log n(log log log n)^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 support estimation of weighted vectors, as in [Stockmeyer, 1983], and weighted graph reconstruction algorithms, as in [Bshouty and Mazzawi, 2012].