Learning Spanning Forests Optimally using CUT Queries in Weighted Undirected Graphs
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In this paper we describe a randomized algorithm which returns a maximal spanning forest of an unknown weighted undirected graph making O(n) 𝖢𝖴𝖳 queries in expectation. For weighted graphs, this is optimal due to a result in [Auza and Lee, 2021] which shows an Ω(n) lower bound for zero-error randomized algorithms. this problem where we have upper and lower bounds tight up to constants. These questions have been extensively studied in the past few years, especially due to the problem's connections to symmetric submodular function minimization. We also describe a simple polynomial time deterministic algorithm that makes O(nlog n/loglog n) queries on undirected unweighted graphs and returns a maximal spanning forest, thereby (slightly) improving upon the state-of-the-art.
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