Given an undirected graph $G$ of $n$ weighted edges, stored one edge per processor in a square mesh of $n$ processors, we show how to determine the connected components and a minimal spanning forest in $Θ(\sqrt{n})$ time. More generally, we show how to solve these problems in $Θ(n^{1/d})$ time when the mesh is a $d$-dimensional cube, where the implied constants depend upon $d$.
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