Optimal component labeling algorithms for mesh-connected computers and VLSI

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 $\Theta(\sqrt{n})$ time. More generally, we show how to solve these problems in $\Theta(n^{1/d})$ time when the mesh is a $d$-dimensional cube, where the implied constants depend upon $d$.