<p><b>Abstract</b>—A reconfigurable mesh, R-mesh for short, is a two-dimensional array of processors connected by a grid-shaped reconfigurable bus system. Each processor has four I/O ports that can be locally connected during execution of algorithms. This paper considers the <it>d</it>-dimensional Euclidean Minimum Spanning Tree (EMST) and the All Nearest Neighbors (ANN) problem. Two results are reported. First, we show that a minimum spanning tree of <it>n</it> points in a fixed <it>d</it>-dimensional space can be constructed in <it>O</it>(1) time on a <tmath>$\sqrt {n^3}\times \sqrt {n^3}$</tmath> R-mesh. Second, all nearest neighbors of <it>n</it> points in a fixed <it>d</it>-dimensional space can be constructed in <it>O</it>(1) time on an <it>n</it>×<it>n</it> R-mesh. There is no previous <it>O</it>(1) time algorithm for the EMST problem; ours is the first such algorithm. A previous R-mesh algorithm exists for the two-dimensional ANN problem; we extend it to any <it>d</it>-dimensional space. Both of the proposed algorithms have a time complexity independent of <it>n</it> but growing with <it>d</it>. The time complexity is <it>O</it>(1) if <it>d</it> is a constant.</p>