Time-Optimal Proximity Graph Computations on Enhanced Meshes

In pattern recognition, morphology, and cellular system design one is interested in objects routinely represented by a polygonP. Distance properties among the vertices ofPare one of the fundamental descriptors useful in shape analysis and clustering. In order to capture perceptually relevant features ofPone associates a variety of proximity graphs with the polygonP. Typically such a proximity graphG(P) has the same vertices asP: two vertices ofPare adjacent inG(P) if they satisfy an application-specific predicate.Our main contribution is to propose time-optimal algorithms for constructing the Euclidian minimum spanning tree, the all-nearest neighbor graph, the relative neighborhood graph, and the symmetric farthest neighbor graph of ann-vertex unimodal polygon. All our algorithms run on meshes enhanced with row and column buses. We begin by establishing a ?(logn) time lower bound for the task of computing the Euclidian minimum spanning tree of ann-vertex unimodal polygon. This lower bound holds for both the CREW-PRAM and for meshes with multiple broadcasting, regardless of the number of processors available.Next, we show that this time lower bound is tight by exhibiting an algorithm for the Euclidian minimum spanning tree problem running in ?(logn) time on a mesh with multiple broadcasting of sizen×n. We also show that the all-nearest neighbor graph, the relative neighborhood graph, and the symmetric farthest neighbor graph of ann-vertex unimodal polygon can be computed inO(1) time on a mesh with multiple broadcasting of sizen×n.