Some Problems in Distributed Computational Geometry

A geometric network is a distributed network where each processor is identified by two numbers, representing the coordinates of the point in the plane where the processor is located. The edges of the network correspond to straight line segments such that no two of them intersect. In this paper we introduce the study of distributed computing in geometric networks. We study several computational geometry problems from the distributed computing point of view, such as finding convex hulls of geometric networks and identification of the external face. In particular, we obtain a O(n log n) message complexity algorithm to find the convex hull of a planar geometric graph, and a O(n logn) algorithm to identify the external face of a geometric graph. We also prove that the message complexity of leader election in an asynchronous geometric ring of n processors is Ω(n logn).