We present parallel algorithms for computing cycle orders and cycle perimeters in relative neighborhood graphs. This parallel algorithm has wide-ranging applications from microscopic to macroscopic domains, e.g., in histopathological image analysis and wireless network routing. Our algorithm consists of the following steps (sub-algorithms): (1) Uniform partitioning of the graph vertices across processes, (2) Parallel Delaunay triangulation and (3) Parallel computation of the relative neighborhood graph and the cycle orders and perimeters. We evaluated our algorithm on a large dataset with 6.5 Million points and demonstrate excellent fixed-size scalability. We also demonstrate excellent isogranular scalability up to 131K processes. Our largest run was on a dataset with 13 billion points on 131K processes on ORNL's Cray XK7 "Titan" supercomputer.
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