Parallel algorithms for all maximal equally-spaced collinear sets and all maximal regular coplanar lattices

The All Maximal Equally-Spaced Collinear Subset (AMESCS) Problem is defined as follows. Given a set P of n points in a Euclidean space E^d, find all maximal equally-spaced collinear subsets of P. An optimal Θ(n²) time sequential solution to the problem is given in Kahng and Robins (1991). A related problem is the All Maximal Regularly-Spaced Subsets (AMRSS) Problem, defined as follows. Given a set P of n points in E^d, find all maximal regularly-spaced coplanar subsets of P. An optimal O(n³) time sequential solution to the problem is given in Kahng and Robins (1991). In this paper, we consider parallel solutions to the AMESCS and AMRSS Problems. Optimal sequential running times of O(n²) and O(n³), respectively, make parallel solutions of these problems desirable of large n. While the optimal sequential algorithms of Kahng and Robins (1991) are dominated by repetitions of a procedure that requires Θ(n) time per repetition, and appears inherently sequential, the algorithms we give are quite different. Our parallel algorithms solve both of these problems to within a logarithmic factor of optimality on the Arbitrary CRCW PRAM, and in optimal time on the mesh-connected computer.