Computational geometry on hypercube computers

This paper gives hypercube algorithms for some simple problems involving geometric properties of sets of points. The properties considered emphasize aspects of convexity and domination. Efficient algorithms are given for both fine-grain and medium-grain hypercube computers. For both serial and parallel computers, sorting plays an important role in geometric algorithms for determining simple properties, often being the dominant component of the time. On a hypercube computer the time required to sort is still not fully understood, so the times of some of our algorithms for unsorted data are not completely determined. For the fine-grain model using worst case timing we show that if the data is presorted then faster algorithms are possible, if sorting one item per processor requires time growing faster than the dimension of the hypercube. For both models we show that faster algorithms are possible for point sets generated randomly, when time is measured using expected time. Our algorithms are developed for sets of planar points, with several of them extending to sets of points in spaces of higher dimension.