P-complete geometric problems

In this paper we show that it is impossible to solve a number of “natural” 2-dimensional geometric problems in polylog time with a polynomial number of processors (unless P = NC). Thus, we disprove a popular belief that there are no natural P-complete geometric problems in the plane. The problems we address include instances of polygon triangulation, planar partitioning, and geometric layering. Our results are based on non-trivial reductions from the monotone circuit value and planar circuit value problems.

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