# Parallel Volume Computation of Massive Polyhedron Union

W. Randolph Franklin.
**Parallel volume computation of massive polyhedron union.**
In *23rd Fall Workshop on Computational Geometry*. City College, New York City, USA, 25–26 Oct 2013.
(extended abstract).

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## Abstract

We present a parallel implementation of UNION3, an algorithm for computing the volume, surface area, center of mass, or similar properties of a boolean combination of many polyhedra. UNION3 has been implemented on the special case of the union of congruent isothetic cubes, and tested on 100, 000, 000 cubes. The algorithm uses a volume formula that does not require first computing the explicit union. All that is needed is the set of output vertices, including each vertex’s position and neighborhood. UNION3’s computation does not use randomization or sampling, and its output is exact. The implementation is parallel, using OpenMP. When using 32 threads, UNION3’s elapsed time is only a few minutes, and the speedup, compared to using one thread, is a factor of ten. When executed on uniform random i.i.d input, UNION3’s execution time is linear in the number of output vertices. UNION3 is intended as an example of a parallel geometry algorithm whose techniques should be broadly applicable.

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