On the parallel decomposability of geometric problems

There is a large and growing body of literature concerning the solution of geometric problems on mesh-connected arrays of processors [5,9,14,17]. Most of these algorithms are optimal (i.e., run in time <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>1/d</supscrpt>) on a <italic>d</italic>-dimensional <italic>n</italic>-processor array), and they all assume that the parallel machine is trying to solve a problem of size <italic>n</italic> on an <italic>n</italic>-processor array. What happens when we have parallel machine for efficiently solving a problem of size <italic>p</italic>, and we are interested in using it to solve a problem of size <italic>n</italic> < <italic>p</italic>? The answer to that question has to do with a fundamental, and yet (at least so far) little-studied property of geometric problems: their <italic>parallel-decomposability</italic>. More specifically, given that a problem of size <italic>p</italic> can be solved on a parallel machine P faster by a factor of (say) <italic>s</italic>(<italic>p</italic>) than on a RAM alone, then that problem is <italic>fully parallel-decomposable for</italic> P if a RAM to which the parallel machine P is attached can solve arbitrarily large problems with a speedup of also <italic>s</italic>(<italic>p</italic>) when compared to a RAM alone. The issue has been settled for the sorting problem when P is a linear systolic array [1,2,3,11]. Here we show that many geometric problems are fully parallel-decomposable for (multidimensional) mesh-connected arrays of processors.

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