## Finding all nearest neighbors for convex polygons in parallel: A new lower bound technique and a matching algorithm.(English)Zbl 0707.68040

Summary: A novel technique for proving lower bounds in parallel computation is presented. The technique is based on mapping any algorithm for the problem being considered to an algorithm for another problem, for which a good lower bound is known. The mapping is done by careful application of Ramsey-like arguments.
Specifically, we study the parallel complexity of the following problem. Given an input convex polygon $$P=(v_ 0,...,v_{n-1})$$, where $$(v_ i,v_{i+1})$$ (the indices are taken modulo n) is an edge of P, for $$i=0,...,n-1$$, find the nearest neighbor of each vertex $$v_ i$$ of P. that is, find a vertex $$v_ j$$, $$j\neq i$$, $$0\leq j<n$$, whose (Euclidean) distance from $$v_ i$$ is minimal. We present a parallel algorithm for the problem which runs in O(log log n) time using n/log log n processors on a CRCW PRAM. We prove that $$\Omega$$ (log log n) time is needed for solving the problem on a CRCW PRAM with O(n log$${}^ c n)$$ processors, for any constant c.

### MSC:

 68W15 Distributed algorithms 05C35 Extremal problems in graph theory 68Q25 Analysis of algorithms and problem complexity 68Q05 Models of computation (Turing machines, etc.) (MSC2010) 68R10 Graph theory (including graph drawing) in computer science
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### References:

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