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**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.

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 |

### Keywords:

lower bound; matching algorithm; parallel computation; parallel complexity; nearest neighbor; CRCW PRAM
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\textit{B. Schieber} and \textit{U. Vishkin}, Discrete Appl. Math. 29, No. 1, 97--111 (1990; Zbl 0707.68040)

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### References:

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