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Abstract

Let $S$ be a set of $n$ points in $D$-dimensional space, where $D$ is a constant, and let $k$ be an integer between $1$ and $n \choose 2$. A new and simpler proof is given of Salowe's theorem, i.e., a sequential algorithm is given that computes the $k$ closest pairs in the set $S$ in $O(n \log n + k)$ time, using $O(n+k)$ space. The algorithm fits in the algebraic decision tree model and is, therefore, optimal. Salowe's algorithm seems difficult to parallelize. A parallel version of our algorithm is given for the CRCW-PRAM model. This version runs in $O((\log n)^{2} \log\log n )$ expected parallel time and has an $O(n \log n \log\log n +k)$ time-processor product.