We study the problem of computing the Voronoi diagram of a set of $n^2$ points with $O(\log n)$-bit coordinates in the Euclidean plane in a substantially sublinear in $n$ number of rounds in the congested clique model with $n$ nodes. Recently, Jansson et al. have shown that if the points are uniformly at random distributed in a unit square then their Voronoi diagram within the square can be computed in $O(1)$ rounds with high probability (w.h.p.). We show that if a very weak smoothness condition is satisfied by an input set of $n^2$ points with $O(\log n)$-bit coordinates in the unit square then the Voronoi diagram of the point set within the unit square can be computed in $O(\log n)$ rounds in this model.

翻译：我们研究在欧几里得平面上，计算由$n$个节点组成的拥塞团簇模型中，具有$O(\log n)$比特坐标的$n^2$个点集的Voronoi图问题，要求轮数显著亚线性于$n$。近期，Jansson等人已证明：若这些点在单位正方形内均匀随机分布，则其在该正方形内的Voronoi图可在$O(1)$轮内以高概率（w.h.p.）计算得出。我们证明：若单位正方形内具有$O(\log n)$比特坐标的$n^2$个点构成的输入点集满足极弱的平滑条件，则在该模型下，点集在单位正方形内的Voronoi图可在$O(\log n)$轮内计算得出。