We consider geometric problems on planar $n^2$-point sets in the congested clique model. Initially, each node in the $n$-clique network holds a batch of $n$ distinct points in the Euclidean plane given by $O(\log n)$-bit coordinates. In each round, each node can send a distinct $O(\log n)$-bit message to each other node in the clique and perform unlimited local computations. We show that the convex hull of the input $n^2$-point set can be constructed in $O(\min\{ h,\log n\})$ rounds, where $h$ is the size of the hull, on the congested clique. We also show that a triangulation of the input $n^2$-point set can be constructed in $O(\log^2n)$ rounds on the congested clique. Finally, we demonstrate that the Voronoi diagram of $n^2$ points with $O(\log n)$-bit coordinates drawn uniformly at random from a unit square can be computed within the square with high probability in $O(1)$ rounds on the congested clique.

翻译：我们考虑拥塞团簇模型下平面$n^2$点集的几何问题。初始时，$n$团簇网络中的每个节点持有欧氏平面上一批$n$个不同点，这些点由$O(\log n)$比特坐标表示。在每个通信轮次中，每个节点可向团簇中其他节点发送不同的$O(\log n)$比特消息，并执行无限本地计算。我们证明，输入$n^2$点集的凸包可在拥塞团簇上以$O(\min\{ h,\log n\})$轮构建，其中$h$为凸包规模。同时，输入$n^2$点集的三角剖分可在拥塞团簇上以$O(\log^2 n)$轮构建。最后，我们证明，在单位正方形内均匀随机抽取的$n^2$个具有$O(\log n)$比特坐标点的沃罗诺伊图，可在拥塞团簇上以高概率在$O(1)$轮内完成计算。