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：我们研究拥堵团簇模型下平面 $n^2$ 点集上的几何问题。初始状态下，$n$-团簇网络中的每个节点持有欧几里得平面中 $n$ 个不同点（由 $O(\log n)$ 位坐标表示）组成的批次。每轮中，每个节点可向团簇中其他节点发送独立的 $O(\log n)$ 位消息，并执行无限本地计算。我们证明，在拥堵团簇中，输入 $n^2$ 点集的凸包可在 $O(\min\{ h,\log n\})$ 轮内构造完成，其中 $h$ 为凸包尺寸。同时，输入 $n^2$ 点集的三角剖分可在拥堵团簇上以 $O(\log^2n)$ 轮构造完成。最后，我们证明，在单位正方形内均匀随机选取的、具有 $O(\log n)$ 位坐标的 $n^2$ 个点的沃罗诺伊图，在拥堵团簇上能以高概率在 $O(1)$ 轮内完成该正方形内的计算。