Deterministic polylog-round Voronoi diagram on the congested clique

Determine a low polylogarithmic upper bound on the number of communication rounds required to deterministically construct the Voronoi diagram of a planar set of n^2 points with O(log n)-bit coordinates in the Euclidean plane on the congested clique model with n nodes, when the points are not necessarily randomly distributed.

Background

The congested clique model studies the number of communication rounds needed to solve problems when each of the n nodes can send O(log n)-bit messages to all others per round, with unlimited local computation. For uniformly random point sets in a unit square, prior work has shown O(1) expected rounds suffices to construct the Voronoi diagram.

This paper achieves O(log n) deterministic rounds for point sets satisfying a very weak smoothness condition, but does not cover arbitrary (non-random) distributions. The core difficulty historically lies in efficiently merging Voronoi diagrams in parallel/distributed settings without randomness. Establishing a deterministic polylogarithmic-round bound for arbitrary planar point sets on the congested clique would close this gap.

References

The remaining major open problem is the derivation of a low polylogarithmic upper bound on the number of rounds sufficient to deterministically construct the Voronoi diagram of $n2$ points with $O(\log n)$-bit coordinates in the Euclidean plane (when the points are not necessarily randomly distributed) on the congested clique with $n$ nodes.

— The Voronoi Diagram of Weakly Smooth Planar Point Sets in $O(\log n)$ Deterministic Rounds on the Congested Clique  (2404.06068 - Jansson et al., 2024) in Section 4 (Final remarks)