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The Voronoi Diagram of Weakly Smooth Planar Point Sets in $O(\log n)$ Deterministic Rounds on the Congested Clique

Jesper Jansson, Christos Levcopoulos, Andrzej Lingas

TL;DR

This work addresses deterministic construction of the Voronoi diagram and its Delaunay triangulation for $n^2$ points with $O(\log n)$-bit coordinates inside the unit square on a congested clique of $n$ nodes, under a very weak $(\varepsilon,d)$-smoothness condition. It introduces a local, quadtree-based approach (DT-SQUARE) that computes the Delaunay triangulation within the unit square in $O(\log n)$ rounds, and then derives the Voronoi diagram in $O(1)$ rounds by angularly ordering the DT edges around each point. The main contributions are the $O(\log n)$-round deterministic DT construction and the subsequent $O(1)$-round Voronoi assembly, with a total message complexity of $O(n^2\log n)$. This advances deterministic geometric computation on the congested clique, linking to MPC/NC-type perspectives and leaving open the challenge of removing the smoothness assumption for general inputs.

Abstract

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.

The Voronoi Diagram of Weakly Smooth Planar Point Sets in $O(\log n)$ Deterministic Rounds on the Congested Clique

TL;DR

This work addresses deterministic construction of the Voronoi diagram and its Delaunay triangulation for points with -bit coordinates inside the unit square on a congested clique of nodes, under a very weak -smoothness condition. It introduces a local, quadtree-based approach (DT-SQUARE) that computes the Delaunay triangulation within the unit square in rounds, and then derives the Voronoi diagram in rounds by angularly ordering the DT edges around each point. The main contributions are the -round deterministic DT construction and the subsequent -round Voronoi assembly, with a total message complexity of . This advances deterministic geometric computation on the congested clique, linking to MPC/NC-type perspectives and leaving open the challenge of removing the smoothness assumption for general inputs.

Abstract

We study the problem of computing the Voronoi diagram of a set of points with -bit coordinates in the Euclidean plane in a substantially sublinear in number of rounds in the congested clique model with 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 rounds with high probability (w.h.p.). We show that if a very weak smoothness condition is satisfied by an input set of points with -bit coordinates in the unit square then the Voronoi diagram of the point set within the unit square can be computed in rounds in this model.
Paper Structure (4 sections, 8 theorems, 3 figures)

This paper contains 4 sections, 8 theorems, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The local approach
  4. Final remarks

Key Result

lemma 1

Let $R$ be a basic square in a grid $G_i(U)$ within the unit square $U$. Consider a finite set $S$ of points within the unit square. If $R$ contains a point in $S$ then the Voronoi diagram of $S$ within $R$ can be computed by taking into account only the points of $S$ within $TL_i(U).$ Hence, in par

Figures (3)

  • Figure 1: An example of a planar point set, its Voronoi diagram, and the dual Delaunay triangulation.
  • Figure 2: An example of the configuration in the proof of Lemma \ref{['lem: first']}.
  • Figure 3: An example of the configuration in the proof of Lemma \ref{['lem: correct']}.

Theorems & Definitions (16)

  • definition 1
  • definition 2
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof
  • lemma 4
  • proof
  • ...and 6 more