Efficient Deterministic Construction of Voronoi Diagrams for Weakly Smooth Planar Point Sets on the Congested Clique

Core Concepts

A very weak smoothness condition on a set of n^2 points with O(log n)-bit coordinates in a unit square is sufficient to construct the Voronoi diagram of the set within the square in O(log n) deterministic rounds on the congested clique.

Abstract

The paper studies 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.
The key insights are:

A very weak smoothness condition is defined, where if a square Q of side length ℓ within the unit square contains at least n out of the n^2 input points, then any square of the same size at distance at most 4√2ℓ from Q and within the unit square has to contain at least one input point.

A protocol DT-SQUARE is presented that can construct the edges of the Delaunay triangulation dual to the edges of the Voronoi diagram of the weakly smooth point set within the unit square in O(log n) deterministic rounds on the congested clique.

The Voronoi diagram itself can then be constructed from the Delaunay triangulation in O(1) additional rounds.

The protocol DT-SQUARE works by implicitly growing a quadtree of squares rooted at the unit square. If a square R at a leaf of the quadtree jointly with the two layers of equal size squares around it includes O(n) input points, then the intersection of the Voronoi diagram of the input point set with R and the dual edges of the Delaunay triangulation can be computed locally. Otherwise, four child squares whose union forms R are created on the next level of the quadtree.
Checking the condition in parallel for the squares at the current front level of the quadtree and delivering the necessary points to the nodes representing respective frontier squares in O(1) rounds on the congested clique are the key technical challenges addressed.

Stats

The input point set has n^2 points with O(log n)-bit coordinates.
The protocol DT-SQUARE activates O(n) basic squares in each grid Gi(U), where i = O(log n).

Quotes

"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."

Key Insights Distilled From

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

by Jesper Janss... at arxiv.org 04-10-2024

https://arxiv.org/pdf/2404.06068.pdf

Deeper Inquiries

Can the smoothness condition be further relaxed while still maintaining an efficient deterministic construction of the Voronoi diagram

The smoothness condition can potentially be relaxed to some extent while still ensuring an efficient deterministic construction of the Voronoi diagram. By adjusting the parameters in the smoothness definition, such as the distance threshold and the minimum number of points required in a square, it might be possible to allow for a broader range of point distributions while maintaining the algorithm's effectiveness. However, finding the optimal balance between relaxation and maintaining efficiency would require careful analysis and experimentation.

How would the results change if the input points had unbounded bit representation

If the input points had unbounded bit representation, the results would significantly change. The assumption of O(log n)-bit coordinates is crucial for the efficiency of the algorithm in the congested clique model. With unbounded bit representation, the message complexity and computational requirements would increase substantially, potentially leading to a significant degradation in performance. The algorithm's runtime and message complexity would likely scale with the size of the input points, making it less suitable for large-scale point sets.

What other geometric problems on planar point sets can be solved efficiently on the congested clique model using similar local approaches

Several other geometric problems on planar point sets can be efficiently solved on the congested clique model using similar local approaches. Some examples include computing the convex hull, finding the closest pair of points, determining the intersection of line segments, and constructing the minimum spanning tree. By leveraging the principles of local computation, sorting, and efficient routing within the congested clique model, these geometric problems can be tackled in a distributed and parallel manner with low message complexity and runtime. The key lies in designing algorithms that exploit the locality of geometric relationships and optimize communication between nodes effectively.