Delaunay Triangulations with Predictions

2026-01-13

Information Technology Convergence and Services

AI Summary

This work introduces, for the first time, a prediction-augmented algorithmic framework to computational geometry, specifically targeting the efficient computation of two-dimensional Delaunay triangulations (DT). Given a point set \( P \) and a predicted triangulation \( G \) that approximates the true DT, the paper proposes an adaptive correction algorithm based on an edge-difference metric \( D \), a violation measure \( d_{\text{vio}} \), and a randomized sampling probability \( \rho \). The main contributions include a deterministic algorithm running in \( O(n + D \log^3 n) \) time and an optimal randomized algorithm with expected time \( O(n + D \log n) \). Under a stochastic model combining edge inclusion and violation degrees, the approach further yields an almost-linear-time solution, which is also extended to related problems such as the Euclidean minimum spanning tree.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

We investigate algorithms with predictions in computational geometry, specifically focusing on the basic problem of computing 2D Delaunay triangulations. Given a set $P$ of $n$ points in the plane and a triangulation $G$ that serves as a"prediction"of the Delaunay triangulation, we would like to use $G$ to compute the correct Delaunay triangulation $\textit{DT}(P)$ more quickly when $G$ is"close"to $\textit{DT}(P)$. We obtain a variety of results of this type, under different deterministic and probabilistic settings, including the following: 1. Define $D$ to be the number of edges in $G$ that are not in $\textit{DT}(P)$. We present a deterministic algorithm to compute $\textit{DT}(P)$ from $G$ in $O(n + D\log^3 n)$ time, and a randomized algorithm in $O(n+D\log n)$ expected time, the latter of which is optimal in terms of $D$. 2. Let $R$ be a random subset of the edges of $\textit{DT}(P)$, where each edge is chosen independently with probability $\rho$. Suppose $G$ is any triangulation of $P$ that contains $R$. We present an algorithm to compute $\textit{DT}(P)$ from $G$ in $O(n\log\log n + n\log(1/\rho))$ time with high probability. 3. Define $d_{\mbox{\scriptsize\rm vio}}$ to be the maximum number of points of $P$ strictly inside the circumcircle of a triangle in $G$ (the number is 0 if $G$ is equal to $\textit{DT}(P)$). We present a deterministic algorithm to compute $\textit{DT}(P)$ from $G$ in $O(n\log^*n + n\log d_{\mbox{\scriptsize\rm vio}})$ time. We also obtain results in similar settings for related problems such as 2D Euclidean minimum spanning trees, and hope that our work will open up a fruitful line of future research.

Problem

• Delaunay triangulation
• algorithms with predictions
• computational geometry
• triangulation refinement

Innovation

• algorithms with predictions
• Delaunay triangulation
• adaptive computation
• computational geometry
• randomized algorithms