Delaunay triangulations with predictions

Cabello, Sergio (author), Chan, Timothy M. (author), Giannopoulos, Panos (author)

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 $DT(P)$ more quickly when $G$ is "close" to $DT(P)$. 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 $DT(P)$. We present a deterministic algorithm to compute $DT(P)$ from $G$ in $O(n + D\log^3n)$ 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 $DT(P)$, 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 $DT(P)$ from $G$ in $O(n\log \log n + n\log(1/\rho))$ time with high probability. 3) Define $d_{\rm vio}$ 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 $DT(P)$). We present a deterministic algorithm to compute $DT(P)$ from $G$ in $O(n\log^*n + n\log d_{\rm vio})$ 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.

• Language: English
• Keywords: Delaunay triangulation, minimum spanning tree, algorithms with predictions
• Work type: Article
• Typology: 1.08 - Published Scientific Conference Contribution
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2026
• Number of pages: Str. 31:1-31:23
• PID: 20.500.12556/RUL-179802
• UDC: 004:519.17
• ISSN on article: 1868-8969
• DOI: 10.4230/LIPIcs.ITCS.2026.31
• COBISS.SI-ID: 269543427
• Publication date in RUL: 25.02.2026

Record is a part of a proceedings

• Title: 17th Innovations in Theoretical Computer Science Conference
• COBISS.SI-ID: 269488387

Record is a part of a journal

• Title: Leibniz international proceedings in informatics
• Shortened title: Leibniz int. proc. inform.
• Publisher: Schloss Dagstuhl, Leibniz-Zentrum für Informatik
• ISSN: 1868-8969
• COBISS.SI-ID: 523260441

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.