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 $ρ$. Suppose $G$ is any triangulation of $P$ that contains $R$. We present an algorithm to compute $\textit{DT}(P)$ from $G$ in $O(n\log\log n + n\log(1/ρ))$ 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.

翻译：本文研究计算几何中基于预测的算法，特别关注计算二维Delaunay三角剖分这一基础问题。给定平面上包含$n$个点的点集$P$，以及作为Delaunay三角剖分"预测"的三角剖分$G$，我们希望在$G$与$\textit{DT}(P)$"接近"时，利用$G$更快地计算出正确的Delaunay三角剖分$\textit{DT}(P)$。我们在不同的确定性与概率设定下获得了多类结果，主要包括：1. 定义$D$为$G$中不属于$\textit{DT}(P)$的边数。我们提出了从$G$计算$\textit{DT}(P)$的确定性算法，时间复杂度为$O(n + D\log^3 n)$；以及随机化算法，期望时间复杂度为$O(n+D\log n)$，后者在$D$的意义下是最优的。2. 设$R$为$\textit{DT}(P)$边的随机子集，其中每条边以概率$ρ$独立选取。假设$G$是包含$R$的任意$P$的三角剖分。我们提出了从$G$计算$\textit{DT}(P)$的算法，在高概率下时间复杂度为$O(n\log\log n + n\log(1/ρ))$。3. 定义$d_{\mbox{\scriptsize\rm vio}}$为$G$中三角形外接圆内严格包含$P$点的最大数量（若$G$等于$\textit{DT}(P)$则该值为0）。我们提出了从$G$计算$\textit{DT}(P)$的确定性算法，时间复杂度为$O(n\log^*n + n\log d_{\mbox{\scriptsize\rm vio}})$。我们还在类似设定下获得了二维欧几里得最小生成树等相关问题的结果，希望本研究能为未来研究开辟富有成果的方向。