Delaunay Triangulations with Predictions

TL;DR

This work studies how predictions can accelerate the computation of the 2D Delaunay triangulation for a point set $P$ by leveraging a predicted triangulation $G$. It develops multiple prediction-aware algorithms with running times that depend on proximity measures between $G$ and $DT(P)$, including deterministic and randomized approaches, and extends the framework to related graphs such as EMST and RNG. The results include a deterministic $O(n+D\log^3 n)$ algorithm, a near-optimal randomized $O(n+D\log n)$ algorithm, and a near-linear-time $O(n\log\log n + n\log(1/\rho))$ method in a probabilistic model, with additional $d_{\text{cross}}$- and $d_{vio}$-sensitive variants. Overall, the paper demonstrates that near-linear time DT computation is achievable when the prediction is close, and it presents a rich set of techniques (r-divisions, planar separators, 3D lifting, conflict lists, and sampling) that may spur further study of predictions in computational geometry.

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 $ρ$. 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.

Figures (6)

• Figure 1: Sorting the $x$-coordinates of $p_1,\ldots,p_n$ reduces to computing the Delaunay triangulation of the $O(n)$ vertices of the triangulation shown (assuming a rescaling to compress the $y$-coordinates).
• Figure 2: Merging $k$$x$-sorted lists $\langle p_{1,1},\ldots,p_{1,n/k}\rangle$, …, $\langle p_{k,1},\ldots,p_{k,n/k}\rangle$ reduces to computing the Delaunay triangulation of the $O(n)$ vertices of the triangulation shown (assuming a rescaling to compress the $y$-coordinates). This triangulation has $D_{\hbox{\scriptsize\rm local}}=\Theta(k)$ non-locally-Delaunay edges shown in dotted lines.
• Figure 3: Proof of Lemma \ref{['lem:conflict']}.
• Figure 4: Proof of Lemma \ref{['lem:vio']}. The left example is in case (i); the right example is in case (ii).
• Figure 5: Left: $N_G(f)$ are marked for the face $f$, and the points of $V(N_G(f))$ are marked with empty squares. The PSLG on the left does not satisfy the conditions of Lemma \ref{['lem:subgraphDT']}, the one on the right does. To verify the assumptions of Lemma \ref{['lem:subgraphDT']} for the shaded triangle on the right we have to check the points marked with empty squares.

Theorems & Definitions (52)

• Theorem 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.4
• proof