The Geodesic Fréchet Distance Between Two Curves Bounding a Simple Polygon

TL;DR

The paper tackles the geodesic Fréchet distance between two polygonal curves that bound a simple polygon P, with distances measured via geodesics inside P. It introduces a near-linear-time $(1+\varepsilon)$-approximation algorithm (and linear-time exactness for convex polygons), by decomposing the problem into near and far slabs using nearest-neighbor fans and transforming far-point decisions into a collection of separated one-dimensional problems. The core technical advances are a decision framework with a partition of the parameter space, a reduction to one-dimensional problems via short separators and anchors, and a propagated reachability analysis built on prefix-minima matchings and greedy paths in a free-space forest, yielding a running time of $O\left(\frac{1}{\varepsilon} (n+m \log n) \log nm \log \frac{1}{\varepsilon}\right)$ for the main result. When P is convex, the authors give a linear-time exact algorithm based on a maximally-parallel Fréchet matching structure derived from rotating calipers. Overall, the work significantly improves prior $2$-approximation and subquadratic results by achieving near-linear-time $(1+\varepsilon)$-approximation and clarifying the geometric structure underlying geodesic Fréchet distance within a simple polygon.

Abstract

The Fréchet distance is a popular similarity measure that is well-understood for polygonal curves in $\mathbb{R}^d$: near-quadratic time algorithms exist, and conditional lower bounds suggest that these results cannot be improved significantly, even in one dimension and when approximating with a factor less than three. We consider the special case where the curves bound a simple polygon and distances are measured via geodesics inside this simple polygon. Here the conditional lower bounds do not apply; Efrat $et$ $al.$ (2002) were able to give a near-linear time $2$-approximation algorithm. In this paper, we significantly improve upon their result: we present a $(1+\varepsilon)$-approximation algorithm, for any $\varepsilon > 0$, that runs in $\mathcal{O}(\frac{1}{\varepsilon} (n+m \log n) \log nm \log \frac{1}{\varepsilon})$ time for a simple polygon bounded by two curves with $n$ and $m$ vertices, respectively. To do so, we show how to compute the reachability of specific groups of points in the free space at once, by interpreting the free space as one between separated one-dimensional curves. We solve this one-dimensional problem in near-linear time, generalizing a result by Bringmann and Künnemann (2015). Finally, we give a linear time exact algorithm if the two curves bound a convex polygon.

Figures (14)

• Figure 1: (left) Points $r$ and $b$ with $b \in \mathit{NN}(r)$. The region shaded in green consists of all points within geodesic distance $\delta$ of $b$. (right) The $(r, b, \delta)$-nearest neighbor fan (orange).
• Figure 2: (left) The mappings (purple) from points on $R$ to their nearest neighbor(s) on $B$. (middle) The partition of the parameter space based on near and far points on $B$. The partly-dashed purple curve indicates the nearest neighbor(s) on $B$ of points on $R$. The beige regions correspond to the $(r, b, \delta)$-nearest neighbor fans. (right) A $\delta$-matching that is greedy on $B$ inside the regions.
• Figure 3: (left) The points $b$ and $b'$ are both nearest neighbor of some point $r$, implying a short separator. (right) Adding anchor points to the separator and snapping the orange geodesic (between arbitrary points on $R$ and $B[b, b']$) to one.
• Figure 4: (left) Snapping a geodesic (orange) to an anchor. (right) The eight regions in the parameter space of $\hat{R}$ and $\hat{B}$ corresponding to the first eight (out of nine) anchors. The orange geodesic lies in region $\mathcal{R}_5$.
• Figure 5: The construction in \ref{['lem:restricted_matching']}. Non-dashed segments are $\overline{r_i r_{i+1}}$ and $\overline{b_j b_{j+1}}$.

Theorems & Definitions (37)

• Theorem 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10