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A near-linear time exact algorithm for the $L_1$-geodesic Fréchet distance between two curves on the boundary of a simple polygon

Thijs van der Horst, Marc van Kreveld, Tim Ophelders, Bettina Speckmann

TL;DR

This work provides an exact near-linear time algorithm for the geodesic $L_1$-Fréchet distance between two boundary curves of a simple polygon. The authors decompose the problem by partitioning the parameter space into axis-aligned regions and reducing subproblems to simpler 1D or $x$-monotone forms, enabling efficient reachability propagation in the free-space diagram. They develop a decision procedure that runs in $O((n+m)\log^3 nm)$ time after preprocessing, and transform this into an exact optimizer via implicit Cartesian sums, achieving a final running time of $O(k \log nm + (n+m)(\log^2 nm \log k + \log^4 nm))$. The result advances exact Fréchet-distance computation in polygonal environments, particularly for curves on the boundary, by achieving near-linear performance in the input size and polygon complexity.

Abstract

Let $P$ be a polygon with $k$ vertices. Let $R$ and $B$ be two simple, interior disjoint curves on the boundary of $P$, with $n$ and $m$ vertices. We show how to compute the Fréchet distance between $R$ and $B$ using the geodesic $L_1$-distance in $P$ in $\mathcal{O}(k \log nm + (n+m) (\log^2 nm \log k + \log^4 nm))$ time.

A near-linear time exact algorithm for the $L_1$-geodesic Fréchet distance between two curves on the boundary of a simple polygon

TL;DR

This work provides an exact near-linear time algorithm for the geodesic -Fréchet distance between two boundary curves of a simple polygon. The authors decompose the problem by partitioning the parameter space into axis-aligned regions and reducing subproblems to simpler 1D or -monotone forms, enabling efficient reachability propagation in the free-space diagram. They develop a decision procedure that runs in time after preprocessing, and transform this into an exact optimizer via implicit Cartesian sums, achieving a final running time of . The result advances exact Fréchet-distance computation in polygonal environments, particularly for curves on the boundary, by achieving near-linear performance in the input size and polygon complexity.

Abstract

Let be a polygon with vertices. Let and be two simple, interior disjoint curves on the boundary of , with and vertices. We show how to compute the Fréchet distance between and using the geodesic -distance in in time.

Paper Structure

This paper contains 9 sections, 15 theorems, 2 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Doubly-separated $x$-monotone curves
  3. Partitioning the parameter space
  4. Vertically-separated $x$-monotone curves
  5. Separated curves on a simple polygon
  6. The general case
  7. Computing the Fréchet distance
  8. The decision algorithm
  9. Obtaining an optimization algorithm

Key Result

Lemma 1

Let $R$ and $B$ be two doubly-separated curves in $\mathbb{R}^2$ with $n$ and $m$ vertices. In $\mathcal{O}(n+m)$ time, we can construct two curves $\bar{R}$ and $\bar{B}$ in $\mathbb{R}$ that are separated by the origin, such that $\mathcal{F}_\delta(R, B) = \mathcal{F}_\delta(\bar{R}, \bar{B})$ fo

Figures (5)

  • Figure 1: The four types of subproblems that arise in our divide-and-conquer scheme. The bottom row shows the parameter spaces corresponding to the top row. Between problem types, the correspondence between some subcurves and parts of the $\delta$-free space is illustrated.
  • Figure 2: (left) A pair of doubly-separated $x$-monotone curves. We may consider the curves as lying on the union of two (degenerate) "histogram" polygons, in which case $\overline{d}$ is equal to the $L_1$-norm. Between any pair of red and blue points, there exists an $L_1$-geodesic through the point $p$. (right) Their $\delta$-free space diagram. For propagating reachability, given are a set of points $S$ (disks) on the bottom and left sides of the parameter space, and a set $T$ (circles and crosses) of points on the top and right sides. The filled circles are $\delta$-reachable, the crosses are not.
  • Figure 3: (left) A pair of $x$-monotone curves separated by a horizontal line, together with a vertical line with roughly half of all vertices on either side. (right) This line represents a partition of the parameter space into four regions, of which the top-left and bottom-right regions correspond to pairs of doubly-separated subcurves. The bottom-left and top-right regions are partitioned further. Green regions signify subproblems of type (\ref{['type_4']}).
  • Figure 4: An illustration of the two settings discussed in \ref{['lem:distance_function_edge']}. On the left is the case where $\mathit{NN}(u) = \mathit{NN}(v)$. On the right is the case where $\mathit{NN}(u) \neq \mathit{NN}(v)$. Marked points on $\overline{uv}$ are points where the functions $\varphi$ and $\psi$ change pieces.
  • Figure 5: Two types of bichromatic chords and the partitions of the parameter space that they induce. All green regions correspond to separated subcurves; the purple regions are split recursively.

Theorems & Definitions (15)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Corollary 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 5 more