Faster Fréchet Distance Approximation through Truncated Smoothing

TL;DR

The paper addresses the continuous Fréchet distance between polygonal curves and the challenge of subquadratic exact computation under SETH-based barriers. It introduces a truncation-based projection simplification (truncated smoothing) that reduces the reachable free space to $O(n^2/α)$ blocks while incurring only additive error $2α$, enabling subquadratic approximation. The authors present an $O((n^2/α)\log n)$ time α-approximation in arbitrary dimensions and an improved $O((n^2/α^3)\log^2 n)$ time α-approximation in one dimension, including the first strongly-subquadratic $n^ε$-approximation for any fixed $ε\in(0,1/2]$. Central to the approach is a linear-time projection simplification and a fast block-propagation technique that leverages ortho-convexity of the free-space within blocks, together with novel approximate exit-set constructions based on δ-signatures. These contributions significantly improve the efficiency of practical Fréchet distance approximations and lay groundwork for further reductions in higher dimensions.

Abstract

The Fréchet distance is a commonly used distance measure for curves. Computing the Fréchet distance between two polygonal curves of $n$ vertices takes roughly quadratic time, and conditional lower bounds suggest that approximating to within a factor $3$ cannot be done in strongly-subquadratic time, even in one dimension. Currently, the best approximation algorithms present trade-offs between approximation quality and running time. At SoCG 2021, Colombe and Fox presented an $O((n^3 / α^2) \log n)$-time $α$-approximate algorithm for curves in arbitrary dimensions, for any $α\in [\sqrt{n}, n]$. In this work, we give an $α$-approximate algorithm with a significantly faster running time of $O((n^2 / α) \log n)$, for any $α\in [1, n]$. In particular, we give the first strongly-subquadratic $n^\varepsilon$-approximation algorithm, for any constant $\varepsilon \in (0, 1/2]$. For curves in one dimension we further improve the running time to $O((n^2 / α^3) \log^2 n)$, for $α\in [1, n^{1/3}]$. Both of our algorithms rely on a linear-time simplification procedure that in one dimension reduces the complexity of the reachable free space to $O(n^2 / α)$ without making sacrifices in the asymptotic approximation factor.

Figures (10)

• Figure 1: An illustration of the approximate decision algorithm.
• Figure 2: An illustration of the approximate decision algorithm for one-dimensional curves. Our main algorithm (top diagram) repeatedly calls a subroutine for constructing approximate exit sets (bottom diagram).
• Figure 3: (left) Two one-dimensional curves (with vertices replaced by vertical segments for clarity) that have only long edges with respect to $\delta$. (right) The $\delta$-free space (white) and all $\delta$-reachable points (green). The reachable points all lie in blocks (cells) close to the diagonal.
• Figure 4: An illustration of truncated smoothings. (left) The vertices of curve $P$ (non-dashed) are drawn as vertical segments for clarity. The minimum edge length of $P$ is realized by $\overline{p_i p_j}$. (right) The result of the smoothing procedure (opaque).
• Figure 5: (left) The sublevel set component of $p_6$, left of the dashed line segment. Points $\ell_i$ and $r_i$ are the minima of the left and right parts of this component. (middle and right) The max-Cartesian tree built on the vertex sequence of $P$.

Theorems & Definitions (30)

• Theorem 1
• Theorem 2
• Remark
• Lemma 3
• Lemma 4
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10
• Lemma 11