Fréchet Distance in the Imbalanced Case

TL;DR

This work investigates Fréchet distance computation under imbalanced input sizes. It delivers tight conditional lower bounds against subquadratic-time approximation using the Orthogonal Vectors Hypothesis, showing that in 1D the discrete Fréchet distance cannot be \(2-\varepsilon\) approximated in \(\mathcal{O}((nm)^{1-\delta})\) time, and in 2D the continuous/discrete variants cannot beat \(1+\sqrt{2}-\varepsilon\) (Euclidean) or \(3-\varepsilon\) (\(L_\infty\)) within the same running time. Complementing these bounds, the authors provide a near-optimal 2-approximation data-structure for 1D discrete Fréchet distance with \(\mathcal{O}(n\log n)\) preprocessing and \(\mathcal{O}(m^2\log m)\) query time, and a universal \((3+\varepsilon)\)-approximation algorithm for all dimensions and all \(L_p\) spaces running in \(\mathcal{O}((n+m^2)\log n)\) time. Collectively, these results tighten our understanding of the trade-offs between accuracy and running time for Fréchet distance in imbalanced regimes and almost close the gap between lower bounds and algorithms in broad settings.

Abstract

Given two polygonal curves $P$ and $Q$ defined by $n$ and $m$ vertices with $m\leq n$, we show that the discrete Fréchet distance in 1D cannot be approximated within a factor of $2-\varepsilon$ in $\mathcal{O}((nm)^{1-δ})$ time for any $\varepsilon, δ>0$ unless OVH fails. Using a similar construction, we extend this bound for curves in 2D under the continuous or discrete Fréchet distance and increase the approximation factor to $1+\sqrt{2}-\varepsilon$ (resp. $3-\varepsilon$) if the curves lie in the Euclidean space (resp. in the $L_\infty$-space). This strengthens the lower bound by Buchin, Ophelders, and Speckmann to the case where $m=n^α$ for $α\in(0,1)$ and increases the approximation factor of $1.001$ by Bringmann. For the discrete Fréchet distance in 1D, we provide an approximation algorithm with optimal approximation factor and almost optimal running time. Further, for curves in any dimension embedded in any $L_p$ space, we present a $(3+\varepsilon)$-approximation algorithm for the continuous and discrete Fréchet distance using $\mathcal{O}((n+m^2)\log n)$ time, which almost matches the approximation factor of the lower bound for the $L_\infty$ metric.

Figures (6)

• Figure 1: A schematic view of the construction of the curves $P$ and $Q$. Here, $Q_{null}$ is drawn in red, $P_0$ and $Q_0$ in green, $P_1$ and $Q_1$ in blue, and $Q_c$ in orange.
• Figure 2: The curves $P_0$, $P_1$, $Q_0$, and $Q_1$. The upper curves are defined in 1D, the middle curves for the $L_2$-metric in 2D and the lower curves for the $L_\infty$-metric in 2D.
• Figure 3: In white: the non-free space of the curve $Q_0$ and $P_0\circ P_0\circ P_1$ on the left and of the curve $Q_1$ and $P_0\circ P_0\circ P_1$ on the right. The red, orange and pink paths visualize the proof of \ref{['lem:2DcontinuousLB_Q01']}.
• Figure 4: The black curve is $P$ and the middle curve is an optimal $3$-simplification for $P$. The red curve below is the compressed curve $P'$. Here, $k_1=3$, $k_2=7$, $k_3=3$ and $z_1=-1$, $z_2=-1$, $z_3=1$.
• Figure 5: The compressed curve $P'$ and the query curve $Q$ are on the left. On the right is the free space matrix of $P'$ and $Q$, where only $1$-entries are added. The dark green cells visualize the parameter $a$ and the light green cells the set $\{u-z\mid z\in A\}$. The hatched cells are the cells that have been in $F_{\text{pre}}$ and $F_{\text{new}}$.

Theorems & Definitions (20)

• Definition 1: Orthogonal Vectors Hypothesis (OVH)
• Lemma 4
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Theorem 8
• Theorem 9
• Theorem 10
• Lemma 10