Fréchet Distance in the Imbalanced Case

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.

💡 Research Summary

The paper studies the computational complexity of the Fréchet distance between two polygonal curves P and Q when their sizes are imbalanced (n ≥ m). While classic results give exact algorithms in O(n m log n) for the continuous version and O(n m) for the discrete version, recent conditional lower bounds (based on SETH) only rule out sub‑quadratic (1 + ε)‑approximation for the case n ≈ m. This work shifts the focus to the “imbalanced” regime, where m may be as small as n^α for any α∈(0,1), and establishes stronger hardness results using the Orthogonal Vectors Hypothesis (OVH).

Key technical contribution – lower bounds.
The authors reduce the Orthogonal Vectors (OV) problem to Fréchet‑distance instances. Given two sets U,V⊂{0,1}^d of sizes m and n (with d=Θ(log n)), they construct two curves P and Q such that the discrete Fréchet distance d_{dF}(P,Q) ≤ 1 if and only if there exists a pair (u∈U, v∈V) with ⟨u,v⟩=0. The construction works in 1‑D by using only integer coordinates −1, 0, 1, which forces any matching of cost < 2 to respect the bit‑wise orthogonality. Consequently, any algorithm that could (2 − ε)‑approximate the 1‑D discrete Fréchet distance in O((nm)^{1‑δ}) time would solve OV faster than OVH allows, contradicting the hypothesis. This improves the previous 1.4‑ε lower bound of Bringmann–Mulzer.

In 2‑D the authors handle both Euclidean (L₂) and Chebyshev (L_∞) metrics. They define eight base points a₁,…,a₈ arranged on a unit circle (L₂) or unit square (L_∞). For each bit of a vector they concatenate a specific sub‑curve (P₀, P₁, Q₀, Q₁) whose geometry encodes the bit value. By adding a “stretch” point ˜a at distance c=√2 (L₂) or c=2 (L_∞) they ensure that any matching with cost < 1 + c forces the corresponding bits to be orthogonal. Hence, unless OVH fails, no algorithm can achieve an approximation factor smaller than 1 + √2 − ε for the continuous or discrete Fréchet distance in L₂, nor smaller than 3 − ε in L_∞, within O((nm)^{1‑δ}) time. These bounds hold for any imbalance m = n^α, strengthening earlier results that required m ≈ n.

Algorithmic contributions.
Despite the strong lower bounds, the paper provides matching upper‑bound algorithms:

1. 1‑D discrete Fréchet distance (2‑approximation).
   The authors preprocess the longer curve P in O(n log n) time, storing a simple data structure of size O(m). For a query curve Q of size m, they compute a 2‑approximation in O(m² log m) time, achieving overall O(n log n + m² log m) time. The algorithm hinges on identifying a “central” point Q_c in Q and matching the bulk of P to this point, guaranteeing that the maximal distance is at most twice the optimum. This matches the lower‑bound factor and is essentially optimal with respect to the Ω(m²) query‑time lower bound shown by Buchin et al.

2. General‑dimensional (any L_p) (3 + ε‑approximation).
   For curves in ℝ^d under any L_p norm, the authors design an algorithm that runs in O((n + m²) log n) time and returns a (3 + ε)‑approximation for both continuous and discrete Fréchet distances. The method builds a “compressed free‑space diagram” by first simplifying P to a curve of size O(n + m²) and then performing a standard reachability search. The extra ε term comes from a standard grid‑based discretisation. The approximation factor 3 + ε is very close to the lower bound of 3 − ε for the L_∞ metric, indicating that the algorithm is essentially optimal for that norm.

Implications and broader impact.
The paper’s lower bounds are conditioned on OVH, a hypothesis that is implied by SETH but is considered weaker and more robust for reductions involving geometric constructions. By showing that even modest approximation factors (2 in 1‑D, 1 + √2 in L₂, 3 in L_∞) cannot be achieved sub‑quadratically when the input sizes are imbalanced, the work closes a gap left by earlier results that only handled the balanced case (m ≈ n). Moreover, the matching algorithms demonstrate that the lower bounds are tight: the 2‑approximation in 1‑D and the (3 + ε)‑approximation in arbitrary dimensions achieve the best possible factors under the given time constraints.

Overall, the paper advances our understanding of Fréchet distance computation by (i) extending conditional hardness to the imbalanced regime, (ii) strengthening approximation‑factor lower bounds, and (iii) delivering practically efficient algorithms that meet these bounds. The techniques—particularly the OV‑based reduction using carefully crafted low‑dimensional geometric gadgets—are likely to be reusable for other similarity measures and may inspire further research on fine‑grained complexity in computational geometry.