度量空间中路径弗雷歇距离的强次二次方$(3+\varepsilon)$近似
A Strongly-Subquadratic $(3+\varepsilon)$-Approximation for the Fréchet Distance for Paths in Metric Spaces
AI总结：
【一句话总结】在度量空间中，计算两条折线间弗雷歇距离通常耗时与顶点数呈二次方关系。本文提出确定性近似算法，改进近似因子与运行时间，给出不同情况下计算近似值的算法及时间复杂度，并引入通用决策算法。
AI中文摘要：
弗雷歇距离是度量空间中路径的一种经过充分研究的距离度量，主要针对$d$维欧几里得空间中的路径进行研究。计算两条折线之间的弗雷歇距离在顶点数量上大致呈二次方时间。在强指数时间假设（SETH）下，无法在强次二次方时间内将其近似到小于$3$的因子。最近有研究表明存在随机算法可在强次二次方期望时间内计算$(7+\varepsilon)$近似。本文提出确定性近似算法，显著改进了近似因子和运行时间。具体算法在$O(nm^{2/3}\log n\cdot\log(\frac{1}{\varepsilon}\log n))$时间内计算$(3+\varepsilon)$近似，几乎达到SETH隐含的近似因子条件下限。对于$\mathbb{R}$中的折线，给出在$O(nm^{2/3}\log^{5/3}n)$时间内运行的$3$近似算法，与条件下限精确匹配。还引入通用的强次二次方时间$3$近似决策算法，该算法对环境度量空间无假设，仅依赖输入路径所谓自由空间的标准假设，在一些温和假设下可导出一般度量空间中的$(3+\varepsilon)$近似算法，这些假设在任何$p\geq1$的度量空间$(\mathbb{R}^d,L_p)$中的折线自动成立。
英文摘要：
The Fréchet distance is a well-studied distance measure for paths in a metric space. It is mostly studied for paths in $d$-dimensional Euclidean space. Here, computing the Fréchet distance between two polylines takes time roughly quadratic in the number of vertices. Assuming the strong exponential time hypothesis (SETH), it cannot be approximated to within a factor less than $3$ in strongly-subquadratic time. Recently, it was shown that for any $\varepsilon>0$, there exists a randomized algorithm that can compute a $(7+\varepsilon)$-approximation in strongly-subquadratic expected time [Cheng, Huang, and Zhang; STOC'25]. For polylines with $n$ and $m$ vertices in a Euclidean space of constant dimension, where $n \geq m$, their algorithm takes $O(nm^{0.99} \log(n/\varepsilon))$ time in expectation. We present a deterministic approximation algorithm that significantly improves upon the approximation factor and running time. Specifically, our algorithm computes a $(3+\varepsilon)$-approximation in $O(nm^{2/3} \log n \cdot \log (\frac{1}{\varepsilon} \log n))$ time. Our algorithm nearly matches the conditional lower bound on the approximation factor implied by SETH. For polylines in $\mathbb{R}$, we present a $3$-approximation algorithm that runs in $O(nm^{2/3} \log^{5/3} n)$ time, and exactly matches the conditional lower bound. For our results, we introduce a general strongly-subquadratic time $3$-approximate decision algorithm. This algorithm makes no assumptions on the ambient metric space, and relies only on standard assumptions on the so-called free space of the input paths. Under some mild assumptions, our decision algorithm leads to a $(3+\varepsilon)$-approximation algorithm in general metric spaces. These assumptions hold automatically for polylines in any metric space $(\mathbb{R}^d, L_p)$ with $p \geq 1$.
