在强次二次时间内对弗雷歇距离的(5 + ε)近似
$(5+ε)$-Approximation of Fréchet Distance in Strongly Subquadratic Time
AI总结：
针对\(\mathbb{R}^d\)中两条多边形曲线的连续和离散弗雷歇距离，提出随机化\((5 + \epsilon)\)近似算法，通过直接验证可达性改进近似，结合双尺度搜索等提升运行效率，改进了现有强次二次常数因子近似算法。
AI中文摘要：
我们针对\(\mathbb{R}^d\)中任意两条分别具有\(n\)和\(m\le n\)个顶点的多边形曲线\(\tau\)和\(\sigma\)，给出了连续和离散弗雷歇距离的随机化\((5 + \epsilon)\)近似算法。连续弗雷歇算法运行时间为\(\widetilde O_{d,\epsilon}(n m^{8/9})\)，离散弗雷歇算法运行时间为\(\widetilde O_{d,\epsilon}(n m^{4/5})\)。这些界限改进了Cheng、Huang和Zhang的近期强次二次常数因子近似算法。近似改进源于通过辅助替代曲线直接验证长边界到边界可达性，运行时间改进源于双尺度宏观替代搜索与二元辅助转移结构相结合。
英文摘要：
We give randomized $(5+ε)$-approximation algorithms for both the continuous and discrete Fréchet distances on arbitrary two polygonal curves $τ$ and $σ$ in $\mathbb R^d$ for fixed $d$, with $n$ and $m\le n$ vertices respectively. Our algorithm for continuous Fréchet runs in $\widetilde O_{d,ε}(n m^{8/9})$ time, and our algorithm for discrete Fréchet runs in $\widetilde O_{d,ε}(n m^{4/5})$ time. These bounds improve the recent strongly subquadratic constant-factor approximation algorithms of Cheng, Huang, and Zhang~\cite{cheng2025constant}, which give $(7+ε)$-approximations. The approximation improvement comes from certifying long boundary-to-boundary reachability directly through auxiliary surrogate curves, avoiding an extra conversion back to input subcurves and hence removing one triangle-inequality loss. The running-time improvement comes from a two-scale macro-surrogate search combined with dyadic auxiliary-transfer structures, with the discrete case gaining a faster bound from exact planar reachability in the discrete free-space graph.
