$(5+ε)$-Approximation of Fréchet Distance in Strongly Subquadratic Time

We give randomized $(5+\epsilon)$-approximation algorithms for both the continuous and discrete Fr\'echet distances on arbitrary two polygonal curves $\tau$ and $\sigma$ in $\mathbb R^d$ for fixed $d$, with $n$ and $m\le n$ vertices respectively. Our algorithm for continuous Fr\'echet runs in $\widetilde O_{d,\epsilon}(n m^{8/9})$ time, and our algorithm for discrete Fr\'echet runs in $\widetilde O_{d,\epsilon}(n m^{4/5})$ time. These bounds improve the recent strongly subquadratic constant-factor approximation algorithms of Cheng, Huang, and Zhang~\cite{cheng2025constant}, which give $(7+\epsilon)$-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.