Fréchet Distance in Unweighted Planar Graphs

The Fr\'echet distance is a distance measure between trajectories in $\Bbb{R}^d$ or walks in a graph $G$. Given constant-time shortest path queries, the Discrete Fr\'echet distance $D_G(P, Q)$ between two walks $P$ and $Q$ can be computed in $O(|P| \cdot |Q|)$ time using a dynamic program. Driemel, van der Hoog, and Rotenberg [SoCG'22] show that for weighted planar graphs this approach is likely tight, as there can be no strongly-subquadratic algorithm to compute a $1.01$-approximation of $D_G(P, Q)$ unless the Orthogonal Vector Hypothesis (OVH) fails. Such quadratic-time conditional lower bounds are common to many Fr\'echet distance variants. However, they can be circumvented by assuming that the input comes from some well-behaved class: There exist $(1+\varepsilon)$-approximations, both in weighted graphs and in $\Bbb{R}^d$, that take near-linear time for $c$-packed or $\kappa$-straight walks in the graph. In $\Bbb{R}^d$ there also exists a near-linear time algorithm to compute the Fr\'echet distance whenever all input edges are long compared to the distance. We consider computing the Fr\'echet distance in unweighted planar graphs. We show that there exist no strongly-subquadratic $1.25$-approximations of the discrete Fr\'echet distance between two disjoint simple paths in an unweighted planar graph in strongly subquadratic time, unless OVH fails. This improves the previous lower bound, both in terms of generality and approximation factor. We subsequently show that adding graph structure circumvents this lower bound: If the graph is a regular tiling with unit-weighted edges, then there exists an $\tilde{O}((|P| + |Q|)^{1.5})$-time algorithm to compute $D_G(P, Q)$. Our result has natural implications in the plane, as it allows us to define a new class of well-behaved curves that facilitate $(1+\varepsilon)$-approximations of their discrete Fr\'echet distance in subquadratic time.