The Fr\’echet distance is a distance measure between trajectories in the
plane or walks in a graph G. Given constant-time shortest path queries in a
graph G, the Discrete Fr\’echet distance $F_G(P, Q)$ 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 $F_G(P, Q)$ unless the Orthogonal Vector
Hypothesis (OVH) fails.
Such quadratic-time conditional lower bounds are common to many Fr\’echet
distance variants. Jedoch, 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 Rd, that take
near-linear time for $c$-packed or $\kappa$-straight walks in the graph. In Rd,
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. Wir
show that there exist no 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 $D_F(P, Q)$. 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.

Dieser Artikel untersucht Zeitreisen und deren Auswirkungen.

PDF herunterladen: