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Computer Science, Computational Geometry, cs.CG
Abstract:
The Fr\'echet distance provides a natural and intuitive measure for the
popular task of computing the similarity of two (polygonal) curves. While a
simple algorithm computes it in near-quadratic time, a strongly subquadratic
algorithm cannot exist unless the Strong Exponential Time Hypothesis fails.
Still, fast practical implementations of the Fr\'echet distance, in particular
for realistic input curves, are highly desirable. This has even lead to a
designated competition, the ACM SIGSPATIAL GIS Cup 2017: Here, the challenge
was to implement a near-neighbor data structure under the Fr\'echet distance.
The bottleneck of the top three implementations turned out to be precisely the
decision procedure for the Fr\'echet distance.
In this work, we present a fast, certifying implementation for deciding the
Fr\'echet distance, in order to (1) complement its pessimistic worst-case
hardness by an empirical analysis on realistic input data and to (2) improve
the state of the art for the GIS Cup challenge. We experimentally evaluate our
implementation on a large benchmark consisting of several data sets (including
handwritten characters and GPS trajectories). Compared to the winning
implementation of the GIS Cup, we obtain running time improvements of up to
more than two orders of magnitude for the decision procedure and of up to a
factor of 30 for queries to the near-neighbor data structure.
