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Computer Science, Computational Geometry, cs.CG
Abstract:
The Frechet distance is a well-studied and very popular measure of similarity
of two curves. Many variants and extensions have been studied since Alt and
Godau introduced this measure to computational geometry in 1991. Their original
algorithm to compute the Frechet distance of two polygonal curves with n
vertices has a runtime of O(n^2 log n). More than 20 years later, the state of
the art algorithms for most variants still take time more than O(n^2 / log n),
but no matching lower bounds are known, not even under reasonable complexity
theoretic assumptions.
To obtain a conditional lower bound, in this paper we assume the Strong
Exponential Time Hypothesis or, more precisely, that there is no
O*((2-delta)^N) algorithm for CNF-SAT for any delta > 0. Under this assumption
we show that the Frechet distance cannot be computed in strongly subquadratic
time, i.e., in time O(n^{2-delta}) for any delta > 0. This means that finding
faster algorithms for the Frechet distance is as hard as finding faster CNF-SAT
algorithms, and the existence of a strongly subquadratic algorithm can be
considered unlikely.
Our result holds for both the continuous and the discrete Frechet distance.
We extend the main result in various directions. Based on the same assumption
we (1) show non-existence of a strongly subquadratic 1.001-approximation, (2)
present tight lower bounds in case the numbers of vertices of the two curves
are imbalanced, and (3) examine realistic input assumptions (c-packed curves).
