The Discrete and Semicontinuous Fréchet Distance with Shortcuts via Approximate Distance Counting and Selection

The Fréchet distance is a well-studied similarity measure between curves. The discrete Fréchet distance is an analogous similarity measure, defined for two sequences of m and n points, where the points are usually sampled from input curves. We consider a variant, called the discrete Fréchet distance with shortcuts, which captures the similarity between (sampled) curves in the presence of outliers. When shortcuts are allowed only in one noise-containing curve, we give a randomized algorithm that runs in O((m + n)6/5 + ϵ) expected time, for any ϵ > 0. When shortcuts are allowed in both curves, we give an O((m2/3n2/3 + m + n)log 3(m + n))-time deterministic algorithm. We also consider the semicontinuous Fréchet distance with one-sided shortcuts, where we have a sequence of m points and a polygonal curve of n edges, and shortcuts are allowed only in the sequence. We show that this problem can be solved in randomized expected time O((m + n)2/3m2/3n1/3log (m + n)). Our techniques are novel and may find further applications. One of the main new technical results is: Given two sets of points A and B in the plane and an interval I, we develop an algorithm that decides whether the number of pairs (x, y) ∈ A × B whose distance dist(x, y) is in I is less than some given threshold L. The running time of this algorithm decreases as L increases. In case there are more than L pairs of points whose distance is in I, we can get a small sample of pairs that contain a pair at approximate median distance (i.e., we can approximately “bisect” I). We combine this procedure with additional ideas to search, with a small overhead, for the optimal one-sided Fréchet distance with shortcuts, using a very fast decision procedure. We also show how to apply this technique for approximating distance selection (with respect to rank), and a somewhat more involved variant of this technique is used in the solution of the semicontinuous Fréchet distance with one-sided shortcuts. In general, the new technique can be applied to optimization problems for which the decision procedure is very fast but standard techniques like parametric search makes the optimization algorithm substantially slower.