Computing the Skorokhod distance between polygonal traces

The Skorokhod distance is a natural metric on traces of continuous and hybrid systems. It measures the best match between two traces, each mapping a time interval [0, T] to a metric space O, when continuous bijective timing distortions are allowed. Formally, it computes the infimum, over all timing distortions, of the maximum of two components: the first component quantifies the timing discrepancy of the timing distortion, and the second quantifies the mismatch (in the metric space O) of the values after the timing distortion. Skorokhod distances appear in various fundamental hybrid systems analysis concerns: from definitions of hybrid systems semantics and notions of equivalence, to practical problems such as checking the closeness of models or the quality of simulations. Despite its extensive use in semantics, the computation problem for the Skorokhod distance between two finite sampled-time hybrid traces remained open. We address the problem of computing the Skorokhod distance between two polygonal traces (these traces arise when sampled-time traces are completed by linear interpolation between sample points). We provide an algorithm to compute the exact Skorokhod distance when trace values are compared using the L1, L2, and L∞ norms in n dimensions. Our algorithm, based on a reduction to Fréchet distances, is fully polynomial-time, and incorporates novel polynomial-time procedures for a set of geometric primitives in IRn over the three norms.