Finding long and similar parts of trajectories

A natural time-dependent similarity measure for two trajectories is their average distance at corresponding times. We give algorithms for computing the most similar subtrajectories under this measure, assuming the two trajectories are given as two polygonal, possibly self-intersecting lines. When a minimum duration is specified for the subtrajectories, and they must start at exactly corresponding times in the input trajectories, we give a linear-time algorithm for computing the starting time and duration of the most similar subtrajectories. The algorithm is based on a result of independent interest: We present a linear-time algorithm to find, for a piece-wise monotone function, an interval of at least a given length that has minimum average value.