Segmentation of trajectories on nonmonotone criteria

Boris Aronov, Anne Driemel, Marc Van Kreveld, Maarten Löffler, Frank Staals

    Research output: Contribution to journalArticlepeer-review


    In the trajectory segmentation problem, we are given a polygonal trajectory with n vertices that we have to subdivide into a minimum number of disjoint segments (subtrajectories) that all satisfy a given criterion. The problem is known to be solvable efficiently for monotone criteria: criteria with the property that if they hold on a certain segment, they also hold on every subsegment of that segment. To the best of our knowledge, no theoretical results are known for nonmonotone criteria. We present a broader study of the segmentation problem, and suggest a general framework for solving it, based on the start-stop diagram: a 2-dimensional diagram that represents all valid and invalid segments of a given trajectory. This yields two subproblems: (1) computing the start-stop diagram, and (2) finding the optimal segmentation for a given diagram. We show that (2) is NP-hard in general. However, we identify properties of the start-stop diagram that make the problem tractable and give a polynomial-time algorithm for this case. We study two concrete nonmonotone criteria that arise in practical applications in more detail. Both are based on a given univariate attribute function f over the domain of the trajectory. We say a segment satisfies an outlier-tolerant criterion if the value of f lies within a certain range for at least a given percentage of the length of the segment. We say a segment satisfies a standard deviation criterion if the standard deviation of f over the length of the segment lies below a given threshold. We show that both criteria satisfy the properties that make the segmentation problem tractable. In particular, we compute an optimal segmentation of a trajectory based on the outlier-tolerant criterion in O(n2 log n + kn2) time and on the standard deviation criterion in O(kn2) time, where n is the number of vertices of the input trajectory and k is the number of segments in an optimal solution.

    Original languageEnglish (US)
    Article number26
    JournalACM Transactions on Algorithms
    Issue number2
    StatePublished - Dec 1 2015


    • Dynamic programming
    • Geometric algorithms
    • Segmentation
    • Trajectory

    ASJC Scopus subject areas

    • Mathematics (miscellaneous)


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