Segmentation of trajectories on nonmonotone criteria

B. Aronov, A. Driemel, M.J. van Kreveld, M. Löffler, F. Staals

Research output: Contribution to journalArticleAcademicpeer-review

12 Citations (Scopus)
3 Downloads (Pure)

Abstract

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
Article number26
JournalACM Transactions on Algorithms
Volume12
Issue number2
DOIs
Publication statusPublished - 1 Dec 2015

Keywords

  • Dynamic programming
  • Geometric algorithms
  • Segmentation
  • Trajectory

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