Segmentation of trajectories for non-monotone criteria

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

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

9 Citations (Scopus)

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 [4]. To the best of our knowledge, no theoretical results are known for non-monotone 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: (i) computing the start-stop diagram, and (ii) finding the optimal segmentation for a given diagram. We show that (ii) is NP-hard in general. However, we identify properties of the start-stop diagram that make the problem tractable, and give polynomial-time algorithm for this case. We study two concrete non-monotone 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(n^2 log n+kn^2) time, and on the standard deviation criterion in O(kn^2) 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
Title of host publicationProceedings 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'13),6-8 january 2013, New Orleans LA, USA, January 6-8, 2013)
Place of PublicationPhiladelphia PA
PublisherSociety for Industrial and Applied Mathematics (SIAM)
Pages1897-1911
ISBN (Print)978-1-61197-251-1
DOIs
Publication statusPublished - 2013
Externally publishedYes
Event24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2013) - Astor Crowne Plaza Hotel, New Orleans, United States
Duration: 6 Jan 20138 Jan 2013
Conference number: 24
http://www.siam.org/meetings/da13/

Conference

Conference24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2013)
Abbreviated titleSODA '13
CountryUnited States
CityNew Orleans
Period6/01/138/01/13
Internet address

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