## 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 language | English (US) |
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Title of host publication | Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 |

Publisher | Association for Computing Machinery |

Pages | 1897-1911 |

Number of pages | 15 |

ISBN (Print) | 9781611972511 |

DOIs | |

State | Published - 2013 |

Event | 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 - New Orleans, LA, United States Duration: Jan 6 2013 → Jan 8 2013 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Other

Other | 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 |
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Country/Territory | United States |

City | New Orleans, LA |

Period | 1/6/13 → 1/8/13 |

## ASJC Scopus subject areas

- Software
- General Mathematics