Name
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(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. |
Year | DOI | Venue |
|---|---|---|
2013 | 10.1145/2660772 | Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms |
Keywords | Field | DocType |
algorithms,design,graph algorithms,path and circuit problems,theory | Discrete mathematics,Combinatorics,Polygon,Disjoint sets,Vertex (geometry),Segmentation,Diagram,Standard deviation,Mathematics,Trajectory,Monotone polygon | Conference |
Volume | Issue | ISSN |
12 | 2 | 1549-6325 |
ISBN | Citations | PageRank |
978-1-61197-251-1 | 2 | 0.51 |
References | Authors | |
7 | 5 | |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Boris Aronov | 1 | 1430 | 149.20 |
| Anne Driemel | 2 | 157 | 11.41 |
| Marc J. van Kreveld | 3 | 1702 | 166.91 |
| Maarten Löffler | 4 | 551 | 62.87 |
| Frank Staals | 5 | 29 | 11.40 |