Trajectory Grouping Structure under Geodesic Distance

Authors Irina Kostitsyna, Marc van Kreveld, Maarten Löffler, Bettina Speckmann, Frank Staals

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Irina Kostitsyna
Marc van Kreveld
Maarten Löffler
Bettina Speckmann
Frank Staals

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Irina Kostitsyna, Marc van Kreveld, Maarten Löffler, Bettina Speckmann, and Frank Staals. Trajectory Grouping Structure under Geodesic Distance. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 674-688, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


In recent years trajectory data has become one of the main types of geographic data, and hence algorithmic tools to handle large quantities of trajectories are essential. A single trajectory is typically represented as a sequence of time-stamped points in the plane. In a collection of trajectories one wants to detect maximal groups of moving entities and their behaviour (merges and splits) over time. This information can be summarized in the trajectory grouping structure. Significantly extending the work of Buchin et al. [WADS 2013] into a realistic setting, we show that the trajectory grouping structure can be computed efficiently also if obstacles are present and the distance between the entities is measured by geodesic distance. We bound the number of critical events: times at which the distance between two subsets of moving entities is exactly epsilon, where epsilon is the threshold distance that determines whether two entities are close enough to be in one group. In case the n entities move in a simple polygon along trajectories with tau vertices each we give an O(tau n^2) upper bound, which is tight in the worst case. In case of well-spaced obstacles we give an O(tau(n^2 + m lambda_4(n))) upper bound, where m is the total complexity of the obstacles, and lambda_s(n) denotes the maximum length of a Davenport-Schinzel sequence of n symbols of order s. In case of general obstacles we give an O(tau min(n^2 + m^3 lambda_4(n), n^2m^2)) upper bound. Furthermore, for all cases we provide efficient algorithms to compute the critical events, which in turn leads to efficient algorithms to compute the trajectory grouping structure.
  • moving entities
  • trajectories
  • grouping
  • computational geometry


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  1. Marc Benkert, Joachim Gudmundsson, Florian Hübner, and Thomas Wolle. Reporting flock patterns. Computational Geometry, 41(3):111-125, 2008. Google Scholar
  2. Kevin Buchin, Maike Buchin, Marc van Kreveld, Bettina Speckmann, and Frank Staals. Trajectory grouping structure. In Proc. 2013 WADS Algorithms and Data Structures Symposium, LNCS, pages 219-230. Springer, 2013. Google Scholar
  3. Maike Buchin, Somayeh Dodge, and Bettina Speckmann. Context-aware similarity of trajectories. In Geographic Information Science, volume 7478 of LNCS, pages 43-56. Springer, 2012. Google Scholar
  4. Maike Buchin, Anne Driemel, and Bettina Speckmann. Computing the Fréchet distance with shortcuts is NP-hard. In Symposium on Computational Geometry, page 367. ACM, 2014. Google Scholar
  5. Bernard Chazelle. Triangulating a simple polygon in linear time. Discrete Comput. Geom., 6(5):485-524, 1991. Google Scholar
  6. Herbert Edelsbrunner and John L. Harer. Computational Topology - an introduction. American Mathematical Society, 2010. Google Scholar
  7. Joachim Gudmundsson and Marc van Kreveld. Computing longest duration flocks in trajectory data. In Proc. 14th ACM International Symposium on Advances in Geographic Information Systems, GIS'06, pages 35-42. ACM, 2006. Google Scholar
  8. Leonidas J. Guibas and John Hershberger. Optimal shortest path queries in a simple polygon. Journal of Computer and System Sciences, 39(2):126-152, 1989. Google Scholar
  9. John Hershberger and Subhash Suri. An Optimal Algorithm for Euclidean Shortest Paths in the Plane. SIAM Journal on Computing, 28(6):2215-2256, 1999. Google Scholar
  10. Hoyoung Jeung, Man Lung Yiu, Xiaofang Zhou, Christian S. Jensen, and Heng Tao Shen. Discovery of convoys in trajectory databases. PVLDB, 1:1068-1080, 2008. Google Scholar
  11. Panos Kalnis, Nikos Mamoulis, and Spiridon Bakiras. On discovering moving clusters in spatio-temporal data. In Advances in Spatial and Temporal Databases, volume 3633 of LNCS, pages 364-381. Springer, 2005. Google Scholar
  12. Patrick Laube, Marc van Kreveld, and Stephan Imfeld. Finding REMO - detecting relative motion patterns in geospatial lifelines. In Developments in Spatial Data Handling, pages 201-215. Springer, 2005. Google Scholar
  13. Nimrod Megiddo. Applying parallel computation algorithms in the design of serial algorithms. J. ACM, 30(4):852-865, 1983. Google Scholar
  14. Salman Parsa. A deterministic O(m log m) time algorithm for the Reeb graph. In Proc. 28th ACM Symposium on Computational Geometry, pages 269-276, 2012. Google Scholar
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