Finding All Maximal Subsequences with Hereditary Properties

Authors Drago Bokal, Sergio Cabello, David Eppstein

Thumbnail PDF


  • Filesize: 0.52 MB
  • 15 pages

Document Identifiers

Author Details

Drago Bokal
Sergio Cabello
David Eppstein

Cite AsGet BibTex

Drago Bokal, Sergio Cabello, and David Eppstein. Finding All Maximal Subsequences with Hereditary Properties. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 240-254, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


Consider a sequence s_1,...,s_n of points in the plane. We want to find all maximal subsequences with a given hereditary property P: find for all indices i the largest index j^*(i) such that s_i,...,s_{j^*(i)} has property P. We provide a general methodology that leads to the following specific results: - In O(n log^2 n) time we can find all maximal subsequences with diameter at most 1. - In O(n log n loglog n) time we can find all maximal subsequences whose convex hull has area at most 1. - In O(n) time we can find all maximal subsequences that define monotone paths in some (subpath-dependent) direction. The same methodology works for graph planarity, as follows. Consider a sequence of edges e_1,...,e_n over a vertex set V. In O(n log n) time we can find, for all indices i, the largest index j^*(i) such that (V,{e_i,..., e_{j^*(i)}}) is planar.
  • convex hull
  • diameter
  • monotone path
  • sequence of points
  • trajectory


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Boris Aronov, Anne Driemel, Marc J. van Kreveld, Maarten Löffler, and Frank Staals. Segmentation of trajectories for non-monotone criteria. In SODA 2013, pages 1897-1911, 2013. Google Scholar
  2. Michael J. Bannister, William E. Devanny, Michael T. Goodrich, and Joe Simons. Windows into geometric events. In CCCG 2014, 2014. Google Scholar
  3. Michael J. Bannister, Christopher DuBois, David Eppstein, and Padhraic Smyth. Windows into relational events: Data structures for contiguous subsequences of edges. In SODA 2013, pages 856-864, 2013. Google Scholar
  4. Kevin Buchin, Maike Buchin, Marc van Kreveld, Maarten Löffler, Rodrigo I. Silveira, Carola Wenk, and Lionov Wiratma. Median trajectories. Algorithmica, 66(3):595-614, 2013. Google Scholar
  5. Kevin Buchin, Maike Buchin, Marc van Kreveld, and Jun Luo. Finding long and similar parts of trajectories. Comput. Geom., 44(9):465-476, 2011. Google Scholar
  6. Maike Buchin, Anne Driemel, Marc J. van Kreveld, and Vera Sacristan. Segmenting trajectories: A framework and algorithms using spatiotemporal criteria. J. Spatial Information Science, 3(1):33-63, 2011. Google Scholar
  7. Timothy M. Chan. Dynamic planar convex hull operations in near-logarithmic amortized time. J. ACM, 48(1):1-12, 2001. Google Scholar
  8. Timothy M. Chan. A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries. J. ACM, 57(3), 2010. Google Scholar
  9. Chen Chen, Hao Su, Qixing Huang, Lin Zhang, and Leonidas Guibas. Pathlet learning for compressing and planning trajectories. In SIGSPATIAL'13, pages 392-395, 2013. Google Scholar
  10. Giuseppe Di Battista and Roberto Tamassia. On-Line planarity testing. SIAM J. Comput., 25(5):956-997, 1996. Google Scholar
  11. David Eppstein. Dynamic Euclidean minimum spanning trees and extrema of binary functions. Discrete Comput. Geom., 13:111-122, 1995. Google Scholar
  12. David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Thomas H. Spencer. Separator based sparsification. I. Planary testing and minimum spanning trees. J. Comput. Syst. Sci., 52(1):3-27, 1996. Google Scholar
  13. David Eppstein, Michael T. Goodrich, and Maarten Löffler. Tracking moving objects with few handovers. In WADS 2011, volume 6844 of LNCS, pages 362-373. Springer, 2011. Google Scholar
  14. Johannes Fischer and Volker Heun. Space-efficient preprocessing schemes for range minimum queries on static arrays. SIAM J. Comput., 40(2):465-492, 2011. Google Scholar
  15. Zvi Galil, Giuseppe F. Italiano, and Neil Sarnak. Fully dynamic planarity testing with applications. J. ACM, 46(1):28-91, 1999. Google Scholar
  16. Joachim Gudmundsson, Jyrki Katajainen, Damian Merrick, Cahya Ong, and Thomas Wolle. Compressing spatio-temporal trajectories. Comput. Geom., 42(9):825-841, 2009. Google Scholar
  17. Joachim Gudmundsson, Marc van Kreveld, and Bettina Speckmann. Efficient detection of patterns in 2D trajectories of moving points. GeoInformatica, 11(2):195-215, 2007. Google Scholar
  18. John Hopcroft and Robert Tarjan. Efficient planarity testing. J. ACM, 21(4):549-568, 1974. Google Scholar
  19. Jakub Ła̧cki and Piotr Sankowski. Reachability in graph timelines. In ITCS 2013, pages 257-268, 2013. Google Scholar
  20. Mark H. Overmars and Jan van Leeuwen. Maintenance of configurations in the plane. J. Comput. Syst. Sci., 23(2):166-204, 1981. Google Scholar
  21. Franco P. Preparata and Michael I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, 1985. Google Scholar
  22. Peter van Emde Boas. Preserving order in a forest in less than logarithmic time and linear space. Inf. Process. Lett., 6(3):80-82, 1977. Google Scholar