Davenport-Schinzel Sequences and their Geometric Applications

  • Micha Sharir
Part of the NATO ASI Series book series (volume 40)

Abstract

Davenport Schinzel sequences are sequences that do not contain forbidden alternating subsequences of certain length. They are a powerful combinatorial tool applicable in contexts which involve the calculation of the pointwise maximum or minimum of a collection of (univariate) continuous functions, and have thus many applications in computational geometry and related areas. We review recent progress in the theory of Davenport Schinzel sequences, and present some of their geometric applications. These applications include efficient algorithms for the following geometric problems: (i) Preprocessing of a 2-D polyhedral terrain so as to support fast ray shooting queries from a fixed point. (ii) Determining whether two disjoint interlocking simple polygons can be separated from one another by a sequence of translations. (iii) Determining whether a given convex polygon can be translated and rotated so as to fit into another given polygonal region. (iv) Motion planning for a convex polygon in the plane amidst po ygonal barriers. (v) We also present some results on the complexity of the pointwise minimum or maximum of bivariate functions, and describe some of their applications.

Keywords

Hull Azimuth 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [ASS]
    P. Agarwal, M. Sharir and P. Shor, Tight bounds on the length of Davenport Schinzel sequences, in preparation.Google Scholar
  2. [Atl]
    M. Atallah, Dynamic computational geometry, Proc. 24th Symp. on Foundations of Computer Science ,1983, pp.92–99.Google Scholar
  3. [At2]
    M. Atallah, Some dynamic computational geometry problems, Comp. and Maths. with Appls. 11 (1985) pp. 1171–1181.MATHMathSciNetGoogle Scholar
  4. [Au]
    F. Aurenhammer, Improved algorithms for discs and balls using power diagrams, Tech. Rept., Technical University of Graz.Google Scholar
  5. [BS]
    A. Baitsan and M. Sharir, On shortest paths between two convex polyhedra, Tech. Rept. 180, Comp. Science Dept., Courant Institute, Sept. 1985.Google Scholar
  6. [BT]
    Ð.K. Bhattacharya and G.T. Toussaint, A linear algorithm for determining translation separability of two simple polygons, Tech. Rept. SOCS-86.1, McGill University, 1986.Google Scholar
  7. [Ch]
    B. Chazelle, A theorem on polygon cutting with applications, Proc. 23th IEEE Symp. on Foundations of Computer Science ,1982, pp. 339–349.Google Scholar
  8. [Ch2]
    B. Chazelle, The polygon containment problem, in Advances in Computing Research, Vol. I: Computational Geometry ,(F.P. Preparata, Ed.), JAI Press, Greenwich, Connecticut (1983), pp. 1–33.Google Scholar
  9. [CG]
    B. Chazelle and L. Guibas, Visibility and intersection problems in plane geometry, Proc. ACM Symposium on Computational Geometry ,1985, pp. 135–146.CrossRefGoogle Scholar
  10. [CG2]
    B. Chazelle and L. Guibas, Fractional cascading: A data structuring technique with geometric applications, Proc. 12th Int. Colloq. on Automata, Languages and Programming ,1985, Lecture Notes in Computer Science, Springer Verlag, pp. 90–99.CrossRefGoogle Scholar
  11. [Co]
    R. Cole, Searching and storing similar lists, J. Algorithms (in press).Google Scholar
  12. [CS]
    R. Cole and M. Sharir, Visibility of a polyhedral surface from a point, Tech. Rept. 266, Comp. Science Dept., Courant Institute, December 1986.Google Scholar
  13. [Da]
    H. Davenport, A combinatorial problem connected with differential equations, II, Acta Arithmetica 17(1971) pp. 363–372.MATHMathSciNetGoogle Scholar
  14. [DS]
    H. Davenport and A. Schinzel, A combinatorial problem connected with differential equations, Amer. J. Math. 87(1965) pp. 684–694.CrossRefMATHMathSciNetGoogle Scholar
  15. [DFPAN]
    L. De Floriani, B. Falcidieno, C. Pienovi, D. Allen and G. Nagy, A visibility-based model for terrain features, Proc. Int. Symp. on Spatial Data Handling ,Seattle, July 1986.Google Scholar
  16. [DSST]
    J. Driscoll, N. Sarnak, D. Sleator and R.E. Tarjan, Making data structures persistent, Proc. 18th Symp. on Theory of Computing ,1986, pp. 109–121.Google Scholar
  17. [GS]
    L. Guibas and M. Sharir, Computing a single face in an arrangement of segments, in preparation.Google Scholar
  18. [HS]
    S. Hart and M. Sharir, Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes, Combinatorica 6 (1986) pp. 151–177.CrossRefMATHMathSciNetGoogle Scholar
  19. [KLPS]
    K. Kedem, R. Livne, J. Pach and M. Sharir, On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles, Discrete and Computational Geometry 1 (1986) pp. 59–71.CrossRefMATHMathSciNetGoogle Scholar
  20. [KS]
    K. Kedem and M. Sharir, An efficient algorithm for planning collision-free translational motion of a convex polygonal object in 2-dimensional space amidst polygonal obstacles, Proc. ACM Symp. on Computational Geometry 1985, pp. 75–80.Google Scholar
  21. [KS2]
    K. Kedem and M. Sharir, An efficient motion planning Algorithm for a convex rigid polygonal object in 2-dimensional polygonal space, Tech. Rept. 253, Comp. Science Dept., Courant Institute, October 1986.Google Scholar
  22. [LS]
    D. Leven and M. Sharir, An Efficient and Simple Motion Planning Algorithm for a Ladder Moving in Two-Dimensional Space Amidst Polygonal Barriers, Proc. ACM Symp. on Computational Geometry 1985, pp. 221–227.Google Scholar
  23. [LS2]
    D. Leven and M. Sharir, Planning a Purely Translational Motion for a Convex Object in Two-dimensional Space Using Generalized Voronoi Diagrams, Discrete and Computational Geometry 2 (1987) pp. 9–31.CrossRefMATHMathSciNetGoogle Scholar
  24. [LS3]
    D. Leven and M. Sharir, On the number of critical free contacts of a convex polygonal object moving in 2-dimensional polygonal space, Tech. Rept. 187, Comp. Sci. Dept., Courant Institute, October 1985. (to appear in Discrete and Computational Geometry) Google Scholar
  25. [LPW]
    Lozano-Perez, T. and Wesley, M., An algorithm for planning collision-free paths among polyhedral obstacles, Comm. ACM 22 (1979), pp. 560–570.CrossRefGoogle Scholar
  26. [MK]
    M. McKenna, Worst-case optimal hidden-surface removal, Tech. Rept. JHU/EECS-86/05, The Johns Hopkins University, 1986.Google Scholar
  27. [OSY]
    O’Dunlaing, C., Sharir, M. and Yap, C, Generalized Voronoi Diagrams for a Ladder: Efficient Construction of the Diagram, Algorithmica 2 (1987) pp. 27–59.CrossRefMATHMathSciNetGoogle Scholar
  28. [PS]
    J. Pach and M. Sharir, The upper envelope of a piecewise linear function and the boundary of a region enclosed by convex plates: Combinatorial analysis, Tech. Rept. 279, Comp. Sci. Dept., Courant Institute, March 1987.Google Scholar
  29. [PD]
    W.H. Plantinga and C.R. Dyer, An algorithm for constructing the aspect graph, Proc. 27th FOCS Symposium ,1986, pp. 123–131.Google Scholar
  30. [PSS]
    R. Pollack, M. Sharir and S. Sifrony, Separating two simple polygons by a sequence of translations, Tech. Rept. 215, Comp. Sci. Dept., Courant Institute, April 1986.Google Scholar
  31. [PM]
    F.P. Preparata and D.E. Muller, Finding the intersection of n half spaces in time 0(n log n), Theoretical Computer Science 8 (1979) pp. 44–55.CrossRefMathSciNetGoogle Scholar
  32. [SS]
    J.T. Schwartz and M. Sharir, On the two-dimensional Davenport Schinzel problem, in preparation.Google Scholar
  33. [Sh1]
    M. Sharir, Almost Linear Upper Bounds on the Length of General Davenport-Schinzel Sequences, Tech. Report 29/85, The Eskenasy Institute of Computer Sciences, Tel Aviv University, February 1985.Google Scholar
  34. [Sh2]
    M. Sharir, Improved lower bounds on the length of Davenport Schinzel sequences, Tech. Rept. 204, Comp. Sci. Dept., Courant Institute, february 1986.Google Scholar
  35. [SL]
    M. Sharir and R. Livne, On Minima of Functions, Intersection Patterns of Curves, and Davenport-Schinzel Sequences, Proc. 26th FOCS Symposium ,1985, pp. 312–320.Google Scholar
  36. [SSi]
    M. Sharir and S. Sifrony, On the general motion planning problem with two degrees of freedom, in preparation.Google Scholar
  37. [Sho]
    P. Shor, Simplified geometric realizations of superlinear Davenport Schinzel sequences, in preparation.Google Scholar
  38. [SiS]
    S. Sifrony and M. Sharir, An Efficient Motion Planning Algorithm for a Rod Moving in Two-dimensional Polygonal Space, Proc. 2nd ACM Symposium on Computational Geometry 1986, pp. 178–186.CrossRefGoogle Scholar
  39. [Sz]
    E. Szemeredi, On a Problem by Davenport and Schinzel, Acta Arithmetica 25(1974), pp. 213–224.MATHMathSciNetGoogle Scholar
  40. [To]
    G.T. Toussaint, Movable separability of sets, in Computational Geometry (G.T. Toussaint, ed.), North-Holland, 1985, pp. 335–375.Google Scholar
  41. [WS]
    A. Wiernik, Planar realization of non-linear Davenport-Schinzel sequences by segments, Tech. Rept. 224, Comp. Science Dept., Courant Institute, June 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Micha Sharir
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityUSA

Personalised recommendations