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.
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P. Agarwal, M. Sharir and P. Shor, Tight bounds on the length of Davenport Schinzel sequences, in preparation.
M. Atallah, Dynamic computational geometry, Proc. 24th Symp. on Foundations of Computer Science ,1983, pp.92–99.
M. Atallah, Some dynamic computational geometry problems, Comp. and Maths. with Appls. 11 (1985) pp. 1171–1181.
F. Aurenhammer, Improved algorithms for discs and balls using power diagrams, Tech. Rept., Technical University of Graz.
A. Baitsan and M. Sharir, On shortest paths between two convex polyhedra, Tech. Rept. 180, Comp. Science Dept., Courant Institute, Sept. 1985.
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.
B. Chazelle, A theorem on polygon cutting with applications, Proc. 23th IEEE Symp. on Foundations of Computer Science ,1982, pp. 339–349.
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.
B. Chazelle and L. Guibas, Visibility and intersection problems in plane geometry, Proc. ACM Symposium on Computational Geometry ,1985, pp. 135–146.
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.
R. Cole, Searching and storing similar lists, J. Algorithms (in press).
R. Cole and M. Sharir, Visibility of a polyhedral surface from a point, Tech. Rept. 266, Comp. Science Dept., Courant Institute, December 1986.
H. Davenport, A combinatorial problem connected with differential equations, II, Acta Arithmetica 17(1971) pp. 363–372.
H. Davenport and A. Schinzel, A combinatorial problem connected with differential equations, Amer. J. Math. 87(1965) pp. 684–694.
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.
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.
L. Guibas and M. Sharir, Computing a single face in an arrangement of segments, in preparation.
S. Hart and M. Sharir, Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes, Combinatorica 6 (1986) pp. 151–177.
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.
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.
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.
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.
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.
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)
Lozano-Perez, T. and Wesley, M., An algorithm for planning collision-free paths among polyhedral obstacles, Comm. ACM 22 (1979), pp. 560–570.
M. McKenna, Worst-case optimal hidden-surface removal, Tech. Rept. JHU/EECS-86/05, The Johns Hopkins University, 1986.
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.
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.
W.H. Plantinga and C.R. Dyer, An algorithm for constructing the aspect graph, Proc. 27th FOCS Symposium ,1986, pp. 123–131.
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.
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.
J.T. Schwartz and M. Sharir, On the two-dimensional Davenport Schinzel problem, in preparation.
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.
M. Sharir, Improved lower bounds on the length of Davenport Schinzel sequences, Tech. Rept. 204, Comp. Sci. Dept., Courant Institute, february 1986.
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.
M. Sharir and S. Sifrony, On the general motion planning problem with two degrees of freedom, in preparation.
P. Shor, Simplified geometric realizations of superlinear Davenport Schinzel sequences, in preparation.
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.
E. Szemeredi, On a Problem by Davenport and Schinzel, Acta Arithmetica 25(1974), pp. 213–224.
G.T. Toussaint, Movable separability of sets, in Computational Geometry (G.T. Toussaint, ed.), North-Holland, 1985, pp. 335–375.
A. Wiernik, Planar realization of non-linear Davenport-Schinzel sequences by segments, Tech. Rept. 224, Comp. Science Dept., Courant Institute, June 1986.
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© 1988 Springer-Verlag Berlin Heidelberg
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Sharir, M. (1988). Davenport-Schinzel Sequences and their Geometric Applications. In: Earnshaw, R.A. (eds) Theoretical Foundations of Computer Graphics and CAD. NATO ASI Series, vol 40. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83539-1_9
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DOI: https://doi.org/10.1007/978-3-642-83539-1_9
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