Combinatorics and Algorithms of Arrangements

In this chapter we study the combinatorial structure of arrangements of algebraic curves or surfaces in low-dimensional Euclidean space. Such arrangements arise in many geometric problems, as will be exemplified below. To introduce the class of problems we will be interested in, we begin with the following concrete example, taken from the theory of motion planning in robotics.

[1]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[2]  Richard Cole,et al.  Visibility Problems for Polyhedral Terrains , 2018, J. Symb. Comput..

[3]  E. Szemerédi.  On a problem of Davenport and Schinzel , 1974 .

[4]  Leonidas J. Guibas,et al.  A Singly Exponential Stratification Scheme for Real Semi-Algebraic Varieties and its Applications , 1991, Theor. Comput. Sci..

[5]  Micha Sharir,et al.  On minima of function, intersection patterns of curves, and davenport-schinzel sequences , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[6]  H. Davenport.  A combinatorial problem connected with differential equations II , 1971 .

[7]  Micha Sharir,et al.  Almost linear upper bounds on the length of general davenport—schinzel sequences , 1987, Comb..

[8]  J. Oden,et al.  The Mathematics of Surfaces II , 1988 .

[9]  Micha Sharir,et al.  On the Two-Dimensional Davenport Schinzel Problem , 2018, J. Symb. Comput..

[10]  Leonidas J. Guibas,et al.  The complexity of many cells in arrangements of planes and related problems , 1990, Discret. Comput. Geom..

[11]  Micha Sharir,et al.  The upper envelope of piecewise linear functions and the boundary of a region enclosed by convex plates: Combinatorial analysis , 2011, Discret. Comput. Geom..

[12]  Leonidas J. Guibas,et al.  The complexity and construction of many faces in arrangements of lines and of segments , 1990, Discret. Comput. Geom..

[13]  M. Atallah.  Some dynamic computational geometry problems , 1985 .

[14]  Bernard Chazelle,et al.  A deterministic view of random sampling and its use in geometry , 1990, Comb..

[15]  Micha Sharir,et al.  Triangles in space or building (and analyzing) castles in the air , 1990, Comb..

[16]  David Haussler,et al.  ɛ-nets and simplex range queries , 1987, Discret. Comput. Geom..

[17]  Herbert Edelsbrunner,et al.  The upper envelope of piecewise linear functions: Tight bounds on the number of faces , 1989, Discret. Comput. Geom..

[18]  Klara Kedem,et al.  Placing the largest similar copy of a convex polygon among polygonal obstacles , 1989, SCG '89.

[19]  Leonidas J. Guibas,et al.  The upper envelope of piecewise linear functions: Algorithms and applications , 2015, Discret. Comput. Geom..

[20]  Leonidas J. Guibas,et al.  Topologically sweeping an arrangement , 1986, STOC '86.

[21]  Bernard Chazelle,et al.  The power of geometric duality , 1985, BIT Comput. Sci. Sect..

[22]  Leonidas J. Guibas,et al.  Lines in space - combinatorics, algorithms and applications , 1989, Symposium on the Theory of Computing.

[23]  Micha Sharir,et al.  Nonlinearity of davenport—Schinzel sequences and of generalized path compression schemes , 1986, FOCS.

[24]  R. Canham.  A theorem on arrangements of lines in the plane , 1969 .

[25]  Micha Sharir,et al.  Improved lower bounds on the length of Davenport-Schinzel sequences , 1988, Comb..

[26]  Micha Sharir,et al.  An Efficient and Simple Motion Planning Algorithm for a Ladder Amidst Polygonal Barriers , 1987, J. Algorithms.

[27]  Micha Sharir,et al.  Improved Combinatorial Bounds and Efficient Techniques for Certain Motion Planning Problems with Three Degrees of Freedom , 1991, Comput. Geom..

[28]  Micha Sharir,et al.  On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles , 1986, Discret. Comput. Geom..

[29]  Micha Sharir,et al.  Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences , 2015, J. Comb. Theory, Ser. A.

[30]  Jirí Matousek.  Cutting hyperplane arrangements , 1991, Discret. Comput. Geom..

[31]  Micha Sharir,et al.  On the shortest paths between two convex polyhedra , 2018, JACM.

[32]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[33]  Marco Pellegrini,et al.  Stabbing and ray shooting in 3 dimensional space , 1990, SCG '90.

[34]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[35]  Raimund Seidel,et al.  Constructing Arrangements of Lines and Hyperplanes with Applications , 1986, SIAM J. Comput..

[36]  John Hershberger,et al.  Finding the Upper Envelope of n Line Segments in O(n log n) Time , 1989, Inf. Process. Lett..

[37]  Micha Sharir,et al.  On the number of critical free contacts of a convex polygonal object moving in two-dimensional polygonal space , 1987, Discret. Comput. Geom..

[38]  Micha Sharir,et al.  Separating two simple polygons by a sequence of translations , 2015, Discret. Comput. Geom..

[39]  Franz Aurenhammer,et al.  Power Diagrams: Properties, Algorithms and Applications , 1987, SIAM J. Comput..

[40]  Micha Sharir,et al.  An efficient and simple motion planning algorithm for a ladder moving in two-dimensional space amidst polygonal barriers (extended abstract) , 1985, SCG '85.

[41]  Boris Aronov,et al.  Counting facets and incidences , 1992, Discret. Comput. Geom..

[42]  Micha Sharir,et al.  Planar realizations of nonlinear davenport-schinzel sequences by segments , 1988, Discret. Comput. Geom..

[43]  Endre Szemerédi,et al.  Extremal problems in discrete geometry , 1983, Comb..

[44]  Thomas Ottmann,et al.  Algorithms for Reporting and Counting Geometric Intersections , 1979, IEEE Transactions on Computers.

[45]  M. Sharir,et al.  Improved bounds on the complexity of many faces in arrangements of segments , 1992 .

[46]  Kenneth L. Clarkson,et al.  New applications of random sampling in computational geometry , 1987, Discret. Comput. Geom..

[47]  Micha Sharir,et al.  On Vertical Visibility in Arrangements of Segments and the Queue Size in the Bentley-Ottmann Line Sweeping Algorithm , 1991, SIAM J. Comput..

[48]  H. Davenport,et al.  A Combinatorial Problem Connected with Differential Equations , 1965 .

[49]  Leonidas J. Guibas,et al.  Combinatorial complexity bounds for arrangements of curves and spheres , 1990, Discret. Comput. Geom..

[50]  P. Erdös.  On Sets of Distances of n Points , 1946 .

[51]  Michael McKenna.  Worst-case optimal hidden-surface removal , 1987, TOGS.

[52]  Jirí Matousek.  Construction of epsilon nets , 1989, SCG '89.

[53]  Leonidas J. Guibas,et al.  Arrangements of Curves in the Plane - Topology, Combinatorics and Algorithms , 2018, Theor. Comput. Sci..

[54]  Leonidas J. Guibas,et al.  On the general motion-planning problem with two degrees of freedom , 2015, SCG '88.

[55]  Micha Sharir,et al.  An efficient motion-planning algorithm for a convex polygonal object in two-dimensional polygonal space , 1990, Discret. Comput. Geom..

[56]  David E. Muller,et al.  Finding the Intersection of n Half-Spaces in Time O(n log n) , 1979, Theor. Comput. Sci..