On Levels in Arrangements of Curves

Abstract. Analyzing the worst-case complexity of the k -level in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O( nk^ 1-1/(9· 2s-3) ) ) for curves that are graphs of polynomial functions of an arbitrary fixed degree s . Previously, nontrivial results were known only for the case s=1 and s=2 . We also improve the earlier bound for pseudo-parabolas (curves that pairwise intersect at most twice) to O( nk7/9 log2/3k) . The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudo-parabolas into pseudo-segments, as well as a new observation for cutting pseudo-segments into pieces that can be extended to pseudo-lines. We mention applications to parametric and kinetic minimum spanning trees.

[1]  Micha Sharir,et al.  Arrangements and Their Applications , 2000, Handbook of Computational Geometry.

[2]  Leonidas J. Guibas,et al.  Points and triangles in the plane and halving planes in space , 1991, Discret. Comput. Geom..

[3]  B.,et al.  Halfspace Range Search : An Algorithmic Application of k-Sets , 2022 .

[4]  Hisao Tamaki,et al.  How to cut pseudo-parabolas into segments , 1995, SCG '95.

[5]  Ketan Mulmuley,et al.  Computational geometry : an introduction through randomized algorithms , 1993 .

[6]  Leonidas J. Guibas,et al.  Kinetic data structures: a state of the art report , 1998 .

[7]  Hisao Tamaki,et al.  How to Cut Pseudoparabolas into Segments , 1998, Discret. Comput. Geom..

[8]  Artur Andrzejak,et al.  K-sets and j-facets - A tour of discrete geometry , 1997 .

[9]  Timothy M. Chan.  Remarks on k-Level Algorithms in the Plane , 1999 .

[10]  Tamal K. Dey,et al.  Improved bounds on planar k-sets and k-levels , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[11]  Sinisa T. Vrecica,et al.  The Colored Tverberg's Problem and Complexes of Injective Functions , 1992, J. Comb. Theory, Ser. A.

[12]  Mark de Berg,et al.  Cuttings and applications , 1995, Int. J. Comput. Geom. Appl..

[13]  Géza Tóth,et al.  Point Sets with Many k-Sets , 2000, SCG '00.

[14]  Bernard Chazelle,et al.  Halfspace range search: An algorithmic application ofk-sets , 1986, Discret. Comput. Geom..

[15]  Leonidas J. Guibas,et al.  Parametric and kinetic minimum spanning trees , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[16]  J. Matoušek,et al.  On geometric optimization with few violated constraints , 1994, SCG '94.

[17]  Béla Bollobás,et al.  Modern Graph Theory , 2002, Graduate Texts in Mathematics.

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

[19]  Micha Sharir,et al.  Davenport-Schinzel sequences and their geometric applications , 1995, Handbook of Computational Geometry.

[20]  Noga Alon,et al.  The number of small semispaces of a finite set of points in the plane , 1986, J. Comb. Theory, Ser. A.

[21]  Herbert Edelsbrunner,et al.  Counting triangle crossings and halving planes , 1993, SCG '93.

[22]  Micha Sharir,et al.  On the complexity of many faces in arrangements of circles , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[23]  Tamal K. Dey,et al.  Improved Bounds for Planar k -Sets and Related Problems , 1998, Discret. Comput. Geom..

[24]  Sariel Har-Peled,et al.  Taking a walk in a planar arrangement , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[25]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[26]  Gert Vegter,et al.  In handbook of discrete and computational geometry , 1997 .

[27]  P. Erdös,et al.  Dissection Graphs of Planar Point Sets , 1973 .

[28]  Leonidas J. Guibas,et al.  Proximity problems on moving points , 1997, SCG '97.

[29]  Timothy M. Chan,et al.  On levels in arrangements of lines , 1998 .

[30]  H. Everett,et al.  AN OPTIMAL ALGORITHM FOR COMPUTING (≤K)-LEVELS, WITH APPLICATIONS , 1996 .

[31]  R. Pollack,et al.  Allowable Sequences and Order Types in Discrete and Computational Geometry , 1993 .

[32]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[33]  Micha Sharir,et al.  Onk-sets in arrangements of curves and surfaces , 1991, Discret. Comput. Geom..

[34]  Micha Sharir,et al.  On levels in arrangements of lines, segments, planes, and triangles , 1997, SCG '97.

[35]  Takeshi Tokuyama,et al.  Lovasz's lemma for the three-dimensional K-level of concave surfaces and its applications , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[36]  János Pach,et al.  Combinatorial Geometry , 2012 .

[37]  William L. Steiger,et al.  Algorithms for ham-sandwich cuts , 1994, CCCG.

[38]  David Eppstein.  Improved Bounds for Intersecting Triangles and Halving Planes , 1993, J. Comb. Theory, Ser. A.

[39]  J. Pach,et al.  Combinatorial geometry , 1995, Wiley-Interscience series in discrete mathematics and optimization.

[40]  Hisao Tamaki,et al.  A Characterization of Planar Graphs by Pseudo-Line Arrangements , 1997, Algorithmica.

[41]  Takeshi Tokuyama,et al.  On minimum and maximum spanning trees of linearly moving points , 1995, Discret. Comput. Geom..

[42]  Sergey Bereg,et al.  Generalizing Ham Sandwich Cuts to Equitable Subdivisions , 1999, SCG '99.

[43]  Hazel Everett,et al.  An optimal algorithm for the (≤ k)-levels, with applications to separation and transversal problems , 1993, SCG '93.

[44]  David Eppstein.  Geometric Lower Bounds for Parametric Matroid Optimization , 1998, Discret. Comput. Geom..

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

[46]  J. Pach,et al.  An upper bound on the number of planar k-sets , 1989, 30th Annual Symposium on Foundations of Computer Science.

[47]  Richard Pollack,et al.  On the Number of k-Subsets of a Set of n Points in the Plane , 1984, J. Comb. Theory, Ser. A.

[48]  Zoltán Füredi,et al.  On the number of halving planes , 1989, SCG '89.

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