Combinatorial complexity bounds for arrangements of curves and spheres

We present upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike. For example, we prove that the maximum number of edges boundingm cells in an arrangement ofn lines is Θ(m2/3n2/3 +n), and that it isO(m2/3n2/3β(n) +n) forn unit-circles, whereβ(n) (and laterβ(m, n)) is a function that depends on the inverse of Ackermann's function and grows extremely slowly. If we replace unit-circles by circles of arbitrary radii the upper bound goes up toO(m3/5n4/5β(n) +n). The same bounds (without theβ(n)-terms) hold for the maximum sum of degrees ofm vertices. In the case of vertex degrees in arrangements of lines and of unit-circles our bounds match previous results, but our proofs are considerably simpler than the previous ones. The maximum sum of degrees ofm vertices in an arrangement ofn spheres in three dimensions isO(m4/7n9/7β(m, n) +n2), in general, andO(m3/4n3/4β(m, n) +n) if no three spheres intersect in a common circle. The latter bound implies that the maximum number of unit-distances amongm points in three dimensions isO(m3/2β(m)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given.

[1]  J. Wrench.  Table errata: The art of computer programming, Vol. 2: Seminumerical algorithms (Addison-Wesley, Reading, Mass., 1969) by Donald E. Knuth , 1970 .

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

[3]  P. Erdös.  On extremal problems of graphs and generalized graphs , 1964 .

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

[5]  George E. Collins,et al.  Cylindrical Algebraic Decomposition I: The Basic Algorithm , 1984, SIAM J. Comput..

[6]  Bernard Chazelle,et al.  Triangulating a non-convex polytype , 1989, SCG '89.

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

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

[9]  Leonidas J. Guibas,et al.  Implicitly representing arrangements of lines or segments , 2011, SCG '88.

[10]  Paul Erdös.  On some problems of elementary and combinatorial geometry , 1975 .

[11]  Herbert Edelsbrunner,et al.  On the maximal number of edges of many faces in an arrangement , 1986, J. Comb. Theory, Ser. A.

[12]  Paul Erdös,et al.  A problem of Leo Moser about repeated distances on the sphere , 1989 .

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

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

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

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

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

[18]  I. Reiman.  Über ein Problem von K. Zarankiewicz , 1958 .

[19]  Micha Sharir,et al.  Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences , 1989, J. Comb. Theory A.

[20]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

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

[22]  H. Edelsbrunner,et al.  On the number of furthest neighbour pairs in a point set , 1989 .

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

[24]  A. Heppes.  Beweis einer Vermutung von A. Vázsonyi , 1956 .

[25]  Martti Mäntylä,et al.  Introduction to Solid Modeling , 1988 .

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

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

[28]  József Beck,et al.  On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry , 1983, Comb..

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

[30]  George E. Collins,et al.  Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975, Automata Theory and Formal Languages.

[31]  Fan Chung.  Sphere-and-point incidence relations in high dimensions with applications to unit distances and furthest-neighbor pairs , 1989 .

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

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

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

[35]  Paul Erdös,et al.  Repeated distances in space , 1988, Graphs Comb..

[36]  J. Schwartz,et al.  On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds , 1983 .

[37]  Richard Pollack,et al.  Proof of Grünbaum's Conjecture on the Stretchability of Certain Arrangements of Pseudolines , 1980, J. Comb. Theory, Ser. A.

[38]  V. Sós,et al.  On a problem of K. Zarankiewicz , 1954 .

[39]  Bernard Chazelle,et al.  A deterministic view of random sampling and its use in geometry , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

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

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

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

[43]  David P. Dobkin,et al.  Primitives for the manipulation of three-dimensional subdivisions , 1987, SCG '87.

[44]  E. Szemerédi,et al.  Unit distances in the Euclidean plane , 1984 .