Searching and sorting similar lists
Journal of Algorithms
Making data structures persistent
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Voronoi diagrams and arrangements
Discrete & Computational Geometry
On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Sharp upper and lower bounds on the length of general Davenport-Schinzel Sequences
Journal of Combinatorial Theory Series A
On the general motion planning problem with two degrees of freedom
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
The maximum number of ways to stab n convex nonintersecting sets in the plane is 2n - 2
Discrete & Computational Geometry
Convex hulls of finite sets of points in two and three dimensions
Communications of the ACM
On minima of function, intersection patterns of curves, and davenport-schinzel sequences
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
New bounds for lower envelopes in three dimensions, with applications to visibility in terrains
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Almost tight upper bounds for the single cell and zone problems in three dimensions
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Pareto envelopes in R3 under l1 and l∞ distance functions
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
| Hi-index | 0.00 |
We analyse the combinatorial complexity @k(F) of the minimum M(x,y) of a collection F of n continuous bivariate functions f"1(x,y), ... , f"n(x,y), such that each triple of function graphs intersect in at most s points, and each pair of functions intersect in a curve having at most t singular points. The following is proved. (1) If the intersection curve of each pair of functions intersects each plane x = const in exactly one point and s = 1 (but not if s = 2) then @k(F) is at most O(n), and can be calculated in time 0(n log n) by a method extending Shamos' algorithm for the calculation of planar Voronoi diagrams, (2) If s = 2 and the intersection of each pair of functions is connected then @k(F)= 0(n^2). (3) If the intersection curve of each pair of functions intersects every plane x = const in at most two points, then @k(F) is at most O(n@l"s"+"2(n)), where the constant of proportionality depends on s and t, and where @l,(q) is the (almost linear) maximum length of a (q,r) Davenport-Schinzel sequence. We also present an algorithm for calculating M in this case, running in time O(n@l"s"+"2(n) log n). (4) Finally, we present some geometric applications of these results.