Pii: S0925-7721(00)00006-7
نویسندگان
چکیده
A k-set of a finite set S of points in the plane is a subset of cardinality k that can be separated from the rest by a straight line. The question of how many k-sets a set of n points can contain is a long-standing open problem where a lower bound of (n logk) and an upper bound of O(nk1/3) are known today. Under certain restrictions on the set S, for example, if all points lie on a convex curve, the number of k-sets is linear. We generalize this observation by showing that if the points of S lie on a constant number of convex curves, the number of k-sets remains linear in n. 2000 Elsevier Science B.V. All rights reserved.
منابع مشابه
On the Complexity of Optimization Problems for 3-dimensional Convex Polyhedra and Decision Trees
We show that several well-known optimization problems involving 3-dimensional convex polyhedra and decision trees are NP-hard or NP-complete. One of the techniques we employ is a linear-time method for realizing a planar 3-connected triangulation as a convex polyhedron, which may be of independent interest.
متن کاملThe DFS-heuristic for orthogonal graph drawing
In this paper, we present a new heuristic for orthogonal graph drawings, which creates drawings by performing a depth-first search and placing the nodes in the order they are encountered. This DFS-heuristic works for graphs with arbitrarily high degrees, and particularly well for graphs with maximum degree 3. It yields drawings with at most one bend per edge, and a total number of m−n+1 bends f...
متن کاملPii: S0925-7721(00)00023-7
It is shown that in any polygonal art gallery of n sides it is possible to place bn/3c point guards whose range of vision is 180◦ so that every interior point of the gallery can be seen by at least one of them. The guards can be stationed at any point of the art gallery. This settles an open problem posed by J. Urrutia. 2000 Elsevier Science B.V. All rights reserved.
متن کاملPii: S0925-7721(00)00010-9
We improve previous lower bounds on the number of simple polygonizations, and other kinds of crossing-free subgraphs, of a set of N points in the plane by analyzing a suitable configuration. We also prove that the number of crossing-free perfect matchings and spanning trees is minimum when the points are in convex position. 2000 Elsevier Science B.V. All rights reserved.
متن کاملDensest lattice packings of 3-polytopes
Based on Minkowski’s work on critical lattices of 3-dimensional convex bodies we present an efficient algorithm for computing the density of a densest lattice packing of an arbitrary 3-polytope. As an application we calculate densest lattice packings of all regular and Archimedean polytopes.
متن کامل