We study the problem of constructing minimum power-p Euclidean k-Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most k additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of p (where p ≥ 1), and the cost of ...

The rectilinear Steiner tree problem is to nd a minimum-length rectilinear interconnection of a set of points in the plane. A reduction from the rectilinear Steiner tree problem to the graph Steiner tree problem allows the use of exact algorithms for the graph Steiner tree problem to solve the rectilinear problem. Furthermore, a number of more direct, geometric algorithms have been devised for ...

We show that the typical wind-tree model, in the sense of Baire, is recurrent and has a dense set of periodic orbits. The recurrence result also holds for the Lorentz gas : the typical Lorentz gas, in the sense of Baire, is recurrent. These Lorentz gases need not be of finite horizon! In 1912 Paul and Tatiana Ehrenfest proposed the wind-tree model of diffusion in order to study the statistical ...

A classical problem in computational geometry is the planar point location problem. This problem calls for preprocessing a polygonal subdivision of the plane defined by n line segments so that, given a sequence of points, the polygon containing each point can be determined quickly on-line. Several ways of solving this problem in O(log n) query time and O(n) space are known, but they are all rat...

Most heuristics for the Steiner tree problem in the Euclidean plane perform a series of iterative improvements using the minimum spanning tree as an initial solution. We may therefore characterize them as local search heuristics. In this paper, we rst give a survey of existing heuristic approaches from a local search perspective, by setting up solution spaces and neighbourhood structures. Secon...

Given a set of n disjoint line segments in the plane, we show that it is always possible to form a spanning tree of the endpoints of the segments, such that each line segment is an edge of the tree and the tree has no crossing edges. We also show that the maximum degree of a node in the minimum weight spanning tree formed by these line segments is seven, and that there exists a set of line segm...

This paper presents a method to render complex trees on high frame rates while providing accurate occlusion and parallax effects. Based on the recognition that a planar billboard is accurate if the represented polygon is in its plane, we find a billboard for each of those groups of tree leaves that lie approximately in the same plane. The tree is thus represented by a set of billboards, called ...

Finding the shortest path between two positions is a fundamental problem in transportation, routing, and communications applications. In robot motion planning, the robot should pass around the obstacles touching none of them, i.e. the goal is to find a collision-free path from a starting to a target position. This task has many specific formulations depending on the shape of obstacles, allowabl...

The tree-drawing problem is to produce a `tidy' mapping of elements of a tree to points in the plane. In this paper, we derive an eecient algorithm for producing tidy drawings of trees. The speciication, the starting point for the derivations, consists of a collection of intuitively appealing criteria satissed by tidy drawings. The derivation shows constructively that these criteria completely ...