Given a simple polygon P without holes all of whose edges are axis-parallel, a rectilinear path in P is a path that consists of axis-parallel segments only and does not cross any edge of P. The length of such a path is defined as the number of segments it consists of and the rectilinear link distance between two points in P is defined as the length of the shortest path connecting the two points...

This paper deals with the kth nearest rectilinear distance of two regular point patterns: square and diamond lattices. The probability density functions of the kth nearest rectilinear distance are theoretically derived for k = 1, 2, . . . , 8. Upper and lower bounds of the kth nearest distance are also derived. As an application of the kth nearest distance, we consider a facility location probl...

We show the power of posets in computational geometry by solving several problems posed on a set S of n points in the plane: (1) nd the n ? k ? 1 rectilinear farthest neighbors (or, equivalently, k nearest neighbors) to every point of S (extendable to higher dimensions), (2) enumerate the k largest (smallest) rectilinear distances in decreasing (increasing) order among the points of S, (3) give...

Given a set P of n points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. We consider the rectilinear instances of the TSP, i.e. TSP instances where the points lie in the plane and the distance between two points is the l1 distance. We propose a fixed-parameter algorithm for the Rectilinear TSP running in O ( nh7 ...

This paper examines the relationship between road directions and the rectilinear distance in regular point patterns. We derive the distributions of the rectilinear distances to the nearest and the second nearest points in rotated regular point patterns. These distributions demonstrate that road directions significantly affect the rectilinear distances. As an application of the nearest and the s...

Given a signal net N = s; 1; ;n to be the set of nodes, with s the source and the remaining nodes sinks, an MRDPT (minimum-cost rectilinear Steiner distance-preserving tree) has the property that the length of every source to sink path is equal to the rectilinear distance between the source and sink. The minimum-cost rectilinear Steiner distance-preserving tree minimizes the total wire length w...

This paper investigates a multi-period rectilinear distance 1-center location problem considering a line-shaped barrier, in which the starting point of the barrier follows the uniform distribution function. In addition, the existing points are sensitive to demands and locations. The purpose of the presented model is to minimize the maximum barrier distance from the new facility to the existing ...