Let V be a set of n points in R. We study the question whether there exists an orientation such that V is the vertex set of a connected rectilinear graph in that orientation. A graph is rectilinear if its edges are straight line segments in d pairwise perpendicular directions. We prove that at most one such orientation can be possible, up to trivial rotations of 90◦ around some axis. In additio...

The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing (with edges as straight line segments) of G. Zarankiewicz proved in 1952 that cr(Kn1,n2) ≤ Z(n1, n2) := ⌊ n1 2 ⌋ ⌊ n1−1 2 ⌋ ⌊ n2 2 ⌋ ⌊ n2−1 2 ⌋ . We define an ...

Magnetic levitation is becoming widely applicable in magnetic bearings, high-speed ground transportation, vibration isolation, etc., [1]. For example, magnetic bearings support radial and thrust loads in rotating machinery. In addition, magnetic suspension generates levitation action in rectilinear motion devices such as high-speed ground transportation systems. Magnetic levitation is immensely...

In this paper, we consider the weighted rectilinear min-sum facility problem to minimize the sum of the weighted rectilinear distance between the given points and a new added point. The core problem is the problem in its one dimensional cases noted as the weighted median problem. We present efficient algorithms for optimally solving the weighted median problem. The computational experiments dem...

We present a very simple family of traveling salesman instances with n cities where the nearest neighbor rule may produce a tour that is Θ(log n) times longer than an optimum solution. Our family works for the graphic, the euclidean, and the rectilinear traveling salesman problem at the same time. It improves the so far best known lower bound in the euclidean case and proves for the first time ...

We introduce thèsquare tiling group' and use it to give necessary conditions on a planar polygon to be tileable with squares. We deene square tilings on Riemann surfaces, and compute the Euclidean structures on a torus which are square-tileable. We give necessary conditions on surfaces of higher genus to be tileable. A higher-dimensional version of the square-tiling group yields necessary condi...

The rectilinear Steiner tree problem requires a shortest tree spanning a given vertex subset in the plane with rectilinear distance. It was proved that the output length of Zelikovsky's 25] and Berman/Ramaiyer 3] heuristics is at most 1.375 and 97 72 1:347 of the optimal length, respectively. It was claimed that these bounds are not tight. Here we improve these bounds to 1.3125 and 61 48 1:271,...

Physical parameters of the constituent modules and gait parameters affect the overall performance of snake-inspired robots. Hence a system-level optimization model needs to concurrently optimize the module parameters and the gait. Incorporating a physics-based model of rectilinear gaits in the system-level optimization model is a computationally challenging problem. This paper presents a case s...

Computing rectilinear shortest paths in two dimensions has been solved optimally using a number of different techniques. A variety of related problems have been solved, including minimizing the number of bends in the path, the total rectilinear distance, or some combination of both. However, solutions to the 3D versions of these problems are less common. We propose a solution to the 3D minimum-...