J. H. Rubinstein

A variational approach to the Steiner network problem

Supposen points are given in the plane. Their coordinates form a 2n-vectorX. To study the question of finding the shortest Steiner networkS connecting these points, we allowX to vary over a configuration space. In particular, the Steiner ratio conjecture is well suited to this approach and short proofs of the casesn=4, 5 are discussed. The variational approach was used by us to solve other cases of the ratio conjecture (n=6, see [11] and for arbitraryn points lying on a circle). Recently, Du and Hwang have given a beautiful complete solution of the ratio conjecture, also using a configuration space approach but with convexity as the major idea. We have also solved Grahams problem to decide when the Steiner network is the same as the minimal spanning tree, for points on a circle and on any convex polygon, again using the variational method.

Optimising declines in underground mines

Abstract This paper describes a method for optimising the layout of a decline in an underground mine. It models a decline as a mathematical network connecting the access points at each level of the proposed mine to the surface portal. A feasible decline is one satisfying all operational constraints such as gradient and turning radius requirements. The task is to find the decline that minimises a given cost objective. Typically, the cost objective will be some combination of development and operational costs representing a project cost or a life-of-mine cost. The procedure to find the optimal decline has been automated and the paper describes the current capability of Decline Optimisation Tool (DOT) software. A case study on the optimisation of a decline to service the Jandam gold mine in the Pajingo field of Newmont Australia Limited demonstrates the practical application of the technique.

Gradient-constrained minimum networks. I. Fundamentals

In three-dimensional space an embedded network is called gradient-constrained if the absolute gradient of any differentiable point on the edges in the network is no more than a given value m. A gradient-constrained minimum Steiner tree T is a minimum gradient-constrained network interconnecting a given set of points. In this paper we investigate some of the fundamental properties of these minimum networks. We first introduce a new metric, the gradient metric, which incorporates a new definition of distance for edges with gradient greater than m. We then discuss the variational argument in the gradient metric, and use it to prove that the degree of Steiner points in T is either three or four. If the edges in T are labelled to indicate whether the gradients between their endpoints are greater than, less than, or equal to m, then we show that, up to symmetry, there are only five possible labellings for degree 3 Steiner points in T. Moreover, we prove that all four edges incident with a degree 4 Steiner point in T must have gradient m if m is less than 0.38. Finally, we use the variational argument to locate the Steiner points in T in terms of the positions of the neighbouring vertices.

Steiner Trees for Terminals Constrained to Curves

We give a polynomial time algorithm for solving the Euclidean Steiner tree problem when the terminals are constrained to lie on a fixed finite set of disjoint finite-length compact simple smooth curves. The problem is known to be NP-hard in general. We also show it to be NP-hard if the terminals lie on two parallel infinite lines or on a bent line segment provided the bend has an angle of less than

Minimum Networks for Four Points in Space

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Minimal Steiner Trees for Rectangular Arrays of Lattice Points

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Full Minimal Steiner Trees on Lattice Sets

The minimum network problem (Steiner tree problem) in space is much harder than the one in the Euclidean plane. The Steiner tree problem for four points in the plane has been well studied. In contrast, very few results are known on this simple Steiner problem in 3D-space. In the first part of this paper we analyze the difficulties of the Steiner problem in space. From this analysis we introduce a new concept — Simpson intersections, and derive a system of iteration formulae for computing Simpson intersections. Using Simpson intersections the Steiner points can be determined by solving quadratic equations. As well this new computational method makes it easy to check the impossibility of computing Steiner trees on 4-point sets by radicals. At the end of the first part we consider some special cases (planar and symmetric 3D-cases) that can be solved by radicals. The Steiner ratio problem is to find the minimum ratio of the length of a Steiner minimal tree to the length of a minimal spanning tree. This ratio problem in the Euclidean plane was solved by D. Z. Du and F. K. Hwang in 1990, but the problem in 3D-space is still open. In 1995 W.D. Smith and J.M. Smith conjectured that the Steiner ratio for 4-point sets in 3D-space is achieved by regular tetrahedra. In the second part of this paper, using the variational method, we give a proof of this conjecture.

Minimal Steiner Trees for 2k×2kSquare Lattices

We construct minimal Steiner trees for any square or rectangular array of integer lattice points on the Euclidean plane.

Decline design in underground mines using constrained path optimisation

Given a finite set of points P in the Euclidean plane, the Steiner problem asks us to constuct a shortest possible network interconnecting P. Such a network is known as a minimal Steiner tree. The Steiner problem is an intrinsically difficult one, having been shown to be NP-hard [7]; however, it often proves far more tractable if we restrict our attention to points in special geometric configurations. One such restriction which has generated considerable interest is that of finding minimal Steiner trees for nice sets of integer lattice points. The first significant result in this direction was that of Chung and Graham [4], which, in effect, precisely characterized the minimal Steiner trees for any horizontal 2_n array of integer lattice points. In 1989, Chung et al. [3] examined a related problem, which they described as the Checkerboard Problem. They asked how to find a minimal Steiner tree for an n_n square lattice, that is, a collection of n_n points arranged in a regular lattice of unit squares like the corners of the cells of article no. TA962752

Degree‐five Steiner points cannot reduce network costs for planar sets

We prove a conjecture of Chung, Graham, and Gardner (Math. Mag.62(1989), 83?96), giving the form of the minimal Steiner trees for the set of points comprising the vertices of a 2k×2ksquare lattice. Each full component of these minimal trees is the minimal Steiner tree for the four vertices of a square.