Steiner Minimal Trees in Rectilinear and Octilinear Planes

This paper considers the Steiner Minimal Tree （SMT） problem in the rectilinear and octilinear planes. The study is motivated by the physical design of VLSI： The rectilinear case corresponds to the currently used M-architecture, which uses either horizontal or vertical routing, while the octilinear case corresponds to a new routing technique, X-architecture, that is based on the pervasive use of diagonal directions. The experimental studies show that the X-architecture demonstrates a length reduction of more than 10-20%. In this paper, we make a theoretical study on the lengths of SMTs in these two planes. Our mathematical analysis confirms that the length reduction is significant as the previous experimental studies claimed, but the reduction for three points is not as significant as for two points. We also obtain the lower and upper bounds on the expected lengths of SMTs in these two planes for arbitrary number of points.     

Proving Strong Duality for Geometric Optimization Using a Conic Formulation

Geometric optimization¹ is an important class of problems that has many applications, especially in engineering design. In this article, we provide new simplified proofs for the well-known associated duality theory, using conic optimization. After introducing suitable convex cones and studying their properties, we model geometric optimization problems with a conic formulation, which allows us to apply the powerful duality theory of conic optimization and derive the duality results valid for geometric optimization.     

A Sausage Heuristic for Steiner Minimal Trees in Three-Dimensional Euclidean Space

Given a set V of size N≥4 vertices in a metric space, how can one interconnect them with the possible use of a set S of size M vertices not in the set V, but in the same metric space, so that the cumulative cost of the inter-connections between all the vertices is a minimum? When one uses the Euclidean metric to compute these inter-connections, this is referred to as the Euclidean Steiner Minimal Tree Problem. This is an NP-hard problem. The Steiner Ratio ρ of a vertex set is the length of this Steiner Minimal Tree (SMT), divided by the length of the Minimum Spanning Tree (MST), and is a popular and tractable measure of solution quality.The ?-Sausage heuristic described in this paper employs a decomposition technique to explore the point set. The fixed vertices of the set are connected to a set of centroid vertices of Delaunay tetrahedrons. The path topology is preserved as far as possible, together with a cycle prevention rule, where junctions, and deviations from the ?-Sausage structure occur. Furthermore, repeated sweeps, with different root vertices are accommodated.The computational complexity of the heuristic is shown to be O(N²). Experimental results with thousands of vertices are presented. Comparisons with an exponential running time Branch and Bound algorithm are also shown.     

Approximations and Lower Bounds for the Length of Minimal Euclidean Steiner Trees

We give a new lower bound on the length of the minimal Steiner tree with a given topology joining given terminals in Euclidean space, in terms of toroidal images. The lower bound is equal to the length when the topology is full. We use the lower bound to prove bounds on the “error” e in the length of an approximate Steiner tree, in terms of the maximum deviation d of an interior angle of the tree from 120°. Such bounds are useful for validating algorithms computing minimal Steiner trees. In addition we give a number of examples illustrating features of the relationship between e and d, and make a conjecture which, if true, would somewhat strengthen our bounds on the error. J. H. Rubinstein, J. Weng: Research supported by the Australian Research Council N. Wormald: Research supported by the Australian Research Council and the Canada Research Chairs Program. Research partly carried out while the author was in the Department of Mathematics and Statistics, University of Melbourne     

Approximations for Steiner Trees with Minimum Number of Steiner Points

Given n terminals in the Euclidean plane and a positive constant, find a Steiner tree interconnecting all terminals with the minimum number of Steiner points such that the Euclidean length of each edge is no more than the given positive constant. This problem is NP-hard with applications in VLSI design, WDM optical networks and wireless communications. In this paper, we show that (a) the Steiner ratio is 1/ 4, that is, the minimum spanning tree yields a polynomial-time approximation with performance ratio exactly 4, (b) there exists a polynomial-time approximation with performance ratio 3, and (c) there exists a polynomial-time approxi-mation scheme under certain conditions.     

Conic Formulation for l p -Norm Optimization

In this paper, we formulate the l _p -norm optimization problem as a conic optimization problem, derive its duality properties (weak duality, zero duality gap, and primal attainment) using standard conic duality and show how it can be solved in polynomial time applying the framework of interior-point algorithms based on self-concordant barriers.     

Approximate Euclidean Steiner Trees

An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error from \(120^{\circ }\). This notion arises in numerical approximations of minimum Steiner trees. We investigate the worst-case relative error of the length of an approximate Steiner tree compared to the shortest tree with the same topology. It has been conjectured that this relative error is at most linear in the maximum error at the angles, independent of the number of terminals. We verify this conjecture for the two-dimensional case as long as the maximum angle error is sufficiently small in terms of the number of terminals. In the two-dimensional case we derive a lower bound for the relative error in length. This bound is linear in terms of the maximum angle error when the angle error is sufficiently small in terms of the number of terminals. We find improved estimates of the relative error in length for larger values of the maximum angle error and calculate exact values in the plane for three and four terminals.     

Steiner Trees on Curved Surfaces

 The Steiner tree problem on surfaces is more complicated than the corresponding one in the Euclidean plane. There are not many results on it to date. In this paper we first make a comparison of Steiner minimal trees on general curved surfaces with Steiner minimal trees in the Euclidean plane. Then, we focus our study on the Steiner trees on spheres. In particular, we detail the properties of locally minimal Steiner points, and the Steiner points for spherical triangles. Received: August 18, 1997 Final version received: March 16, 1998     

最小树的一般算法

本文根据设计并行算法的基本原则,给出了最小树的两个对偶定理.在此基础上,建立了两种对偶的同步并行算法的雏型.这两种算法恰恰在对偶的意义下,概括了以往的最小树算法.     

A New Formulation for Spanning Trees

Spanning trees are fundamental structures in graph theory. Furthermore, computing them is a central part in many relevant algorithms, used in either practical or theoretical applications. The classical Minimum Spanning Tree problem is solvable in polynomial time but almost all of its variants are NP-Hard. In this paper, a novel polynomial size mixed integer linear programming formulation is introduced for spanning trees. This formulation is based on a new characterization we propose for acyclic graphs. Preliminary computational results show that this formulation is capable of solving small instances of the diameter constrained minimum spanning tree problem. It should be possible to strengthen the formulation to tackle larger instances of that problem. Additionally, our spanning tree formulation may prove to be a more effective model for some related applications.     

Finding a Minimal Efficient Solution of a Convex Multiobjective Program

In this paper, we construct an interior-point homotopy method for solving a minimal efficient solution of a convex multiobjective program. Some examples are shown.     

Locally Minimal Trees in n-Normed Spaces

The locally minimal trees in normed spaces $({\mathbb{R}}^2, \rho)$ , where the unit circle ${x \in {\mathbb{R}}^2 | \rho (x)=1}$ in the norm ρ coincides with the regular m-gon (m = 2n) inscribed in the Euclidean unit circle S ¹, are completely classified.     

A Network-Flow Technique for Finding Low-Weight Bounded-Degree Spanning Trees

Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, the problem of computing a low-weight spanning tree such that the degree of each vertex is at most its specified bound is considered. In particular, modifying a given spanning treeTusingadoptionsto meet the degree constraints is considered. A novel network-flow-based algorithm for finding a good sequence of adoptions is introduced. The method yields a better performance guarantee than any previous algorithm. If the degree constraintd(v) for eachvis at least 2, the algorithm is guaranteed to find a tree whose weight is at most the weight of the given tree times 2 − min{(d(v) − 2)/(deg_T(v) − 2) : deg_T(v) > 2}, where deg_T(v) is the initial degree ofv. Equally importantly, it takes this approach to the limit in the following sense: if any performance guarantee that is solely a function of the topology and edge weights of a given tree holds foranyalgorithm at all, then it also holds for the given algorithm. Examples are provided in which no lighter tree meeting the degree constraint exists. Linear-time algorithms are provided with the same worst-case performance guarantee. ChoosingTto be a minimum spanning tree yields approximation algorithms with factors less than 2 for the general problem on geometric graphs with distances induced by variousL_pnorms. Finally, examples of Euclidean graphs are provided in which the ratio of the lengths of an optimal Traveling Salesman path and a minimum spanning tree can be arbitrarily close to 2.     

求鳞状循环因子矩阵的极小多项式的算法

提出了任意域上鳞状循环因子矩阵 ,利用多项式环的理想的Go bner基的算法给出了任意域上鳞状循环因子矩阵的极小多项式和公共极小多项式的一种算法 .同时给出了这类矩阵逆矩阵的一种求法 .在有理数域或模素数剩余类域上 ,这一算法可由代数系统软件Co CoA4 .0实现 .数值例子说明了算法的有效性     

Upper and Lower Bounds for the Lengths of Steiner Trees in 3-Space

We present a method of determining upper and lower bounds for the length of a Steiner minimal tree in 3-space whose topology is a given full Steiner topology, or a degenerate form of that full Steiner topology. The bounds are tight, in the sense that they are exactly satisfied for some configurations. This represents the first nontrivial lower bound to appear in the literature. The bounds are developed by first studying properties of Simpson lines in both two and three dimensional space, and then introducing a class of easily constructed trees, called midpoint trees, which provide the upper and lower bounds. These bounds can be constructed in quadratic time. Finally, we discuss strategies for improving the lower bound.Supported by a grant from the Australia Research Council.     

Vertex-Disjoint Packing of Two Steiner Trees: polyhedra and branch-and-cut

Consider the problem of routing the electrical connections among two large terminal sets in circuit layout. A realistic model for this problem is given by the vertex-disjoint packing of two Steiner trees (2VPST), which is known to be NP-complete. This work presents an investigation on the 2VPST polyhedra. The main idea is to start from facet-defining inequalities for a vertex-weighted Steiner tree polyhedra. Some of these inequalities are proven to also define facets for the packing polyhedra, while others are lifted to derive new important families of inequalities, including proven facets. Separation algorithms are provided. Branch-and-cut implementation issues are also discussed, including some new practical techniques to improve the performance of the algorithm. The resulting code is capable of solving problems on grid graphs with up to 10000 vertices and 5000 terminals in a few minutes. Received: August 1999 / Accepted: January 2001?Published online April 12, 2001     

Minimal homogeneous Steiner 2-(v,3) trades

A Steiner 2-$(v,3)$ trade is a pair $(T_1,T_2)$ of disjoint partial Steiner triple systems, each on the same set of $v$ points, such that each pair of points occurs in $T_1$ if and only if it occurs in $T_2$. A Steiner 2-$(v,3)$ trade is called d-homogeneous if each point occurs in exactly d blocks of $T_1$ (or $T_2$). In this paper we construct minimal d-homogeneous Steiner 2-$(v,3)$ trades of foundation $v$ and volume $\mathit{dv}/3$ for sufficiently large values of $v$. (Specifically, $v>3(1.75d^2+3)$ if $v$ is divisible by 3 and $v>d(4^{d/3+1}+1)$ otherwise.)     

基于加权绝对值距离Steiner最优树的选址问题

提出基于加权绝对值距离Steiner最优树思想的选址模型,给出了该模型的蚂蚁算法实现策略.在此基础上,分析了电子商务环境下企业配送中心选址问题,并用算例验证了该选址方案的可行性.