Improved computation of plane Steiner Minimal Trees

ASteiner Minimal Tree (SMT) for a given setA = {a ₁,...,a _n } in the plane is a tree which interconnects these points and whose total length, i.e., the sum of lengths of the branches, is minimum. To achieve the minimum, the tree may contain other points (Steiner points) besidesa ₁,...,a _n . Various improvements are presented to an earlier computer program of the authors for plane SMTs. These changes have radically reduced machine times. The existing program was limited in application to aboutn = 30, while the innovations have facilitated solution of many randomly generated 100-point problems in reasonable processing times.This work was supported by the Canadian Natural Sciences and Engineering Council under Grant Numbers A-7544 and A-7558.     

Expansion of Linear Steiner Trees

A Steiner tree T on a given set of points A is called linear if all Steiner points, including those collapsing into their adjacent given points, lie on one path referred to as its trunk. Suppose A is a simple polygonal line. Roughly speaking, T is similar to A if its trunk turns right or left when A does. In this paper we prove that A can be expanded to another polygonal line, and T can be constructed in linear time using this expansion method. Received January 15, 1995; revised November 19, 1995, and February 3, 1996.     

Concatenation-Based Greedy Heuristics for the Euclidean Steiner Tree Problem

We present a class of O(n log n) heuristics for the Steiner tree problem in the Euclidean plane. These heuristics identify a small number of subsets with few, geometrically close, terminals using minimum spanning trees and other well-known structures from computational geometry: Delaunay triangulations, Gabriel graphs, relative neighborhood graphs, and higher-order Voronoi diagrams. Full Steiner trees of all these subsets are sorted according to some appropriately chosen measure of quality. A tree spanning all terminals is constructed using greedy concatenation. New heuristics are compared with each other and with heuristics from the literature by performing extensive computational experiments on both randomly generated and library problem instances. Received October 27, 1997; revised May 7, 1998.     

考虑障碍的多端点最小直角Steiner树构造算法

首先将所有障碍视为不存在,构造初始Steiner树、连接线网所有端点,可采用已有的无障碍Steiner树算法来实现．然后考虑障碍的影响,改造所构造的初始Steiner树：找到初始Steiner树与障碍的相交点,重布某些树边,使它们绕过障碍,并尽量保持树长较短;进一步地,加入预处理和后期处理措施,以更好地处理特殊线网并使算法的结果更优．该算法能够处理多个障碍的情况,并能适应多种形状的障碍;同时,算法有较高的效率,其复杂度为O(m,z),其中,m和n分别是障碍个数和线网端点数．该算法已经在SUN工作站、Unix上利用C语言编程实现,并进行了MCNC电路测试．测试结果表明：文中算法得到的树长结果仅与最优值平均相差5．31％,且算法的执行时间保持在1s以内．     

Grade of Service Steiner Minimum Trees in the Euclidean Plane

Abstract. Let P = {p ₁ , p ₂ , \ldots, p _n } be a set of n {\lilsf terminal points} in the Euclidean plane, where point p _i has a {\lilsf service request of grade} g(p _i ) ∈ {1, 2, \ldots, n} . Let 0 < c(1) < c(2) < ⋅s < c(n) be n real numbers. The {\lilsf Grade of Service Steiner Minimum Tree (GOSST)} problem asks for a minimum cost network interconnecting point set P and some {\lilsf Steiner points} with a service request of grade 0 such that (1) between each pair of terminal points p _i and p _j there is a path whose minimum grade of service is at least as large as \min(g(p _i ), g(p _j )) ; and (2) the cost of the network is minimum among all interconnecting networks satisfying (1), where the cost of an edge with service of grade g is the product of the Euclidean length of the edge with c(g) . The GOSST problem is a generalization of the Euclidean Steiner minimum tree problem where all terminal points have the same grade of service request. When there are only two (three, respectively) different grades of service request by the terminal points, we present a polynomial time approximation algorithm with performance ratio \frac 4 3 ρ (((5+4\sqrt 2 )/7)ρ , respectively), where ρ is the performance ratio achieved by an approximation algorithm for the Euclidean Steiner minimum tree problem. For the general case, we prove that there exists a GOSST that is the minimum cost network under a full Steiner topology or its degeneracies. A powerful interior-point algorithm is used to find a (1+ε) -approximation to the minimum cost network under a given topology or its degeneracies in O(n ^1.5 (log n + log (1/ε))) time. We also prove a lower bound theorem which enables effective pruning in a branch-and-bound method that partially enumerates the full Steiner topologies in search for a GOSST. We then present a k -optimal heuristic algorithm to compute good solutions when the problem size is too large for the branch-and-bound algorithm. Preliminary computational results are presented.     

A proof of the Gilbert-Pollak conjecture on the Steiner ratio

LetP be a set ofn points on the euclidean plane. LetL _s(P) andL _m (P) denote the lengths of the Steiner minimum tree and the minimum spanning tree onP, respectively. In 1968, Gilbert and Pollak conjectured that for anyP,L _s (P)(3/2)L _m (P). We provide a proof for their conjecture in this paper.supported by NSF under grant STC88-09648.supported in part by the National Natural Science Foundation of China.     

On the location of Steiner points in uniformly-oriented Steiner trees

We give a fundamental result on the location of Steiner points for Steiner minimum trees in uniform orientation metrics. As a corollary we obtain a linear time algorithm for constructing a Steiner minimum tree for a given full topology when the number of uniform orientations is λ=3m,m?1.     

The General Steiner Tree-Star problem

The Steiner tree problem is defined as follows—given a graph G=(V,E) and a subset X⊂V of terminals, compute a minimum cost tree that includes all nodes in X. Furthermore, it is reasonable to assume that the edge costs form a metric. This problem is NP-hard and has been the study of many heuristics and algorithms. We study a generalization of this problem, where there is a “switch” cost in addition to the cost of the edges. Switches are placed at internal nodes of the tree (essentially, we may assume that all non-leaf nodes of the Steiner tree have a switch). The cost for placing a switch may vary from node to node. A restricted version of this problem, where the terminal set X cannot be connected to each other directly but only via the Steiner nodes V?X, is referred to as the Steiner Tree-Star problem. The General Steiner Tree-Star problem does not require the terminal set and Steiner node set to be disjoint. This generalized problem can be reduced to the node weighted Steiner tree problem, for which algorithms with performance guarantees of Θ(lnn) are known. However, such approach does not make use of the fact that the edge costs form a metric. In this paper we derive approximation algorithms with small constant factors for this problem. We show two different polynomial time algorithms with approximation factors of 5.16 and 5.     

Prize collecting Steiner trees with node degree dependent costs

We discuss a variant of the prize-collecting Steiner tree problem with node degree dependent costs using a telecommunications setting to motivate these costs. We present and test models which are tailored for this variant of the problem. Results taken from instances with up to 100 nodes are used to evaluate the quality of the proposed models for solving the problem, as well as, in terms of the correspondent linear programming relaxation.     

Improved computation of plane Steiner Minimal Trees

ASteiner Minimal Tree (SMT) for a given setA = {a ₁,...,a _n } in the plane is a tree which interconnects these points and whose total length, i.e., the sum of lengths of the branches, is minimum. To achieve the minimum, the tree may contain other points (Steiner points) besidesa ₁,...,a _n . Various improvements are presented to an earlier computer program of the authors for plane SMTs. These changes have radically reduced machine times. The existing program was limited in application to aboutn = 30, while the innovations have facilitated solution of many randomly generated 100-point problems in reasonable processing times.     

G-Tree:基于引力指向技术减少拐弯的Steiner树算法

提出一种基于引力指向技术、以减少拐弯数为目标的最小直角Steiner树构造算法G-Tree.利用一个节点受到其他节点的引力来决定它的移动方向,并采用引力加权以考虑减少拐弯数,生成Steiner树后对拐弯数进行了进一步优化.减少拐弯数有助于在布线阶段减少可能的通孔,从而增强电路的可靠性和可制造性.实验结果表明,G-Tree算法在减少布线树的拐弯数方面有明显的效果.     

X结构下VLSI多层绕障Steiner最小树算法

Steiner最小树作为VLSI布线的基础模型,应进一步考虑到X结构、障碍物、多层等条件,文中基于粒子群优化提出了多层绕障X结构Steiner最小树算法.首先引入边变换操作以改变布线树的拓扑,使其具有较强的绕障能力;为了避免边变换操作带来的布线树环路问题,结合并查集策略设计新的操作算子;为了保证布线边不违反约束,提出一个与绕障情况及通孔数相关的惩罚函数策略,从而优化了多层布线中布线总代价这一最重要的目标.实验结果表明,相对于同类算法,该算法在布线总代价的优化能力上是最强的.     

Approximations for a Bottleneck Steiner Tree Problem

Abstract. In the design of wireless communication networks, due to a budget limit, suppose we could put totally n+k stations in the plane. However, n of them must be located at given points. Of course, one would like to have the distance between stations as small as possible. The problem is how to choose locations for other k stations to minimize the longest distance between stations. This problem is NP-hard. We show that if NP neq P , no polynomial-time approximation for the problem in the rectilinear plane has a performance ratio less than 2 and no polynomial-time approximation for the problem in the Euclidean plane has a performance ratio less than sqrt 2 and that there exists a polynomial-time approximation with performance ratio 2 for the problem in both the rectilinear plane and the Euclidean plane.     

Canonical Forms and Algorithms for Steiner Trees in Uniform Orientation Metrics

We present some fundamental structural properties for minimum length networks (known as Steiner minimum trees) interconnecting a given set of points in an environment in which edge segments are restricted to λ uniformly oriented directions. We show that the edge segments of any full component of such a tree contain a total of at most four directions if λ is not a multiple of 3, or six directions if λ is a multiple of 3. This result allows us to develop useful canonical forms for these full components. The structural properties of these Steiner minimum trees are then used to resolve an important open problem in the area: does there exist a polynomial time algorithm for constructing a Steiner minimum tree if the topology of the tree is known? We obtain a simple linear time algorithm for constructing a Steiner minimum tree for any given set of points and a given Steiner topology.     

The multi-weighted Steiner tree problem: A reformulation by intersection

We propose a new formulation for the multi-weighted Steiner tree (MWST) problem. This formulation is based on the fact that a previously proposed formulation for the problem is non-symmetric in the sense that the corresponding linear programming relaxation bounds depend on the node selected as a root of the tree. The new formulation (the reformulation by intersection) is obtained by intersecting the feasible sets of the models corresponding to each possible root selection for the underlying directed problem. Theoretical results will show that the linear programming relaxation of the new formulation dominates the linear programming relaxation of each of the rooted formulations and is comparable with the linear programming bounds of the best formulation known for the problem. A Lagrangean relaxation scheme derived from the new formulation is also proposed and tested, with quite favourable results, on instances with up to 500 nodes and 5000 edges.     

On Exact Solutions for the Rectilinear Steiner Tree Problem Part I: Theoretical Results

The rectilinear Steiner tree problem asks for a shortest tree connecting given points in the plane with rectilinear distance. The best theoretically analyzed algorithms for this problem are based on dynamic programming and have a running time of O(n ² . . . 2.62 ⁿ ) (Ganley and Cohoon), resp. (Smith). The first algorithm can solve problems of size 27, the second one is highly impractical because of the large constant in the exponent. The best implementations perform poorly even on small problem instances; the best practical results can be reached using a Branch \& Bound approach (Salowe and Warme); this implementation can solve random problems of size 35 within a day, while the dynamic programming approach of Ganley and Cohoon can handle only 27 point examples. In this paper we improve the theoretical worst-case time bound to O(n ² . . . 2.38 ⁿ ) , for random problem instances we prove a running time of α ⁿ with a constant α < 2 . We have implemented our algorithms and can now solve problems of 40 points in a day using a provably good dynamic programming approach, and can solve problems of 55 points with a Branch \& Bound strategy. For exponential-time algorithms, this is an enormous improvement. Received April 2, 1997; revised January 5, 1998.     

Two new criteria for finding Steiner hulls in Steiner tree problems

The Steiner tree problem considered in this paper is that of finding a network of minimum length connecting a given setK of terminals in a regionR of the Euclidean plane. ASteiner hull forK inR is any subregion ofR known to contain a Steiner tree forK inR. Two new criteria are given for finding Steiner hulls—one for the Steiner tree problem on plane graphs and one for the rectilinear Steiner tree problem—which strengthen known polynomial-time methods of finding Steiner hulls.This research was supported by the Air Force Office of Scientific Research under Grant AFOSR-84-0140. Reproduction in whole or part is permitted for any purpose of the United States Government.     

How to find Steiner minimal trees in euclideand-space

This paper has two purposes. The first is to present a new way to find a Steiner minimum tree (SMT) connectingN sites ind-space,d >- 2. We present (in Appendix 1) a computer code for this purpose. This is the only procedure known to the author for finding Steiner minimal trees ind-space ford > 2, and also the first one which fits naturally into the framework of backtracking and branch-and-bound. Finding SMTs of up toN = 12 general sites ind-space (for anyd) now appears feasible.We tabulate Steiner minimal trees for many point sets, including the vertices of most of the regular and Archimedeand-polytopes with <- 16 vertices. As a consequence of these tables, the Gilbert-Pollak conjecture is shown to be false in dimensions 3–9. (The conjecture remains open in other dimensions; it is probably false in all dimensionsd withd 3, but it is probably true whend = 2.)The second purpose is to present some new theoretical results regarding the asymptotic computational complexity of finding SMTs to precision .We show that in two-dimensions, Steiner minimum trees may be found exactly in exponential time O(C ^N ) on a real RAM. (All previous provable time bounds were superexponential.) If the tree is only wanted to precision , then there is an (N/)^O(N)-time algorithm, which is subexponential if 1/ grows only polynomially withN. Also, therectilinear Steiner minimal tree ofN points in the plane may be found inN ^O(N) time.J. S. Provan devised an O(N ⁶/⁴)-time algorithm for finding the SMT of a convexN-point set in the plane. (Also the rectilinear SMT of such a set may be found in O(N ⁶) time.) One therefore suspects that this problem may be solved exactly in polynomial time. We show that this suspicion is in fact true—if a certain conjecture about the size of Steiner sensitivity diagrams is correct.All of these algorithms are for a real RAM model of computation allowing infinite precision arithmetic. They make no probabilistic or other assumptions about the input; the time bounds are valid in the worst case; and all our algorithms may be implemented with a polynomial amount of space. Only algorithms yielding theexact optimum SMT, or trees with lengths (1 + ) × optimum, where is arbitrarily small, are considered here.     

On the terminal Steiner tree problem

We investigate a practical variant of the well-known graph Steiner tree problem. In this variant, every target vertex is required to be a leaf vertex in the solution Steiner tree. We present hardness results for this variant as well as a polynomial time approximation algorithm with performance ratio ρ+2, where ρ is the best-known approximation ratio for the graph Steiner tree problem.     

Polygonal space carving with geometric anti-aliasing

This paper proposes an algorithm that solves the shape recovery problem from N arbitrary images. By introducing a polygonal carving technique, the proposed algorithm can reconstruct the image-consistent polygonal shape that is patched by input images. This algorithm eliminates the invalid vertices and polygons from the initial polygonal grid space according to the color variance that represents their image consistency. The carved shape is refined by moving the outlier vertices on the boundary of each image. The final reconstructed shape faithfully accounts for the input images, and its textured appearance reflects the similar color property of the target object.