The Steiner problem on the regular tetrahedron (original) (raw)
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A New Heuristic Constructing Minimal Steiner Trees inside Simple Polygons
The Steiner tree problem has numerous applications in urban transportation network, design of electronic integrated circuits, and computer network routing. This problem aims at finding a minimum Steiner tree in the Euclidean space, the distance between each two edges of which has the least cost. This problem is considered as an NP-hard one. Assuming the simple polygon P with m vertices and n terminals, the purpose of the minimum Steiner tree in the Euclidean space is to connect the n terminals existing in p. In the proposed algorithm, obtaining optimal responses will be sought by turning this problem into the Steiner tree problem on a graph.
An algorithm f o r solving the Steiner problem on a f i n i t e undirected graph i s presented. s e t of graph arcs of minimum t o t a l length needed t o connect a specified s e t of k graph nodes. I f the entire graph contains n nodes, the algorithm requires time proportional t o ThiB algorithm computes the 3 The t h e requirement above includes the term n / 2 , which can be eliminated i f the s e t of shortest paths connecting each pair of nodes in the graph i s available. Also, the RAND Corporation, which through the U.S. A i r Force under t h e Project RAND, supported the authors during t h e i r i n i t i a l research on t h i s problem. i n p a r t i s permitted for any purposes of the United S t a t e s Government.
Steiner tree problem leads to solutions in several scientific and business contexts, including computer networks routing and electronic integrated circuits. Computing fields of this problem has become an important research topic in computational geometry. Considering the number of points in the Euclidean plane, called terminal points, a minimum spanning tree is obtained which connects these points. A series of other points (Steiner points) are added to the tree, which makes it shorter in length. The resulting tree is called Euclidean Steiner minimal tree. It is considered as an NP-hard problem. Considering a simple polygon P with m vertices and n terminals, in which you are trying to find a Euclidean Steiner tree that is connected to all n terminals existing inside p. In this paper we propose a solution for several terminals in a simple polygonal in presence of obstacles.
A near linear time approximation scheme for Steiner tree among obstacles in the plane
Computational Geometry, 2010
We present a polynomial-time approximation scheme (PTAS) for the Steiner tree problem with polygonal obstacles in the plane with running time O(n log 2 n), where n denotes the number of terminals plus obstacle vertices. To this end, we show how a planar spanner of size O(n log n) can be constructed that contains a (1 + ǫ)-approximation of the optimal tree. Then one can find an approximately optimal Steiner tree in the spanner using the algorithm of for the Steiner tree problem in planar graphs. We prove this result for the Euclidean metric and also for all uniform orientation metrics, i.e. particularly the rectilinear and octilinear metrics.
New algorithms for the rectilinear Steiner tree problem
IEEE Transactions on Computer-aided Design of Integrated Circuits and Systems, 1990
We discuss a new approach to constructing the rectilinear Steiner tree (RST) of a given set of points in the plane, starting from a minimum spanning tree (MST). The main idea in our approach is to find layouts for the edges of the MST, so as to maximize the overlaps between the layouts, thus minimizing,the cost (i.e., wire length) of the resulting rectilinear Steiner tree. We describe two algorithms for constructing rectilinear Steiner trees from MST's, that are optimal under the conditions that the layout of each edge of the MST is (1) a L-shape, or (2) any staircase, respectively. The first algorithm has linear time complexity and the second algorithm has a higher polynomial time complexity. Steiner trees produced by the second algorithm have a property called stability, which enables the rerouting of any segment of the tree, while maintaining the cost of the tree, and not causing overlaps with the rest of the tree. Stability is a desirable property in VLSI global routing applications.
Steiner Minimal Trees in Rectilinear and Octilinear Planes
Acta Mathematica Sinica, English Series, 2007
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.
On-line steiner trees in the Euclidean plane
Discrete & Computational Geometry, 1993
Suppose we are given a sequence of n points in the Euclidean plane, and our objective is to construct, on-line, a connected graph that connects all of them, trying to minimize the total sum of lengths of its edges. The points appear one at a time, and at each step the on-line algorithm must construct a connected graph that contains all current points by connecting the new point to the previously constructed graph. This can be done by joining the new point (not necessarily by a straight line) to any point of the previous graph, (not necessarily one of the given points). The performance of our algorithm is measured by its competitive ratio: the supremum, over all sequences of points, of the ratio between the total length of the graph constructed by our algorithm and the total length of the best Steiner tree that connects all the points. There are known on-line algorithms whose competitive ratio is O(log n) even for all metric spaces, but the only lower bound known is of [IW] for some contrived discrete metric space. Moreover, for the plane, on-line algorithms could have been more powerful and achieve a better competitive ratio, and no nontrivial lower bounds for the best possible competitive ratio were known. Here we prove an almost tight lower bound of Ω(log n/ log log n) for the competitive ratio of any on-line algorithm. The lower bound holds for deterministic algorithms as well as for randomized ones, and obviously holds in any Euclidean space of dimension greater than 2 as well.
Some structural and geometric properties of two-connected steiner networks
2007
We consider the problem of constructing a shortest Euclidean 2-connected Steiner network (SMN) for a set of terminals. This problem has natural applications in the design of survivable communication networks. A SMN decomposes into components that are full Steiner trees. Winter and Zachariasen proved that all cycles in SMNs with Steiner points must have two pairs of consecutive terminals of degree 2. We use this result and the notion of reduced block-bridge trees of Luebke to show that no component in a SMN spans more than approximately one-third of the terminals. Furthermore, we show that no component spans more than two terminals on the boundary of the convex hull of the terminals; such two terminals must in addition be consecutive on the boundary of this convex hull. Algorithmic implications of these results are discussed.
2014
The Steiner minimum tree (SMT) problem is one of the classic nonlinear combinatorial optimization problems for centuries. The Steiner tree problem finds the minimum cost tree connecting a given set of nodes called required nodes in a given undirected weighted graph. In this minimum cost Steiner tree due to the reduction of path-cost, some nonrequired nodes are also used while finding the SMT, which are called Steiner nodes. The Steiner path problem comes from the Steiner tree problem of finding the minimum cost path connecting a given set of nodes called required nodes in a given undirected weighted graph. The Steiner path problem can be characterized as either a decision or optimization problem. This paper focuses on solving the Steiner path as a decision problem. The Steiner Path decision problem finds whether there is a path connecting a given set of nodes called required nodes in a given binary tree. The decision that whether there exists a Steiner path or not can help in furthe...