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## 算法一览二十二- steiner树问题

(2009-03-27 13:41:00)

### 算法一览二十二- Steiner树问题

Sunday, July 13th, 2008 星期日， 2008年7月13号

The Steiner tree problem, named after Jakob Steiner, is a problem in combinatorial optimization. Steiner树的问题，雅各布施泰纳的名字命名的，是一个问题的组合优化。

TheSteiner tree problem is superficially similar to the minimum spanningtree problem: given a set V of points (vertices), interconnect them bya network (graph) of shortest length, where the length is the sum ofthe lengths of all edges.的Steiner树问题是表面类似最小生成树问题：鉴于一套五点（顶点） ，它们的互连网络（图）的最短长度，如果长度的总和，长度为所有边缘。 Thedifference between the Steiner tree problem and the minimum spanningtree problem is that, in the Steiner tree problem, extra intermediatevertices and edges may be added to the graph in order to reduce thelength of the spanning tree.之间的差额Steiner树问题和最小生成树问题是，在Steiner树问题，额外的中间顶点和边可能被添加到图，以缩短生成树。 These new vertices introduced to decrease the total length of connection are known as Steiner points or Steiner vertices.这些新的顶点介绍，以减少总长度的连接被称为斯坦纳点或施泰纳顶点。 It has been proved that the resulting connection is a tree, known as the Steiner tree.已经证明，由此产生的连接是一种树，被称为Steiner树。 There may be several Steiner trees for a given set of initial vertices.可能有几个斯坦纳树为一组给定的初始点。

Theoriginal problem was stated in the form that has become known as theEuclidean Steiner tree problem: Given N points in the plane, it isrequired to connect them by lines of minimum total length in such a waythat any two points may be interconnected by line segments eitherdirectly or via other points and line segments.原来的问题中所指出的形式，已成为所谓的欧几里德Steiner树问题：鉴于ñ分，飞机，这是需要连接的线路总长度最小的方式，任何两个地点可能是相互关联的线段直接或通过其他点和线段。

Forthe Euclidean Steiner problem, points added to the graph (Steinerpoints) must have a degree of three, and the three edges incident tosuch a point must form three 120 degree angles.施泰纳的欧几里德问题，点添加到图表（斯坦纳点）必须具备一定程度的三个，以及三条边事件，这样一个点要形成三个120度的角度。 Itfollows that the maximum number of Steiner points that a Steiner treecan have is N-2, where N is the initial number of given points.因此，数量的上限施泰纳指出，一个Steiner树可以为N - 2 ，其中n是初次一些特定点。

It may be further generalized to the metric Steiner tree problem.它可以进一步推广到度量Steiner树问题。 Givena weighted graph G(S,E,w) whose vertices correspond to points in ametric space, with edge weights being the distances in the space, it isrequired to find a tree of minimum total length whose vertices are asuperset of set S of the vertices in G.鉴于加权图G （硫，英，西） ，其顶点对应点的度量空间，与边缘的距离正在重量的空间，这是需要找到一种树的最小总长度的顶点的一个超集S的顶点在湾

Themost general version is Steiner tree in graphs: Given a weighted graphG(V,E,w) and a vertices subset S\subseteq V find a tree of minimalweight which includes all vertices in S.最普遍的版本是Steiner树图：鉴于加权图G （五，英，西）和一个顶点子县\ subseteq V找到了一棵树的最小重量包括所有顶点在南

Themetric Steiner tree problem corresponds to the Steiner tree in graphsproblem where the graph has an infinite number of vertices, which areall points of the metric space.公制Steiner树问题相对应的Steiner树问题，即在图图有无限多的顶点，这是所有点的度量空间。

The Steiner tree problem has applications in circuit layout or network design. Steiner树的问题已应用在电路布局或网络设计。 Most versions of the Steiner tree problem are NP-complete, ie, thought to be computationally hard.大多数版本的Steiner树问题是NP -完全的，即认为是难以计算。 In fact, one of these was among Karp’s original 21 NP-complete problems.事实上，这些是斯卡普原来21 NP完全问题。 Some restricted cases can be solved in polynomial time.一些受限制情况下就可以解决在多项式时间。 In practice, heuristics are used.在实践中，采用启发式。

One common approximation to the Euclidean Steiner tree problem is to compute the Euclidean minimum spanning tree.一个共同的近似值欧几里德Steiner树问题是计算欧几里德最小生成树。

Outside the plane飞机外面传来
The Steiner tree problem has also been investigated in multiple dimensions and on various surfaces. Steiner树的问题也已在多个层面的调查和各种表面。 Algorithmsto find the Steiner minimal tree have been found on the sphere, torus,projected plane, wide and narrow cones, and others.算法找到最小斯坦纳树已经发现的领域，花托，预计飞机，宽和窄锥等。

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