修剪最小生成树,通过最小化总边长,剪切连接给定节点数的最长边

Vadim Romanuke
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摘要

背景。在许多设计高效电信网络的任务中,网络节点的数量是有限的,但节点的初始选择却更广泛。为了进一步满足消费者的需求,可能存在的地点要多于事实上需要安放到这些地点的活动工具。这就产生了一个可用节点约束问题。 目标。给定初始平面节点集,问题是建立一棵连接给定节点数的最小生成树,该节点数小于初始节点集的卡入度。因此,可用节点约束问题的目的是通过最小化树的长度,构建一棵最优的最小生成树,以连接小于初始节点数的给定平面节点。 方法对初始节点集进行三角化处理。这样就得到了一组边,计算出这些边的长度并将其作为图权重。在此图形上建立一棵最小生成树。通过修剪连接初始节点数的最小生成树来达到所需的节点数,其中权重最大的自由边会被反复从生成树中删除。另一种方法是切割法,即从初始最小生成树中删除最长的边,而不管这些边是否是自由边。 结果与剪枝法不同,剪切最长边的方法可能会导致最小生成树中连接的节点数少于所需的节点数。但是,剪切法通常会得到一棵更短的树,尤其是当边的长度变化很大时。此外,切割法的速度较慢,因为它会反复重建最小生成树。 结论该问题最初用剪枝法解决。然后使用切割法,并将其解法与剪枝法的解法进行比较。最佳树更短。可以在节点和树长之间进行权衡。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
PRUNING MINIMUM SPANNING TREES AND CUTTING LONGEST EDGES TO CONNECT A GIVEN NUMBER OF NODES BY MINIMIZING TOTAL EDGE LENGTH
Background. Whereas in many tasks of designing efficient telecommunication networks, the number of network nodes is limited, the initial choice of nodes is wider. There are more possible locations than factually active tools to be settled to those locations to further satisfy consumers. This induces an available node constraint problem. Objective. Given an initial set of planar nodes, the problem is to build a minimum spanning tree connecting a given number of the nodes, which is less than the cardinality of the initial set. Therefore, the available node constraint problem aims at building an optimally minimum spanning tree to connect a given number of planar nodes being less than an initial number of nodes by minimizing the tree length. Methods. The initial set of nodes is triangulated. This gives a set of edges, whose lengths are calculated and used as graph weights. A minimum spanning tree is built over this graph. The desired number of nodes is reached by pruning the minimum spanning tree connecting the initial number of nodes, where free edges whose weights are the largest are iteratively removed from the tree. The other approach, the cutting method, removes longest edges off the initial minimum spanning tree, regardless of whether they are free or not. Results. Unlike the pruning method, the method of cutting longest edges may result in a minimum spanning tree connecting fewer nodes than the desired number. However, the cutting method often outputs a shorter tree, especially when the edge length varies much. Besides, the cutting method is slower due to it iteratively rebuilds a minimum spanning tree. Conclusions. The problem is initially solved by the pruning method. Then the cutting method is used and its solution is compared to the solution by the pruning method. The best tree is shorter. A tradeoff for the nodes and tree length is possible.
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