欧氏 k-Steiner 树问题的精确算法

IF 0.4 4区 计算机科学 Q4 MATHEMATICS
Marcus Brazil , Michael Hendriksen , Jae Lee , Michael S. Payne , Charl Ras , Doreen Thomas
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引用次数: 0

摘要

欧几里得 k-Steiner 树问题要求在平面上连接 n 个给定点的最小成本网络中,最多允许 k 个额外的节点(称为 Steiner 点)。在节点数量不受限制的经典斯坦纳树问题中,每个斯坦纳点的度数都必须是 3,而 k-斯坦纳问题的不同之处在于,最优解中可以包含度数为 4 的斯坦纳点。这一简单的变化导致了在尝试创建最优 k-Steiner 树生成算法时的许多复杂性,而事实证明,k-Steiner 树生成算法是解决经典 Steiner 树问题的旗舰算法(即 GeoSteiner)的强大组成部分。在本文中,我们首先扩展了 GeoSteiner 生成算法的基本框架,将 4 度 Steiner 点纳入其中。然后,我们介绍了一系列限制最优 k-Steiner 树的结构和几何特性的新结果,并展示了如何将这些特性用作拓扑剪枝方法,以支持我们的生成算法。最后,我们通过实验数据展示了我们的剪枝方法在减少次优解拓扑数量方面的有效性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
An exact algorithm for the Euclidean k-Steiner tree problem

The Euclidean k-Steiner tree problem asks for a minimum-cost network connecting n given points in the plane, allowing at most k additional nodes referred to as Steiner points. In the classical Steiner tree problem in which there is no restriction on the number of nodes, every Steiner point must be of degree 3. The k-Steiner problem differs in that Steiner points of degree 4 may be included in an optimal solution. This simple change leads to a number of complexities when attempting to create a generation algorithm for optimal k-Steiner trees, which has proven to be a powerful component of the flagship algorithm, namely GeoSteiner, for solving the classical Steiner tree problem. In the present paper we firstly extend the basic framework of GeoSteiner's generation algorithm to include degree-4 Steiner points. We then introduce a number of novel results restricting the structural and geometric properties of optimal k-Steiner trees, and then show how these properties may be used as topological pruning methods underpinning our generation algorithm. Finally, we present experimental data to show the effectiveness of our pruning methods in reducing the number of sub-optimal solution topologies.

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来源期刊
CiteScore
1.60
自引率
16.70%
发文量
43
审稿时长
>12 weeks
期刊介绍: Computational Geometry is a forum for research in theoretical and applied aspects of computational geometry. The journal publishes fundamental research in all areas of the subject, as well as disseminating information on the applications, techniques, and use of computational geometry. Computational Geometry publishes articles on the design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects. Also welcomed are computational geometry solutions to fundamental problems arising in computer graphics, pattern recognition, robotics, image processing, CAD-CAM, VLSI design and geographical information systems. Computational Geometry features a special section containing open problems and concise reports on implementations of computational geometry tools.
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