$L_p$平面上瓶颈2-连通$k$-Steiner网络问题的精确算法
An exact algorithm for the bottleneck 2-connected $k$-Steiner network problem in $L_p$ planes
摘要：
我们提出了构造最优几何瓶颈2-连通Steiner网络的第一个精确多项式时间算法，该网络包含最多$k$ Steiner点，其中$k>2$是常数.给定一组嵌入在$L_p$平面中的$n$顶点，问题的目标是找到一个2-连通网络，该网络跨越给定的顶点和至多$k$附加顶点，使得最长边的长度最小。与此问题的离散形式相反，附加顶点可以位于平面中的任何位置。这个问题的动机是在无线自组织网络中的能量消耗的优化中继增强建模。该算法利用Voronoi图和图的块割点分解的性质，当$1<p<\infty$时在$O(n^k\log^{\frac{5k}{2}}n)$步骤中找到最优解，当$p\in\{1,\infty\}$时在$O(n^2\log^{\frac{7k}{2}+1}n)$步骤中找到最优解。
英文摘要：
We present the first exact polynomial time algorithm for constructing optimal geometric bottleneck 2-connected Steiner networks containing at most $k$ Steiner points, where $k>2$ is a constant. Given a set of $n$ vertices embedded in an $L_p$ plane, the objective of the problem is to find a 2-connected network, spanning the given vertices and at most $k$ additional vertices, such that the length of the longest edge is minimised. In contrast to the discrete version of this problem the additional vertices may be located anywhere in the plane. The problem is motivated by the modelling of relay-augmentation for the optimisation of energy consumption in wireless ad hoc networks. Our algorithm employs Voronoi diagrams and properties of block-cut-vertex decompositions of graphs to find an optimal solution in $O(n^k\log^{\frac{5k}{2}}n)$ steps when $1<p<\infty$ and in $O(n^2\log^{\frac{7k}{2}+1}n)$ steps when $p\in\{1,\infty\}$.
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