距离随机平均场模型中的极大Steiner树
Maximal Steiner Trees in the Stochastic Mean-Field Model of Distance
摘要：
考虑$n$顶点上的完全图，边权重独立于具有单位均值的指数分布绘制。Janson表明，两个顶点之间的典型距离尺度为$\log{n}/n$，而直径（任何两个顶点之间的最大距离）尺度为$3\log{n}/n$。Bollobás等人证明，对于任何固定的k，连接$k$典型顶点的Steiner树的权重为$(k-1)\log{n}/n$，这恢复了Janson对$k=2$的结果。我们扩展这一结果表明，最坏的情况下$k$ -施泰纳树，在所有的选择$k$顶点，有重量缩放为$(2k-1)\log{n}/n$，最后，我们将这一结果推广到施泰纳树的典型和最坏的情况下的顶点的混合物。
英文摘要：
Consider the complete graph on $n$ vertices, with edge weights drawn independently from the exponential distribution with unit mean. Janson showed that the typical distance between two vertices scales as $\log{n}/n$, whereas the diameter (maximum distance between any two vertices) scales as $3\log{n}/n$. Bollobás et al. showed that, for any fixed k, the weight of the Steiner tree connecting $k$ typical vertices scales as $(k-1)\log{n}/n$, which recovers Janson's result for $k=2$. We extend this result to show that the worst case $k$-Steiner tree, over all choices of $k$ vertices, has weight scaling as $(2k-1)\log{n}/n$ and finally, we generalise this result to Steiner trees with a mixture of typical and worst case vertices.
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