基于Poisson线模式的低成本网络中的短路径
Short-length routes in low-cost networks via Poisson line patterns
摘要：
在设计一个网络连接n个城市在一个广场的面积n，一个可能是由以下两个desiderata指导。第一，网络总长度不应比连接所有城市的最短网络长度大大于多少。第二，平均路线长度（来源-目的地对）不应比平均直线距离大大于很多。我们能把这两个差别缩小到什么程度呢？对于典型的配置，最短的网络长度是n阶，平均直线距离是n^1/2阶，因此，我们似乎不可能构建一个网络，其中第一个差是o（n），第二个差是o（n^1/2）。但事实上，我们可以做得更好：对于任意配置，我们可以构造一个网络，其中第一个差值为O（n），第二个差值几乎为O（log n）。构造概念简单：在最小长度连通网络（Steiner树）上，构造一个稀疏平稳各向同性Poisson线过程。其中的关键部分是关于Poisson线过程的一个新结果。考虑两个相距r的点，从直线过程中删除所有分隔这两个点的直线。由此产生的线条图案将平面划分为单元;包含两个点的单元的平均边界长度为2 r + C log r。转向下界，我们表明，在弱等分布假设下，如果第一个差异是O（n），那么第二个差异不能是O（sqrt（log n））。
英文摘要：
In designing a network to link n cities in a square of area n, one might be guided by the following two desiderata. First, the total network length should not be much greater than the length of the shortest network connecting all cities. Second, the average route length (taken over source-destination pairs) should not be much greater than the average straight-line distance. How small can we make these two differences? For typical configurations the shortest network length is order n and the average straight-line distance is order n^1/2, so it seems implausible that one can construct a network in which the first difference is o(n) and the second difference is o(n^1/2). But in fact one can do better: for an arbitrary configuration one can construct a network where the first difference is o(n) and the second difference is almost as small as O(log n). The construction is conceptually simple: over the minimum-length connected network (Steiner tree) superimpose a sparse stationary and isotropic Poisson line process. The key ingredient is a new result about the Poisson line process. Consider two points at distance r apart, and delete from the line process all lines which separate these two points. The resulting pattern of lines partitions the plane into cells; the cell containing the two points has mean boundary length 2r + C log r. Turning to lower bounds we show that, under a weak equidistribution assumption, if the first difference is O(n) then the second difference cannot be o(sqrt(log n)).
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