arXiv 1510.074282015-10-26math.PR

几何Achlioptas过程

A geometric Achlioptas process

Tobias Müller, Reto Spöhel

备注：: Published at http://dx.doi.org/10.1214/14-AAP1074 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：: Annals of Applied Probability 2015, Vol. 25, No. 6, 3295-3337
DOI：: 10.1214/14-AAP1074

摘要：

随机几何图是通过从单位正方形（均匀随机和独立地）采样$n$点，并且对于给定的$r=r(n)$，只要两点的距离至多为$r$，就连接两点来获得的。我们考虑随机几何图的以下变化：在总共的每一轮$n$中，玩家从单位正方形中获得两个随机点，并且必须选择这两个点中的一个以包含在演化的几何图中。我们研究的问题，避免线性大小（或“巨人”）的组件在这种设置。具体来说，我们表明，对于任何$r\ll(n\log\log n)^{-1/3}$有一个策略，成功地保持所有组件的大小次线性，概率趋于一个$n\to\infty$。我们还表明，这是紧在以下意义上：对于任何$r\gg(n\log\log n)^{-1/3}$，球员将被迫创建一个组件的大小$(1-o(1))n$，无论他如何发挥，再次与概率倾向于一$n\to\infty$。我们还证明了相应的离线问题表现出类似的阈值行为在$r(n)=Θ(n^{-1/3})$。这些发现应与现有的结果（普通）随机几何图：有一个巨大的组件出现的概率很高，一旦$r$是为了$n^{-1/2}$。因此，我们的结果表明，特别是，在几何设置的选择的权力可以利用到一个更大的程度比在经典的Erdens-Rényi随机图，在那里的一个巨大的组件的出现只能延迟一个常数因子。

英文摘要：

The random geometric graph is obtained by sampling $n$ points from the unit square (uniformly at random and independently), and connecting two points whenever their distance is at most $r$, for some given $r=r(n)$. We consider the following variation on the random geometric graph: in each of $n$ rounds in total, a player is offered two random points from the unit square, and has to select exactly one of these two points for inclusion in the evolving geometric graph. We study the problem of avoiding a linear-sized (or "giant") component in this setting. Specifically, we show that for any $r\ll(n\log\log n)^{-1/3}$ there is a strategy that succeeds in keeping all component sizes sublinear, with probability tending to one as $n\to\infty$. We also show that this is tight in the following sense: for any $r\gg(n\log\log n)^{-1/3}$, the player will be forced to create a component of size $(1-o(1))n$, no matter how he plays, again with probability tending to one as $n\to\infty$. We also prove that the corresponding offline problem exhibits a similar threshold behaviour at $r(n)=Θ(n^{-1/3})$. These findings should be compared to the existing results for the (ordinary) random geometric graph: there a giant component arises with high probability once $r$ is of order $n^{-1/2}$. Thus, our results show, in particular, that in the geometric setting the power of choices can be exploited to a much larger extent than in the classical Erdős-Rényi random graph, where the appearance of a giant component can only be delayed by a constant factor.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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