连通性约束下网络鲁棒性的在线算法
Online Algorithms for Network Robustness under Connectivity Constraints
摘要：
在本文中，我们提出了设计网络的算法，是强大的节点故障，最小或有限数量的链接。我们提出了算法的静态网络设置和动态网络设置，设置新的节点可以在未来到达。对于静态设置，我们提出了算法，用于构建最优网络的链接数量用于一个给定的节点大小和节点的数量，可以失败。然后，我们考虑动态设置，其中删除任何旧链接都是破坏性的。对于这种设置，我们提出了两种情况下的在线算法：（i）当可以失败的节点的数量保持不变，（ii）当只有可以失败的节点的比例保持不变。我们表明，所提出的算法为第一种情况下保存近$3/4$ th的总可能的链接在任何时间点。然后，我们提出了算法的节点，可以失败的分数和表征他们的链路使用的各个级别。我们表明，当$1/2$的节点数量可以在任何时间点失败，该算法节省了近$1/2$的总可能的链接在任何时间点。我们表明，当节点的数量，可以失败的是有限的分数$1/(2m)$（$m \in \mathbb{N}$），该算法节省了几乎一样多的$(1-1/2m)$的总可能的链接在任何时间点。我们还表明，当节点的数量，可以在任何时间点失败的是$1/2$的节点的数量加上$n$，$n \in \mathbb{N}$，所提出的算法保存的链接的数量只线性减少$n$。我们推测，我们提出的算法实现的节省率是最佳的动态设置。
英文摘要：
In this paper, we present algorithms for designing networks that are robust to node failures with minimal or limited number of links. We present algorithms for both the static network setting and the dynamic network setting; setting where new nodes can arrive in the future. For the static setting, we present algorithms for constructing the optimal network in terms of the number of links used for a given node size and the number of nodes that can fail. We then consider the dynamic setting where it is disruptive to remove any of the older links. For this setting, we present online algorithms for two cases: (i) when the number of nodes that can fail remains constant and (ii) when only the proportion of the nodes that can fail remains constant. We show that the proposed algorithm for the first case saves nearly $3/4$th of the total possible links at any point of time. We then present algorithms for various levels of the fraction of the nodes that can fail and characterize their link usage. We show that when $1/2$ the number of nodes can fail at any point of time, the proposed algorithm saves nearly $1/2$ of the total possible links at any point of time. We show that when the number of nodes that can fail is limited to the fraction $1/(2m)$ ($m \in \mathbb{N}$), the proposed algorithm saves nearly as much as $(1-1/2m)$ of the total possible links at any point of time. We also show that when the number of nodes that can fail at any point of time is $1/2$ of the number of nodes plus $n$, $n \in \mathbb{N}$, the number of links saved by the proposed algorithm reduces only linearly in $n$. We conjecture that the saving ratio achieved by the algorithms we present is optimal for the dynamic setting.
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