动态图的T-区间连通性的有效检验
Efficiently Testing T-Interval Connectivity in Dynamic Graphs
摘要：
许多类型的动态网络都是由持久的实体组成的，这些实体的联系会随着时间的推移而演变。当从{ \em全局}和{ \em离散}的观点考虑时，这些网络通常被建模为演化图，即图序列${\cal G}=(G_1,G_2,...,G_δ)$，使得$G_i=(V,E_i)$表示时间步长$i$处的网络拓扑。这样的序列称为$T$ -区间连通的，如果对于任何$t\in [1, δ-T+1]$，$\{G_t,G_{t+1},...,G_{t+T-1}\}$中的所有图共享一个公共的连通生成子图。本文考虑了给定序列${\cal G}$对于给定的$T$是否是$T$ -区间连通的问题.我们还考虑了相关的问题，找到最大的$T$，其中一个给定的${\cal G}$是$T$ -区间连接。我们假设两个连续图之间的变化是任意的，并且两个操作，{ \em二进制交叉}和{ \em连通性测试}，可用于解决问题。我们表明，$Ω(δ)$这样的操作需要解决这两个问题，我们提出了最佳的$O(δ)$在线算法这两个问题。我们将我们的在线算法扩展到一个动态的设置，其中连接是基于最近的网络演变。
英文摘要：
Many types of dynamic networks are made up of durable entities whose links evolve over time. When considered from a {\em global} and {\em discrete} standpoint, these networks are often modelled as evolving graphs, i.e. a sequence of graphs ${\cal G}=(G_1,G_2,...,G_δ)$ such that $G_i=(V,E_i)$ represents the network topology at time step $i$. Such a sequence is said to be $T$-interval connected if for any $t\in [1, δ-T+1]$ all graphs in $\{G_t,G_{t+1},...,G_{t+T-1}\}$ share a common connected spanning subgraph. In this paper, we consider the problem of deciding whether a given sequence ${\cal G}$ is $T$-interval connected for a given $T$. We also consider the related problem of finding the largest $T$ for which a given ${\cal G}$ is $T$-interval connected. We assume that the changes between two consecutive graphs are arbitrary, and that two operations, {\em binary intersection} and {\em connectivity testing}, are available to solve the problems. We show that $Ω(δ)$ such operations are required to solve both problems, and we present optimal $O(δ)$ online algorithms for both problems. We extend our online algorithms to a dynamic setting in which connectivity is based on the recent evolution of the network.
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