极小弱$\mathcal{ST}$ -可达有向无圈图的自稳定构造
Self-Stabilizing Construction of a Minimal Weakly $\mathcal{ST}$-Reachable Directed Acyclic Graph
摘要：
我们提出了一个自稳定算法来构造一个最小的弱$\mathcal{ST}$ -可达有向无环图（DAG），它适用于无线网络上的消息路由。给定一个任意的简单连通无向图$G=(V, E)$和两组节点，即节点$\mathcal{S} (\subset V)$和目标$\mathcal{T} (\subset V)$，$G$的有向子图$\vec{G}$是$G$上的弱$\mathcal{ST}$ -可达DAG，如果$\vec{G}$是一个DAG，并且每个发送者都能到达至少一个目标，并且每个目标都能从$\vec{G}$中的至少一个发送者到达。我们称$G$上的弱$\mathcal{ST}$ -可达DAG $\vec{G}$是极小的，如果$\vec{G}$的任意真子图不再是弱$\mathcal{ST}$ -可达DAG.此DAG是原始（或强）$\mathcal{ST}$可达DAG的宽松版本，其中每个目标都可以从每个发送方到达。这是因为强$\mathcal{ST}$可达的DAG $G$并不总是存在;即使在$|\mathcal{S}|=|\mathcal{T}|=2$的情况下，某些图也没有强$\mathcal{ST}$可达的DAG。另一方面，对于任意的$|\mathcal{S}|$和$|\mathcal{T}|$，该算法总是构造出一个弱$\mathcal{ST}$可达的DAG.此外，所提出的算法是自稳定的，即使构造的DAG偏离可达性要求的故障或耗尽电池的一个节点有一个弧在DAG中，该算法自动重建的DAG，以满足要求。算法的收敛时间为$O(D)$异步轮，其中$D$为给定图的直径。我们进行小的模拟，以评估所提出的算法的性能。仿真结果表明，当发送节点或目标节点的数目较大时，该算法的执行时间缩短。
英文摘要：
We propose a self-stabilizing algorithm to construct a minimal weakly $\mathcal{ST}$-reachable directed acyclic graph (DAG), which is suited for routing messages on wireless networks. Given an arbitrary, simple, connected, and undirected graph $G=(V, E)$ and two sets of nodes, senders $\mathcal{S} (\subset V)$ and targets $\mathcal{T} (\subset V)$, a directed subgraph $\vec{G}$ of $G$ is a weakly $\mathcal{ST}$-reachable DAG on $G$, if $\vec{G}$ is a DAG and every sender can reach at least one target, and every target is reachable from at least one sender in $\vec{G}$. We say that a weakly $\mathcal{ST}$-reachable DAG $\vec{G}$ on $G$ is minimal if any proper subgraph of $\vec{G}$ is no longer a weakly $\mathcal{ST}$-reachable DAG. This DAG is a relaxed version of the original (or strongly) $\mathcal{ST}$-reachable DAG, where every target is reachable from every sender. This is because a strongly $\mathcal{ST}$-reachable DAG $G$ does not always exist; some graph has no strongly $\mathcal{ST}$-reachable DAG even in the case $|\mathcal{S}|=|\mathcal{T}|=2$. On the other hand, the proposed algorithm always constructs a weakly $\mathcal{ST}$-reachable DAG for any $|\mathcal{S}|$ and $|\mathcal{T}|$. Furthermore, the proposed algorithm is self-stabilizing; even if the constructed DAG deviates from the reachability requirement by a breakdown or exhausting the battery of a node having an arc in the DAG, this algorithm automatically reconstructs the DAG to satisfy the requirement again. The convergence time of the algorithm is $O(D)$ asynchronous rounds, where $D$ is the diameter of a given graph. We conduct small simulations to evaluate the performance of the proposed algorithm. The simulation result indicates that its execution time decreases when the number of sender nodes or target nodes is large.
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