无线传感器网络虚拟骨干网容错的近似算法
Approximation Algorithm for Fault-Tolerant Virtual Backbone in Wireless Sensor Networks
摘要：
为了在无线传感器网络中节省能量和减少干扰，提出了使用虚拟骨干网的方法。由于意外损坏或能量耗尽，需要构造一个容错的虚拟骨干网，它可以在图中建模为一个$k$ -连通的$m$ -折叠支配集（简称$(k,m)$ -CDS）.一个节点集$C\subseteq V(G)$是图$G$的一个$(k,m)$，如果$V(G)\backslash C$中的每个节点至少与$C$中的$m$个节点相邻，并且由$C$导出的$G$的子图是$k$ -连通的.本文提出了一个用$m\geq3$求解最小$(3,m)$-CDS问题的近似算法.性能比至多为$γ$，其中$α\geq4$为$γ=α+8+2\ln(2α-6)$，$α<4$为$γ=3α+2\ln2$，$α$为最小$(2,m)$-CDS问题的性能比。使用当前最佳已知的$α$值，性能比是$\lnδ+o(\lnδ)$，其中$δ$是图的最大度，鉴于问题的非近似性，这是渐近最佳可能的。这是第一个性能保证算法的最小$(3,m)$ -CDS问题的一般图。此外，将我们的算法应用于一个同构无线传感器网络的单位圆盘图上，性能比小于27，大大提高了以前的单位圆盘图上的$(3,m)$ -CDS问题的性能比62.3。
英文摘要：
To save energy and alleviate interferences in a wireless sensor network, the usage of virtual backbone was proposed. Because of accidental damages or energy depletion, it is desirable to construct a fault tolerant virtual backbone, which can be modeled as a $k$-connected $m$-fold dominating set (abbreviated as $(k,m)$-CDS) in a graph. A node set $C\subseteq V(G)$ is a $(k,m)$-CDS of graph $G$ if every node in $V(G)\backslash C$ is adjacent with at least $m$ nodes in $C$ and the subgraph of $G$ induced by $C$ is $k$-connected. In this paper, we present an approximation algorithm for the minimum $(3,m)$-CDS problem with $m\geq3$. The performance ratio is at most $γ$, where $γ=α+8+2\ln(2α-6)$ for $α\geq4$ and $γ=3α+2\ln2$ for $α<4$, and $α$ is the performance ratio for the minimum $(2,m)$-CDS problem. Using currently best known value of $α$, the performance ratio is $\lnδ+o(\lnδ)$, where $δ$ is the maximum degree of the graph, which is asymptotically best possible in view of the non-approximability of the problem. This is the first performance-guaranteed algorithm for the minimum $(3,m)$-CDS problem on a general graph. Furthermore, applying our algorithm on a unit disk graph which models a homogeneous wireless sensor network, the performance ratio is less than 27, improving previous ratio 62.3 by a large amount for the $(3,m)$-CDS problem on a unit disk graph.
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