随机网格ad-hoc网络中的最小能量广播：近似和分布式算法
Minimum-energy broadcast in random-grid ad-hoc networks: approximation and distributed algorithms
摘要：
最小能量广播问题在于将传输范围分配给ad-hoc网络的节点，以便保证从给定源节点的有向生成树，并且同时最小化由范围分配产生的能量消耗（即能量成本）。最小能量广播已知是NP难的。我们考虑随机网格网络中的节点是独立地随机选择的$n$点的一个$\sqrt n \times \sqrt n$正方形网格在平面上。在网格的给定点处存在节点的概率确实取决于该点，即概率分布可以是非均匀的。通过使用信息理论的参数，我们证明了一个下界$(1-ε) \frac nπ$的能量成本的任何可行的解决方案，这个问题。然后，我们提供了一个有效的解决方案，能源成本不大于$1.1204 \frac nπ$。最后，我们提出了一个完全分布式的协议，构建了一个广播范围分配的能量成本不大于$8n$，从而仍然产生恒定的近似。能量负载很好地平衡，同时，工作复杂度（即，由于协议的所有消息传输的能量）是渐近最优的。协议的完成时间仅比最佳值慢$O(\log n)$因子。我们的分布式解决方案的近似质量也进行了实验评估。所有边界至少以$1-1/n^{Θ(1)}$的概率保持。
英文摘要：
The Min Energy broadcast problem consists in assigning transmission ranges to the nodes of an ad-hoc network in order to guarantee a directed spanning tree from a given source node and, at the same time, to minimize the energy consumption (i.e. the energy cost) yielded by the range assignment. Min energy broadcast is known to be NP-hard. We consider random-grid networks where nodes are chosen independently at random from the $n$ points of a $\sqrt n \times \sqrt n$ square grid in the plane. The probability of the existence of a node at a given point of the grid does depend on that point, that is, the probability distribution can be non-uniform. By using information-theoretic arguments, we prove a lower bound $(1-ε) \frac nπ$ on the energy cost of any feasible solution for this problem. Then, we provide an efficient solution of energy cost not larger than $1.1204 \frac nπ$. Finally, we present a fully-distributed protocol that constructs a broadcast range assignment of energy cost not larger than $8n$,thus still yielding constant approximation. The energy load is well balanced and, at the same time, the work complexity (i.e. the energy due to all message transmissions of the protocol) is asymptotically optimal. The completion time of the protocol is only an $O(\log n)$ factor slower than the optimum. The approximation quality of our distributed solution is also experimentally evaluated. All bounds hold with probability at least $1-1/n^{Θ(1)}$.
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