[论文解读] SINR Diagrams: Towards Algorithmically Usable SINR Models of Wireless Networks

Chen Avin, Yuval Emek+4 more|ArXiv.org|Nov 20, 2008

Mobile Ad Hoc Networks参考文献 16被引用 5

一句话总结

本文提出SINR图作为无线网络接收区域的几何框架，基于信号干扰加噪声比（SINR）模型。证明在统一功率传输下，这些接收区域是凸的且形状良好，从而支持一种$O(n^3\tilde{\rho}^{-1})$时间的高效预处理算法，利用Voronoi图和每站专用数据结构，实现$O(\log n)$查询时间的$\epsilon$-近似点定位。

ABSTRACT

The rules governing the availability and quality of connections in a wireless network are described by physical models such as the signal-to-interference & noise ratio (SINR) model. For a collection of simultaneously transmitting stations in the plane, it is possible to identify a reception zone for each station, consisting of the points where its transmission is received correctly. The resulting SINR diagram partitions the plane into a reception zone per station and the remaining plane where no station can be heard. SINR diagrams appear to be fundamental to understanding the behavior of wireless networks, and may play a key role in the development of suitable algorithms for such networks, analogous perhaps to the role played by Voronoi diagrams in the study of proximity queries and related issues in computational geometry. So far, however, the properties of SINR diagrams have not been studied systematically, and most algorithmic studies in wireless networking rely on simplified graph-based models such as the unit disk graph (UDG) model, which conveniently abstract away interference-related complications, and make it easier to handle algorithmic issues, but consequently fail to capture accurately some important aspects of wireless networks. The current paper focuses on obtaining some basic understanding of SINR diagrams, their properties and their usability in algorithmic applications. Specifically, based on some algebraic properties of the polynomials defining the reception zones we show that assuming uniform power transmissions, the reception zones are convex and relatively well-rounded. These results are then used to develop an efficient approximation algorithm for a fundamental point location problem in wireless networks.

研究动机与目标

• 建立SINR图的系统性理解，作为无线网络算法设计的基础。
• 解决简化图模型（如单位圆图）的局限性，后者无法捕捉干扰和信号质量变化。
• 为SINR图中的点定位开发高效的算法框架，这是无线网络管理中的基础问题。
• 证明在统一功率下，接收区域是凸的且具有有界纵横比，从而支持几何近似。

提出的方法

• 将接收区域$\mathcal{H}_i$定义为满足$\mathrm{SINR}(p, s_i) \geq \beta$的点集，基于路径损耗模型推导出多项式不等式。
• 基于SINR多项式的代数性质，证明每个$\mathcal{H}_i$是凸的且有界的，且站点$s_i$位于其内部。
• 建立$\mathcal{H}_i$内切圆半径和外接圆半径的下界$\tilde{\delta}$与上界$\tilde{\Delta}$，并证明$\tilde{\Delta}/\tilde{\delta} = O(\sqrt{n})$。
• 使用二分查找将这些边界细化至$\Theta(r)$，其中$r$为区域的尺度，耗时$O(n\log n)$。
• 为每站构建数据结构$\mathtt{QDS}_i$，将平面划分为$\mathcal{H}_i^+$、$\mathcal{H}_i^-$和$\mathcal{H}_i^?$，满足$\mathrm{area}(\mathcal{H}_i^?) \leq \epsilon \cdot \mathrm{area}(\mathcal{H}_i)$，耗时$O(n^2\epsilon^{-1})$。
• 将大小为$O(n)$的Voronoi图与所有$\mathtt{QDS}_i$结合，构建大小为$O(n\epsilon^{-1})$的全局数据结构$\mathtt{DS}$，支持$O(\log n)$时间查询。

实验结果

研究问题

• RQ1 在统一功率传输下，SINR接收区域是否为凸且行为良好？
• RQ2 SINR图的几何结构能否支持高效的点定位算法？
• RQ3 SINR接收区域的纵横比是多少？能否以网络规模$n$表示其有界性？
• RQ4 如何以可控误差近似接收区域边界，以实现快速查询？
• RQ5 Voronoi图能否与SINR专用数据结构有效结合，以加速点定位？

主要发现

• 在SINR模型中，当所有站以统一功率传输时，接收区域$\mathcal{H}_i$是凸集。
• 每个$\mathcal{H}_i$的外接圆半径与内切圆半径之比有界于$O(\sqrt{n})$，确保区域不会过度拉长。
• 可通过类似二分查找的程序在$O(n\log n)$时间内计算$\mathcal{H}_i$的内切圆半径和外接圆半径的改进边界。
• 每个站的$\epsilon$-近似点定位数据结构$\mathtt{QDS}_i$可在$O(n^2\epsilon^{-1})$时间内构建，大小为$O(\epsilon^{-1})$。
• 大小为$O(n\epsilon^{-1})$的全局数据结构$\mathtt{DS}$支持$O(\log n)$时间查询，判断某点是否属于正区域、负区域或不确定区域。
• 完整数据结构的总体预处理时间为$O(n^3\epsilon^{-1})$，适用于实际部署。