Improved algorithms in directional wireless sensor networks

Recent advances in wireless sensor networks (WSNs) allow directional antennas to be used instead of omni-directional antennas. However, the problem of maintaining (symmetric) connectivity in directional wireless sensor networks is significantly harder. Contributing to this field of research, in this paper, we study two problems in WSNs equipped with k directional antennas (3≤k≤4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3 \le k \le 4$$\end{document}). The first problem, called antenna orientation (AO) is that given a set S of nodes equipped with omni-directional antennas of unit range, the goal is to replace omni-directional antennas by directional antennas with beam-width θ≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \ge 0$$\end{document} and to find a way to orient them such that the required range to yield a symmetric connected communication graph (SCCG) is minimized. For this problem, we propose an O(nlogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n \log n)$$\end{document} time algorithm yielding r=2sin180∘k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r = 2sin\left( \frac{180^\circ }{k}\right) $$\end{document}. The second problem, called antenna orientation and power assignment (AOPA) is to determine for each node u an orientation of its antennas and a range ru\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_u$$\end{document} in order to induce an SCCG such that the total power assignment ∑uruβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _u r_u^\beta $$\end{document} is minimized, where β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} is a distant-power gradient. We show that our solution for the AO problem also induces an O(1)-approximation algorithm for the AOPA problem. Simulation results demonstrate that our algorithms have better performance than previous approaches, especially in case k=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k = 3$$\end{document}.