Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints

We introduce a new structure for a set of points in the plane and an angle $\alpha$, which is similar in flavor to a bounded-degree MST. We name this structure $\alpha$-MST. Let $P$ be a set of points in the plane and let $0 < \alpha \le 2\pi$ be an angle. An $\alpha$-ST of $P$ is a spanning tree of the complete Euclidean graph induced by $P$, with the additional property that for each point $p \in P$, the smallest angle around $p$ containing all the edges adjacent to $p$ is at most $\alpha$. An $\alpha$-MST of $P$ is then an $\alpha$-ST of $P$ of minimum weight. For $\alpha < \pi/3$, an $\alpha$-ST does not always exist, and, for $\alpha \ge \pi/3$, it always exists. In this paper, we study the problem of computing an $\alpha$-MST for several common values of $\alpha$. Motivated by wireless networks, we formulate the problem in terms of directional antennas. With each point $p \in P$, we associate a wedge $W_p$ of angle $\alpha$ and apex $p$. The goal is to assign an orientation and a radius $r_p$ to each wedge $W_p$, such that the resulting graph is connected and its MST is an $\alpha$-MST. (We draw an edge between $p$ and $q$ if $p \in W_q$, $q \in W_p$, and $|pq| \le r_p, r_q$.) Unsurprisingly, the problem of computing an $\alpha$-MST is NP-hard, at least for $\alpha=\pi$ and $\alpha=2\pi/3$. We present constant-factor approximation algorithms for $\alpha = \pi/2, 2\pi/3, \pi$. One of our major results is a surprising theorem for $\alpha = 2\pi/3$, which, besides being interesting from a geometric point of view, has important applications. For example, the theorem guarantees that given any set $P$ of $3n$ points in the plane and any partitioning of the points into $n$ triplets, one can orient the wedges of each triplet {\em independently}, such that the graph induced by $P$ is connected. We apply the theorem to the {\em antenna conversion} problem

Impact of directional antennas on routing and neighbor discovery in wireless ad-hoc networks

Wireless ad-hoc networks are data networks that are deployed without a fixed infrastructure nor central controllers such as access points or base stations. In these networks, data packets are forwarded directly to the destination node if they are within the transmission range of the sender or sent through a multi-hop path of intermediary nodes that act as relays. This paradigm where a fixed infrastructure is not needed, is tolerant to topology changes and allows a fast deployment have been considered as a promissory technology that is suitable for a large number of network implementations, such as mobile hand-held devices, wireless sensors, disaster recovery networks, etc. Recently, smart directional antennas have been identified as a robust technology that can boost the performance of wireless ad-hoc networks in terms of coverage, connectivity, and capacity. Contrary to omnidirectional antennas, which can radiate energy in all directions, directional antennas can focus the energy in a specific direction, extending the coverage range for the same power level. Longer ranges provide shorter paths to destination nodes and also improve connectivity. Moreover, directional antennas can reduce the number of collisions in a contention-based access scheme as they can steer the main lobe in the desired direction and set nulls in all the others, thereby they minimize the co-channel interference and reduce the noise level. Connections are more reliable due to the increased link stability and spatial diversity. Shorter paths, as well as alternative paths, are also available as a consequence of the use of directional antennas. All these features combined results in a higher network capacity. Most of the previous research has focused on adapting the existing medium access control and routing protocols to utilize directional communications. This research work is novel because it improves the neighbor discovery process as it allows to discover nodes in the second neighborhood of a given node using a gossip based procedure and by sharing the relative position information obtained during this stage with the routing protocol with the aim of reducing the number of hops between source and destination. We have also developed a model to evaluate the energy consumed by the nodes when smart directional antennas are used in the ad-hoc network. This study has demonstrated that by adapting the beamwidth of the antennas nodes are able to reach furthest nodes and consequently, reduce the number of hops between source and destination. This fact not only reduces the end-to-end delay and improves the network throughput but also reduces the average energy consumed by the whole network