Constructing minimum energy mobile wireless networks

Energy conservation is a critical issue in designing wireless ad hoc networks, as the nodes are powered by batteries only. Given a set of wireless network nodes, the directed weighted transmission graph G<inf>t</inf> has an edge <i>uv</i> if and only if node <i>v</i> is in the transmission range of node <i>u</i> and the weight of <i>uv</i> is typically defined as ||<i>uv</i>||<sup>α</sup> + <b>c</b> for a constant <i>2</i> ≤ α ≤ <i>5</i> and c > <i>0</i>. The minimum power topology G<inf>m</inf> is the smallest subgraph of G<inf>t</inf> that contains the shortest paths between all pairs of nodes, i.e., the union of all shortest paths. In this paper, we described a distributed position-based networking protocol to construct an enclosure graph G<inf>e</inf>, which is an approximation of G<inf>m</inf>. The time complexity of each node <i>u</i> is O<i>(min(</i>d<inf>G<inf>t</inf></inf> <i>(u)</i>d<inf>G<inf>e</inf></inf> <i>(u),</i> d<inf>G<inf>t</inf></inf> <i>(u)</i> <i>log</i> d<inf>G<inf>t</inf></inf> <i>(u))),</i> where d<inf>G</inf> <i>(u)</i> is the degree of node <i>u</i> in a graph G. The space required at each node to compute the minimum power topology is O<i>(</i>d<inf>G<inf>t</inf></inf> <i>(u)).</i> This improves the previous result that computes G<inf>m</inf> in O<i>(</i>d<inf>G<inf>t</inf></inf> <i>(u)<sup>3</sup>)</i> time using O<i>(</i>d<inf>G<inf>t</inf></inf> <i>(u)<sup>2</sup>)</i> spaces. We also show that the average degree d<inf>G<inf>e</inf></inf> <i>(u)</i> is usually a constant, which is at most <i>6.</i> Our result is first developed for stationary network and then extended to mobile networks.

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