No abstract
A power assignment is an assignment of transmission power to each of the nodes of a wireless network, so that the induced communication graph has some desired properties. The cost of a power assignment is the sum of the powers. The energy of a transmission path from node u to node v is the sum of the squares of the distances between adjacent nodes along the path. For a constant t [ 1, an energy t-spanner is a graph G 0 , such that for any two nodes u and v, there exists a path from u to v in G 0 , whose energy is at most t times the energy of a minimum-energy path from u to v in the complete Euclidean graph. In this paper, we study the problem of finding a power assignment, such that (1) its induced communication graph is a 'good' energy spanner, and (2) its cost is 'low'. We show that for any constant t [ 1, one can find a power assignment, such that its induced communication graph is an energy t-spanner, and its cost is bounded by some constant times the cost of an optimal power assignment (where the sole requirement is strong connectivity of the induced communication graph). This is a significant improvement over the previous result due to Shpungin and
Let P be a set of points in the plane, representing transceivers equipped with a directional antenna of angle α and range r. The coverage area of the antenna at point p is a circular sector of angle α and radius r, whose orientation can be adjusted. For a given assignment of orientations, the induced symmetric communication graph (SCG) of P is the undirected graph, in which two vertices (i.e., points) u and v are connected by an edge if and only if v lies in u's sector and vice versa. In this paper we ask what is the smallest angle α for which there exists an integer n = n(α), such that for any set P of n antennas of angle α and unbounded range, one can orient the antennas so that (i) the induced SCG is connected, and (ii) the union of the corresponding wedges is the entire plane. We show (by construction) that the answer to this problem is α = π/2, for which n = 4. Moreover, we prove that if Q 1 and Q 2 are two quadruplets of antennas of angle π/2 and unbounded range, separated by a line, to which one applies the above construction, independently, then the induced SCG of Q 1 ∪ Q 2 is connected. This latter result enables us to apply the construction locally, and to solve the following two further problems.In the first problem (replacing omni-directional antennas with directional antennas), we are given a connected unit disk graph, corresponding to a set P of omni-directional antennas of range 1, and the goal is to replace the omni-directional antennas by directional antennas of angle π/2 and range r = O(1) and to orient them, such that the induced SCG is connected, and, moreover, is an O(1)-spanner of the unit disk graph, w.r.t. hop distance. In our solution r = 14 √ 2 and the spanning ratio is 8. In the second problem (orientation and power assignment), we are given a set P of directional antennas of angle π/2 and adjustable range. The goal is to assign to each antenna p, an orientation and a range r p , such that the resulting SCG is (i) connected, and (ii) p∈P r β p is minimized, where β ≥ 1 is a constant. For this problem, we present an O(1)-approximation algorithm.
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