Fault-tolerant spanners in networks with symmetric directional antennas

Let P be a set of points in the plane, each equipped with a directional antenna that covers a sector of angle α and range r. In the symmetric model of communication, two antennas u and v can communicate to each other, if and only if v lies in u's coverage area and vice versa. In this paper, we introduce the concept of fault-tolerant spanners for directional antennas, which enables us to construct communication networks that retain their connectivity and spanning ratio even if a subset of antennas are removed from the network. We show how to orient the antennas with angle α and range r to obtain a k-fault-tolerant spanner for any positive integer k. For α≥π, we show that the range 13 for the antennas is sufficient to obtain a k-fault-tolerant 3-spanner. For π/2<α<π, we show that using range 6δ+19 for δ=⌈4/|cos⁡α|⌉, one can direct antennas so that the induced communication graph is a k-fault-tolerant 7-spanner.