UBC Theses and Dissertations
Approximating barrier resilience and related notions for disk sensors in a two-dimensional plane Chan, David Yu Cheng
Let A be an arrangement of n sensors constituting a barrier, the resilience of A with respect to two regions S and T, denoted ρ(A,S,T), is defined as the number of sensors in A that must be removed in order for there to be a path from S to T that is not detected by any sensor. Previous work by Bereg and Kirkpatrick proved that a 3-approximation of the resilience could be guaranteed by a polynomial time algorithm when sensors are unit disks in the plane. This was tightened to a 5/3-approximation when S and T are well-separated. In this thesis, we define and analyze a Multi-Path Algorithm (MPA) which improves on these bounds to get approximation ratios of 2 and 1.5 respectively. Additionally, we define a new notion of barrier strength called Dc-Resilience, which is the resilience of the barrier if regions covered by more than c distinct sensors in the original arrangement are treated as inaccessible, in the sense that no part of any path is permitted to intersect with such a region. We consider arrangements of disk sensors with radii in range [1 - ξ, 1] for any 0 ≤ ξ < 1 independent of n. We prove that for any such arrangement and any constant c, the MPA guarantees a constant approximation of the Dc-resilience in time polynomial in n for any constant c. Furthermore, we show that this tightens to a 2-approximation under certain conditions which are usually fulfilled when disk sensors are close to equal size.
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