Maximal Breach and Support: Geometric Characterisation and Algorithms

Coverage is an important issue in Wireless Sensors Networks (WSNs). It gives a measure of the quality of surveillance a WSN provides over a field it is designed to monitor. Different measures of coverage capture different aspects of surveillance. In this paper, we study two measures of coverage maximal breach/support, and all-pairs average maximal breach/support. The geometric characterisation of these measures is one of the principal goals of our study. We have treated the coverage problem from two angles from the intruder’s point of view, and from the defender’s point of view. These two view-points give rise to two generic combinatorial optimisation problems. I Searching for “safe” paths in the field (important for the intruder). II Optimising the degree of coverage over all parts of the field (important for the defender). Our study of the geometric and combinatorial properties of single-pair and average maximal breach have led to exact polynomial time algorithms that solve problem I above when applied to these two specific measures. We have also framed and solved problem II for single-pair maximal breach. We have proved a “relaxed” version of problem II, applied to maximal breach, to be NP-Hard and designed an approximation algorithm for it. We have devised an efficient heuristic for solving some insertion based problems on maximal breach. We have stated and proved four basic theorems describing the behaviour of single-pair maximal breach under sensor insertion and deletion. These theorems have been used to develop the heuristic mentioned above. They have also led to important lower-bound results for single-pair and all-pairs average maximal breach.