**10. Taming the exponential state space of the maximum lifetime sensor cover problem**

If we consider the LD graph, it is quickly obvious that even creating this graph will take exponential time since there are 2*<sup>n</sup>* cover sets to consider where, *n* is the number of sensors. However, the target coverage problem has a useful property - if the local targets for every sensor are covered, then globally, all targets are also covered. In [35], we make use of this property to look at the LD graph locally (fixed 1-2 hop neighbors), and are able to construct all the local covers for the local targets and then model their dependencies. Based on these dependencies, a sensor can then prioritize its covers and negotiate these with its neighbors. Simple heuristics based on properties of this graph were presented in [35] and showed a 10-15% improvement over comparable algorithms in the literature. [15] built on this work by examining how an optimal sequence would pick covers in the LD graph and designing heuristics that behave in a similar fashion. Though the proposed heuristics are efficient in practice, the running time is a function of the number of neighbors and the number of local targets. Both of these are relatively small for most graphs but theoretically are exponential in the number of targets and sensors.

A key issue that remains unresolved is the question of how to deal with this exponential space of cover sets. In this paper we present a reduction of this exponential space to a linear one based on grouping cover sets into *equivalence classes*. We use [*Ci*] to denote the equivalence class of a cover *Ci*. The partition defined by the equivalence relation on the set of all sensor covers Given a set *C* and an equivalence relation �, the equivalence class of an element *Ci* ∈ *C* is the subset of all elements in *C* which are equivalent to *Ci*. The notation used to represent the equivalence class of *Ci* is [*Ci*]. In the context of the problem being studied, *C* is the set of all sensor covers and for any single cover *Ci*, [*Ci*] represents all other covers which are *equivalent* to *Ci* as given by the definition of some equivalence relation �. Our approach stems from the understanding that from the possible exponential number of sensor covers, several covers are very similar, being only minor variations of each other. In Section 11, we present the definition of the relation �, based on a grouping that considers cover sets equivalent if their lifetime is bounded by the same sensor. We then show the use of this relation to collapse the exponential LD Graph into an *Equivalence Class* (EC) Graph with linear number of nodes. This theoretical insight allows us to design a sampling scheme that selects a subset of all local covers based on their equivalence class properties and presents this as an input to our simple LD graph degree-based heuristic. Simulation results show that class based sampling cuts the running time of these heuristics by nearly half, while only resulting in a less than 10% loss in quality.
