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A key problem in Wireless Sensor Networks is that of scheduling sensors into sleep-sense cycles that maximize the lifetime of the network while ensuring coverage of a set of targets. This is a known NP-complete problem, due to the exponential number of possible ways to select a subset of sensors to turn on. Our earlier work had presented a unique model for this problem by introducing a lifetime dependency (LD) graph in which these possible cover sets are nodes and edges represent shared sensors between them. Using the graph properties, we presented a range of effective distributed heuristics. Even though the local subgraphs are practically tractable, their theoretically exponential growth has remained a persistent issue. In this paper, we present a theoretical model for representing this exponential space of possible cover sets that groups related covers together, thereby reducing this exponential space virtually to a linear one. We then present an equivalence class (EC) graph as a model for this reduction. We use these underlying theoretical properties to develop a smart linear time sampling algorithm of this exponential space. To demonstrate the effectiveness of our sampling technique, we employ the sampled localized LD subgraph as input to our previous heuristics. Simulation studies show about twofold speedup while reducing solution quality (network lifetime) only by under 10% when compared to our previous heuristics that operate on the exponential space. We also compare our algorithms to two other state-of-art greedy algorithms and show that ours still manage to achieve improvements of around 8-10% over them.