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Due to the application-specific nature of wireless sensor networks, the overall performance can be sensitive to sensor coverage and data reporting latency, thus requiring intelligent algorithm and protocol design paradigms. In this paper, we propose a family of three novel coverage-adaptive random sensor selection (CANSEE) schemes for controlling data-gathering latency in wireless sensor networks, with a goal to increase energy conservation rate and hence network lifetime. The underlying concept is to select in each round k sensors (i.e., data reporters) which can cover the desired sensing coverage (DSC) specified by the users/applications. The selection of such k sensors is based on a geometric probability theory and a randomization technique with constant computational complexity but without requiring exchange of control (location) information with local neighbors. Only the selected k sensors transmit data to the gathering point while others cache their sensed data and wait for the next reporting round. This incurs some delay but saves energy. Each sensor has an equal opportunity to report data periodically so the entire monitored area is covered within a fixed delay. Simulation results show that the proposed three CANSEE schemes lead to a significant conservation of energy with a small tradeoff in data reporting latency while meeting the coverage requirement specified by the users/applications.