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In this paper, we investigate how to maximize network lifetime, while meeting the constraints on data collecting time, or data convergence latency (DCL), in networks consisting of N energy-constrained sensor nodes and a central base station (BS). Noting the trade-off relation that a data packet delivered to the BS requires less transmission energy through multi-hop relays but more DCL, we first formulate an optimal routing algorithm that maximizes the network lifetime while satisfying the DCL constraint determined by latency-sensitivity of the application or network stability conditions. As this optimal solution is too complex to implement in practice, we modify it to a suboptimal latency-constrained cost minimization (LCCM) algorithm that is close to the optimal algorithm in performance but is; much lighter in complexity. The LCCM algorithm is designed to minimize the total link cost, while satisfying the DCL constraint. Simulation results show that the performance of the proposed LCCM algorithm is slightly marginal to that of the optimal algorithm in terms of the lifetime, in various network environments.