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We study the network lifetime problem by considering not only the maximized time until the first node fails, but also the maximized lifetime for all the nodes in the network which we define as the lexicographic max-min (LMM) node lifetime problem. The main contributions of this paper are two-fold. First, we develop a polynomial-time algorithm to derive the LMM-optimal node lifetime vector, which effectively circumvents the computational complexity problem associated with an existing state-of-the-art approach, which is exponential. Second, we present a simple (also polynomial-time) algorithm to calculate the flow routing schedule such that the LMM-optimal node lifetime vector can be achieved. Our results in this paper advance the state-of-the-art algorithmic design to network-wide node lifetime problems.