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In ad hoc wireless networks, to compute the transmission power of each wireless node such that the resulting network is connected and the total energy consumption is minimized is defined as a Minimum Energy Network Connectivity (MENC) problem, which is an NP-complete problem. In this paper, we consider the approximated solutions for the MENC problem in ad hoc wireless networks. We present a theorem that reveals the relation between the energy consumption of an optimal solution and that of a spanning tree and propose an optimization algorithm that can improve the result of any spanning tree-based topology. Two polynomial time approximation heuristics are provided in the paper that can be used to compute the power assignment of wireless nodes in both static and low mobility ad hoc wireless networks. The two heuristics are implemented and the numerical results verify the theoretical analysis.