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Distributed resource allocation is an important problem in wireless ad hoc networks, in which there is no centralized scheduler and the resource allocation is carried out in a distributed way. Information exchange in the distributed resource allocation incurs overhead since it does not convey data information. The communication complexity, defined as the minimum number of exchanged messages needed for computing a common function with distributed inputs, is studied and the resource allocation is considered to be the procedure of computing a common function whose inputs are the parameters of multiple communication links. A lower bound for the communication complexity is provided based on the first order differentiation of the output function of resource allocation by extending the two-input-single-output case in  to multi-input-multi-output case. The conclusion is then applied to a concrete example of distributed resource allocation.