We study the cooperative data exchange problem for fully connected networks. In this problem, nodes make broadcast transmissions to recover a file consisting of K independent packets. Each node initially only possesses a subset of the packets. We propose (d,K) -Basis Searching, a deterministic polynomial-time minimization approach, to calculate the minimum rate for this problem. (d,K) -Basis Searching has strictly reduced complexity compared with the state-of-the-art algorithms, which are based on submodular function minimization. We extend our algorithm to a generalized problem: the so-called successive local omniscience problem.