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We have designed an algorithm for a problem in multicast communication. The problem is to construct a multicast tree while minimizing its cost, which is known to be NP-complete. Our algorithm, which employs new concepts defined as potential cost and spanning cost, generates a multicast tree more efficiently than the well-known heuristic called Takahashi and Matsuyama (TM)  in terms of tree cost. The time complexity of our algorithm is O(kn2) for an n-node network with k members in the multicast group and is comparable to the TM. Our empirical performance evaluation comparing the proposed algorithm with TM shows that the enhancement is up to 1.25%∼4.23% for each best case.