We study the optimal structure for group broadcast problem where the key tree model is extensively used. The objective is usually to find an optimal key tree to minimize the cost based on certain assumptions. Under the assumption that n members arrive in the initial setup period and only member deletions are allowed after that period, previous works show that when only considering the deletion cost, the optimal tree can be computed in O(n2) time. In this paper, we first prove a semi-balance property for the optimal tree and use it to improve the running time from O (n2) to O(log log n). Then we study the optimal tree structure when insertion cost is also considered. We show that the optimal tree is such a tree where any internal node has at most degree 7 and children of nodes with degree not equal to 2 or 3 are all leaves. Based on this result we give a dynamic programming algorithm with O(n2) time to compute the optimal tree.
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