Abstract
We study the probabilistic model in the key tree management problem. Users have different behaviors. Normal users have probability p to issue join/leave request while the loyal users have probability zero. Given the numbers of such users, our objective is to construct a key tree with minimum expected updating cost. We observe that a single LUN (Loyal User Node) is enough to represent all loyal users. When 1−p≤0.57 we prove that the optimal tree that minimizes the cost is a star. When 1−p>0.57, we try to bound the size of the subtree rooted at every non-root node. Based on the size bound, we construct the optimal tree using dynamic programming algorithm in O(n⋅K+K 4) time where K=min {4(log (1−p)−1)−1,n} and n is the number of normal users.
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Chan, YK., Li, M. & Wu, W. Optimal tree structure with loyal users and batch updates. J Comb Optim 22, 630–639 (2011). https://doi.org/10.1007/s10878-010-9312-z
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DOI: https://doi.org/10.1007/s10878-010-9312-z