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We consider the problem of finding a multicast tree rooted at the source node and including all the destination nodes such that the maximum weight of the tree arcs is minimized. It is of paramount importance for many optimization problems, e.g., the maximum-lifetime multicast problem in multihop wireless networks, in the data networking community. We explore some important properties of this problem from a graph theory perspective and obtain a min-max-tree max-min-cut theorem, which provides a unified explanation for some important while separated results in the recent literature. We also apply the theorem to derive an algorithm that can construct a global optimal min-max multicast tree in a distributed fashion. In random networks with n nodes and m arcs, our theoretical analysis shows that the expected communication complexity of our distributed algorithm is in the order of O(m). Specifically, the average number of messages is 2(n - 1 - γ) -2 ln(n-1) + m at most, in which γ is the Euler constant. To our best knowledge, this is the first contribution that possesses the distributed and scalable properties for the min-max multicast problem and is especially desirable to the large-scale resource-limited multihop wireless networks, like sensor networks.
Date of Publication: Aug. 2010