On Packing Low-Diameter Spanning Trees
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Edge connectivity of a graph is one of the most fundamental graph-theoretic concepts. The celebrated tree packing theorem of Tutte and Nash-Williams from 1961 states that every k-edge connected graph G contains a collection T of ⌊ k/2 ⌋ edge-disjoint spanning trees, that we refer to as a tree packing; the diameter of the tree packing T is the largest diameter of any tree in T. A desirable property of a tree packing, that is both sufficient and necessary for leveraging the high connectivity of a graph in distributed communication, is that its diameter is low. Yet, despite extensive research in this area, it is still unclear how to compute a tree packing, whose diameter is sublinear in |V(G)|, in a low-diameter graph G, or alternatively how to show that such a packing does not exist. In this paper we provide first non-trivial upper and lower bounds on the diameter of tree packing. First, we show that, for every k-edge connected n-vertex graph G of diameter D, there is a tree packing T of size Ω(k), diameter O((101klog n)^D), that causes edge-congestion at most 2. Second, we show that for every k-edge connected n-vertex graph G of diameter D, the diameter of G[p] is O(k^D(D+1)/2) with high probability, where G[p] is obtained by sampling each edge of G independently with probability p=Θ(log n/k). This provides a packing of Ω(k/log n) edge-disjoint trees of diameter at most O(k^(D(D+1)/2)) each. We then prove that these two results are nearly tight. Lastly, we show that if every pair of vertices in a graph has k edge-disjoint paths of length at most D connecting them, then there is a tree packing of size k, diameter O(Dlog n), causing edge-congestion O(log n). We also provide several applications of low-diameter tree packing in distributed computation.
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