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((101klogn)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(kD(D+1)/2) with high
probability, where G[p] is obtained by sampling each edge of G
independently with probability p=Θ(logn/k). This provides a packing of
Ω(k/logn) 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(Dlogn), causing edge-congestion O(logn). We also provide several
applications of low-diameter tree packing in distributed computation