A sharp threshold for minimum bounded-depth and bounded-diameter spanning trees and Steiner trees in random networks

In the complete graph on n vertices, when each edge has a weight which is an exponential random variable, Frieze proved that the minimum spanning tree has weight tending to zeta(3)=1/1^3+1/2^3+1/3^3+... as n goes to infinity. We consider spanning trees constrained to have depth bounded by k from a specified root. We prove that if k > log_2 log n+omega(1), where omega(1) is any function going to infinity with n, then the minimum bounded-depth spanning tree still has weight tending to zeta(3) as n -> infinity, and that if k < log_2 log n, then the weight is doubly-exponentially large in log_2 log n - k. It is NP-hard to find the minimum bounded-depth spanning tree, but when k < log_2 log n - omega(1), a simple greedy algorithm is asymptotically optimal, and when k > log_2 log n+omega(1), an algorithm which makes small changes to the minimum (unbounded depth) spanning tree is asymptotically optimal. We prove similar results for minimum bounded-depth Steiner trees, where the tree must connect a specified set of m vertices, and may or may not include other vertices. In particular, when m = const * n, if k > log_2 log n+omega(1), the minimum bounded-depth Steiner tree on the complete graph has asymptotically the same weight as the minimum Steiner tree, and if 1 <= k <= log_2 log n-omega(1), the weight tends to (1-2^{-k}) sqrt{8m/n} [sqrt{2mn}/2^k]^{1/(2^k-1)} in both expectation and probability. The same results hold for minimum bounded-diameter Steiner trees when the diameter bound is 2k; when the diameter bound is increased from 2k to 2k+1, the minimum Steiner tree weight is reduced by a factor of 2^{1/(2^k-1)}.