A sharp threshold for minimum bounded-depth/diameter spanning and Steiner trees

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 ζ(3) = 1/13 + 1/23 + 1/33 + · · · as n → ∞. We consider spanning trees constrained to have depth bounded by k from a specified root. We prove that if k ≥ log2 log n+ω(1), where ω(1) is any function going to ∞ with n, then the minimum bounded-depth spanning tree still has weight tending to ζ(3) as n → ∞, and that if k < log2 log n, then the weight is doubly-exponentially large in log2 log n − k. It is NPhard to find the minimum bounded-depth spanning tree, but when k ≤ log2 log n−ω(1), a simple greedy algorithm is asymptotically optimal, and when k ≥ log2 log n + ω(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 ≥ log2 log n+ ω(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 ≤ log2 log n− ω(1), the weight tends to (1−2−k) √ 8m/n [√ 2mn/2k ]1/(2k−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 21/(2 k−1).