## Abstract

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) = 1/1^{3} + 1/2^{3} + 1/3^{3} +… as *n* → ∞. We consider spanning trees constrained to have depth bounded by *k* from a specified root. We prove that if *k* ≥ log_{2} 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* < 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*−*ω*(1), a simple greedy algorithm is asymptotically optimal, and when *k* ≥ log_{2} 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*≥log_{2} 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* ≤ log_{2} log*n*−*ω*(1), the weight tends to \((1 - 2^{ - k} )\sqrt {8m/n} \left[ {\sqrt {2mn} /2^k } \right]^{1/(2^k - 1)}\) in both expectation and probability. The same results hold for minimum bounded-diameter Steiner trees when the diameter bound is 2*k*; when the diameter bound is increased from 2*k* to 2*k*+1, the minimum Steiner tree weight is reduced by a factor of \(2^{1/(2^k - 1)}\).

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## Additional information

Earlier affiliation: University of Toronto.

Earlier affiliation: Microsoft Research

Omer Angel was supported by the University of Toronto and NSERC. Abraham D. Flaxman was supported by Microsoft Research.

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Angel, O., Flaxman, A.D. & Wilson, D.B. A sharp threshold for minimum bounded-depth and bounded-diameter spanning trees and Steiner trees in random networks.
*Combinatorica* **32, **1–33 (2012). https://doi.org/10.1007/s00493-012-2552-z

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### Mathematics Subject Classification (2010)

- 05C80
- 90C27
- 05C05
- 60C05
- 82B26
- 68W40
- 68R10
- 68W25