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

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) = 1/13 + 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 logn+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 logn, then the weight is doubly-exponentially large in log2 lognk. It is NP-hard to find the minimum bounded-depth spanning tree, but when k≤log2 lognω(1), a simple greedy algorithm is asymptotically optimal, and when k ≥ log2 logn+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 logn+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 lognω(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 2k; when the diameter bound is increased from 2k to 2k+1, the minimum Steiner tree weight is reduced by a factor of \(2^{1/(2^k - 1)}\).

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Correspondence to Omer Angel.

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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