# 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 $$2^{1/(2^k - 1)}$$.

This is a preview of subscription content, access via your institution.

## References

1. [1]

Florin Avram and Dimitris Bertsimas: The minimum spanning tree constant in geometrical probability and under the independent model: a unified approach, Ann. Appl. Probab. 2(1) (1992), 113–130.

2. [2]

L. Addario-Berry, N. Broutin and B. Reed: Critical random graphs and the structure of a minimum spanning tree, Random Structures Algorithms 35(3) (2009), 323–347.

3. [3]

N. R. Achuthan and L. Caccetta: Minimum weight spanning trees with bounded diameter, Australas. J. Combin. 5 (1992), 261–276.

4. [4]

N. R. Achuthan and L. Caccetta: Addendum: “Minimum weight spanning trees with bounded diameter”, Australas. J. Combin. 8 (1993), 279–281.

5. [5]

Ayman Abdalla and Narsingh Deo: Random-tree diameter and the diameterconstrained MST, Int. J. Comput. Math. 79(6) (2002), 651–663.

6. [6]

A. Abdalla, N. Deo and R. Franceschini: Parallel heuristics for the diameterconstrained MST problem, in: Proceedings of the Thirtieth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1999), volume 136, 97–118, 1999.

7. [7]

Ernst Althaus, Stefan Funke, Sariel Har-Peled, Jochen Knemann, Edgar A. Ramos and Martin Skutella: Approximating k-hop minimumspanning trees, Oper. Res. Lett. 33(2) (2005), 115–120.

8. [8]

David Aldous and Allon G. Percus: Scaling and universality in continuous length combinatorial optimization, Proc. Natl. Acad. Sci. USA 100(20) (2003), 11211–11215 (electronic).

9. [9]

M. Bayati, C. Borgs, A. Braunstein, J. Chayes, A. Ramezanpour and R. Zecchina: Statistical mechanics of Steiner trees, Physical Review Letters 101(3) (2008), 037208.

10. [10]

Bla Bollobs, David Gamarnik, Oliver Riordan and Benny Sudakov: On the value of a random minimum weight Steiner tree, Combinatorica 24(2) (2004), 187–207.

11. [11]

Judit Bar-Ilan, Guy Kortsarz and David Peleg: Generalized submodular cover problems and applications, Theoret. Comput. Sci. 250(1–2) (2001), 179–200.

12. [12]

Alysson M. Costa, Jean-Franois Cordeauc and Gilbert Laporte: Fast heuristics for the Steiner tree problem with revenues, budget and hop constraints, European Journal of Operational Research 1906(1) (2008), 68–78.

13. [13]

Alysson M. Costa, Jean-Franois Cordeau and Gilbert Laporte: Models and branch-and-cut algorithms for the Steiner tree problem with revenues, budget and hop constraints, Networks 53 (2009), 141–159.

14. [14]

Andrea E. F. Clementi, Miriam Di Ianni, Massimo Lauria, Angelo Monti, Gianluca Rossi and Riccardo Silvestri: On the bounded-hop MST problem on random Euclidean instances, Theor. Comput. Sci. 384(2–3) (2007), 161–167.

15. [15]

Geir Dahl, Luis Gouveia and Cristina Requejo: On formulations and methods for the hop-constrained minimum spanning tree problem, in: Mauricio G. C. Resende and Panos M. Pardalos, editors, Handbook of Optimization in Telecommunications, 493–516, 2006.

16. [16]

A. M. Frieze and C. J. H. McDiarmid: On random minimum length spanning trees, Combinatorica 9(4) (1989), 363–374.

17. [17]

A. M. Frieze: On the value of a random minimum spanning tree problem, Discrete Appl. Math. 10(1) (1985), 47–56.

18. [18]

Michael R. Garey and David S. Johnson: Computers and Intractability: A guide to the theory of NP-completeness, W. H. Freeman and Co., San Francisco, Calif., 1979.

19. [19]

Luis Gouveia and Thomas L. Magnanti: Network flow models for designing diameter-constrained minimum-spanning and Steiner trees, Networks 41(3) (2003), 159–173.

20. [20]

Luis Gouveia: Using the Miller-Tucker-Zemlin constraints to formulate a minimal spanning tree problem with hop constraints, Computers & OR 22(9) (1995), 959–970.

21. [21]

L. Gouveia: Multicommodity flow models for spanning trees with hop constraints. European Journal of Operational Research 95, 178–190, 22 November 1996.

22. [22]

M. Gruber and G. R. Raidl: A new 0–1 ILP approach for the bounded diameter minimum spanning tree problem, in: 2nd Int. Network Optimization Conference, vol. 1, 178–185, 2005.

23. [23]

M. Gruber and G. R. Raidl: Variable neighborhood search for the bounded diameter minimum spanning tree problem, in: Proc. of the 18th Mini Euro Conference on Variable Neighborhood Search, Tenerife, Spain, 2005.

24. [24]

M. Gruber, J. van Hemert and G. R. Raidl: Neighborhood searches for the bounded diameter minimum spanning tree problem embedded in a VNS, EA, and ACO, in: Proc. of the Genetic and Evolutionary Computation Conference, Seattle, volume 2. ACM Press, 2006.

25. [25]

Svante Janson: The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph, Random Structures Algorithms 7(4) (1995), 337–355.

26. [26]

Svante Janson: One, two and three times logn/n for paths in a complete graph with random weights, Combin. Probab. Comput. 8(4) (1999), 347–361. Random graphs and combinatorial structures (Oberwolfach, 1997).

27. [27]

B. A. Julstrom and G. R. Raidl: A permutation-coded evolutionary algorithm for the bounded-diameter minimum spanning tree problem, in: 2003 GECCO Workshops Proc., Workshop on Analysis and Design of Representations (ADoRO), Chicago, 2–7, 2003.

28. [28]

Svante Janson and Johan Wstlund: Addendum to: “The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph” Structures Algorithms 7 (1995), no. 4, Janson, Random Structures Algorithms 28(4) (2006), 511–512.

29. [29]

Boris Kopinitsch: An ant colony optimisation algorithm for the bounded diameter minimum spanning tree problem, Master’s thesis, Vienna University of Technology, Institute of Computer Graphics and Algorithms, 2006, supervised by G. Raidl and M. Gruber.

30. [30]

Guy Kortsarz and David Peleg: Approximating the weight of shallow Steiner trees, Discrete Appl. Math. 93(2–3) (1999), 265–285.

31. [31]

Colin McDiarmid: On the method of bounded differences, in: Surveys in combinatorics, 1989 (Norwich, 1989), volume 141 of London Math. Soc. Lecture Note Ser., 148–188. Cambridge Univ. Press, Cambridge, 1989.

32. [32]

Jrme Monnot: The maximum f-depth spanning tree problem, Inform. Process. Lett. 80(4) (2001), 179–187.

33. [33]

Mathew Penrose: Random Geometric Graphs, volume 5 of Oxford Studies in Probability, Oxford University Press, Oxford, 2003.

34. [34]

Peter Putz: Subgradient optimization based lagrangian relaxation and relax-andcut approaches for the bounded diameter minimum spanning tree problem, Master’s thesis, Vienna University of Technology, Institute of Computer Graphics and Algorithms, 2007, supervised by G. Raidl.

35. [35]

Günther R. Raidl and Bryant A. Julstrom: Greedy heuristics and an evolutionary algorithm for the bounded-diameter minimum spanning tree problem, in: SAC’ 03: Proc. of the 2003 ACM Symposium on Applied Computing, Melbourne, FL, 747–752, 2003.

36. [36]

A. Rnyi and G. Szekeres: On the height of trees, J. Austral. Math. Soc. 7 (1967), 497–507.

37. [37]

J. Michael Steele: On Frieze’s ζ(3) limit for lengths of minimal spanning trees, Discrete Appl. Math. 18(1) (1987), 99–103.

38. [38]

G. Szekeres: Distribution of labelled trees by diameter, in: Combinatorial mathematics, X (Adelaide, 1982), volume 1036 of Lecture Notes in Math., 392–397. Springer, Berlin, 1983.

39. [39]

Stefan Vo: The Steiner tree problem with hop constraints, Ann. Oper. Res. 86 (1999), 321–345. Advances in combinatorial optimization (London, 1996).

40. [40]

Ferdinand Zaubzer: Lagrangian relax-and-cut and hybrid methods for the bounded diameter and the hop constrained minimum spanning tree problems, Master’s thesis, Vienna University of Technology, Institute of Computer Graphics and Algorithms, 2008, supervised by G. Raidl and M. Gruber.

## Author information

Authors

### Corresponding author

Correspondence to Omer Angel.

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.

## Rights and permissions

Reprints and Permissions

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

• Revised:

• Published:

• Issue Date:

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