Combinatorica

, Volume 32, Issue 1, pp 1–33 | Cite as

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

  • Omer Angel
  • Abraham D. Flaxman
  • David B. Wilson
Original Paper

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 \(2^{1/(2^k - 1)}\).

Mathematics Subject Classification (2010)

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

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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.MathSciNetCrossRefGoogle Scholar
  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.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    N. R. Achuthan and L. Caccetta: Minimum weight spanning trees with bounded diameter, Australas. J. Combin. 5 (1992), 261–276.MathSciNetMATHGoogle Scholar
  4. [4]
    N. R. Achuthan and L. Caccetta: Addendum: “Minimum weight spanning trees with bounded diameter”, Australas. J. Combin. 8 (1993), 279–281.MathSciNetMATHGoogle Scholar
  5. [5]
    Ayman Abdalla and Narsingh Deo: Random-tree diameter and the diameterconstrained MST, Int. J. Comput. Math. 79(6) (2002), 651–663.MathSciNetMATHCrossRefGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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.MathSciNetMATHCrossRefGoogle Scholar
  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).MathSciNetMATHCrossRefGoogle Scholar
  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.CrossRefGoogle Scholar
  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.MathSciNetCrossRefGoogle Scholar
  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.MathSciNetMATHCrossRefGoogle Scholar
  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.CrossRefGoogle Scholar
  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.MathSciNetMATHCrossRefGoogle Scholar
  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.MATHCrossRefGoogle Scholar
  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.Google Scholar
  16. [16]
    A. M. Frieze and C. J. H. McDiarmid: On random minimum length spanning trees, Combinatorica 9(4) (1989), 363–374.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    A. M. Frieze: On the value of a random minimum spanning tree problem, Discrete Appl. Math. 10(1) (1985), 47–56.MathSciNetMATHCrossRefGoogle Scholar
  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.MATHGoogle Scholar
  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.MathSciNetMATHCrossRefGoogle Scholar
  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.MATHCrossRefGoogle Scholar
  21. [21]
    L. Gouveia: Multicommodity flow models for spanning trees with hop constraints. European Journal of Operational Research 95, 178–190, 22 November 1996.MATHCrossRefGoogle Scholar
  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.Google Scholar
  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.Google Scholar
  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.Google Scholar
  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.MathSciNetMATHCrossRefGoogle Scholar
  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).MathSciNetMATHCrossRefGoogle Scholar
  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.Google Scholar
  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.Google Scholar
  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.Google Scholar
  30. [30]
    Guy Kortsarz and David Peleg: Approximating the weight of shallow Steiner trees, Discrete Appl. Math. 93(2–3) (1999), 265–285.MathSciNetMATHCrossRefGoogle Scholar
  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.Google Scholar
  32. [32]
    Jrme Monnot: The maximum f-depth spanning tree problem, Inform. Process. Lett. 80(4) (2001), 179–187.MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    Mathew Penrose: Random Geometric Graphs, volume 5 of Oxford Studies in Probability, Oxford University Press, Oxford, 2003.MATHCrossRefGoogle Scholar
  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.Google Scholar
  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.Google Scholar
  36. [36]
    A. Rnyi and G. Szekeres: On the height of trees, J. Austral. Math. Soc. 7 (1967), 497–507.MathSciNetCrossRefGoogle Scholar
  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.MathSciNetMATHCrossRefGoogle Scholar
  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.CrossRefGoogle Scholar
  39. [39]
    Stefan Vo: The Steiner tree problem with hop constraints, Ann. Oper. Res. 86 (1999), 321–345. Advances in combinatorial optimization (London, 1996).MathSciNetCrossRefGoogle Scholar
  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.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2012

Authors and Affiliations

  • Omer Angel
    • 1
  • Abraham D. Flaxman
    • 2
  • David B. Wilson
    • 3
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Institute for Health Metrics and EvaluationUniversity of WashingtonSeattleUSA
  3. 3.RedmondUSA

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