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On the Length of a Random Minimum Spanning Tree

Part of: Graph theory

Published online by Cambridge University Press:  23 January 2015

COLIN COOPER ,
ALAN FRIEZE ,
NATE INCE ,
SVANTE JANSON  and
JOEL SPENCER
COLIN COOPER
Affiliation:
Department of Computer Science, King's College, University of London, London WC2R 2LS, UK (e-mail: colin.cooper@kcl.ac.uk)
ALAN FRIEZE
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA15217, USA (e-mail: alan@random.math.cmu.edu, incenate@gmail.com)
NATE INCE
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA15217, USA (e-mail: alan@random.math.cmu.edu, incenate@gmail.com)
SVANTE JANSON
Affiliation:
Department of Mathematics, Uppsala University, SE-75310 Uppsala, Sweden (e-mail: svante@math.uu.se)
JOEL SPENCER
Affiliation:
Courant Institute, New York, NY 10012, USA (e-mail: spencer@cims.nyu.edu)
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Abstract

We study the expected value of the length Ln of the minimum spanning tree of the complete graph Kn when each edge e is given an independent uniform [0, 1] edge weight. We sharpen the result of Frieze [6] that limn→∞$\mathbb{E}$(Ln) = ζ(3) and show that

$$ \mathbb{E}(L_n)=\zeta(3)+\frac{c_1}{n}+\frac{c_2+o(1)}{n^{4/3}}, $$
where c1, c2 are explicitly defined constants.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C85: Graph algorithms
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 1: Oberwolfach Special Issue Part 2 , January 2016 , pp. 89 - 107
Copyright
Copyright © Cambridge University Press 2015 

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References

[1]Alon, N. and Spencer, J. H. (2008) The Probabilistic Method, third edition, Wiley.CrossRefGoogle Scholar
[2]Beveridge, A., Frieze, A. M. and McDiarmid, C. J. H. (1998) Minimum length spanning trees in regular graphs. Combinatorica 18 311–333.CrossRefGoogle Scholar
[3]Bollobás, B. (2001) Random Graphs, second edition, Cambridge University Press.Google Scholar
[4]Fill, J. A. and Steele, J. M. (2005) Exact expectations of minimal spanning trees for graphs with random edge weights. In Stein's Method and Applications, Singapore University Press, pp. 169–180.CrossRefGoogle Scholar
[5]Flaxman, A. (2007) The lower tail of the random minimum spanning tree. Electron. J. Combin. 14 N3.Google Scholar
[6]Frieze, A. M. (1985) On the value of a random minimum spanning tree problem. Discrete Appl. Math. 10 47–56.CrossRefGoogle Scholar
[7]Frieze, A. M. and McDiarmid, C. J. H. (1989) On random minimum length spanning trees. Combinatorica 9 363–374.CrossRefGoogle Scholar
[8]Frieze, A. M., Ruszinkó, M. and Thoma, L. (2000) A note on random minimum length spanning trees. Electron. J. Combin. 7 R41.CrossRefGoogle Scholar
[9]Gamarnik, D. (2005) The expected value of random minimal length spanning tree of a complete graph. In Proc. Sixteenth Annual ACM–SIAM Symposium on Discrete Algorithms: SODA 2005, ACM, pp. 700–704.Google Scholar
[10]Janson, S. (1993) Multicyclic components in a random graph process. Random Struct. Alg. 4 71–84.Google Scholar
[11]Janson, S. (1995) The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph. Random Struct. Alg. 7 337–355.CrossRefGoogle Scholar
[12]Janson, S. (2007) Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas. Probab. Surveys 3 80–145.Google Scholar
[13]Janson, S. and Chassaing, P. (2004) The center of mass of the ISE and the Wiener index of trees. Electron. Comm. Probab. 9 178–187.Google Scholar
[14]Janson, S., Knuth, D. E., Łuczak, T. and Pittel, B. (1993) The birth of the giant component. Random Struct. Alg. 3 233–358.CrossRefGoogle Scholar
[15]Janson, S. and Louchard, G. (2007) Tail estimates for the Brownian excursion area and other Brownian areas. Electron. J. Probab. 12 1600–1632.CrossRefGoogle Scholar
[16]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley.CrossRefGoogle Scholar
[17]Janson, S. and Spencer, J. (2007) A point process describing the component sizes in the critical window of the random graph evolution. Combin. Probab. Comput. 16 631–658.CrossRefGoogle Scholar
[18]Li, W. and Zhang, X. (2009) On the difference of expected lengths of minimum spanning trees. Combin. Probab. Comput. 18 423–434.CrossRefGoogle Scholar
[19]Louchard, G. (1984) Kac's formula, Lévy's local time and Brownian excursion. J. Appl. Probab. 21 479–499.CrossRefGoogle Scholar
[20]Louchard, G. (1984) The Brownian excursion area: A numerical analysis. Comput. Math. Appl. 10 413–417. Erratum: Comput. Math. Appl. A 12 (1986) 375.Google Scholar
[21]Nishikawa, J., Otto, P. T. and Starr, C. (2012) Polynomial representation for the expected length of minimal spanning trees. Pi Mu Epsilon J. 13 357–365.Google Scholar
[22]NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/Google Scholar
[23]Penrose, M. (1998) Random minimum spanning tree and percolation on the n-cube. Random Struct. Alg. 12 63–82.Google Scholar
[24]Read, N. (2005) Minimum spanning trees and random resistor networks in d dimensions. Phys. Rev. E 72 036114.Google Scholar
[25]Rényi, A. (1959) Some remarks on the theory of trees. Publ. Math. Inst. Hungar. Acad. Sci. 4 73–85.Google Scholar
[26]Spencer, J. (1997) Enumerating graphs and Brownian motion. Comm. Pure Appl. Math. 50 291–294.Google Scholar
[27]Steele, J. M. (1987) On Frieze's ζ(3) limit for lengths of minimal spanning trees. Discrete Appl. Math. 18 99–103.CrossRefGoogle Scholar
[28]Steele, J. M. (2002) Minimum spanning trees for graphs with random edge lengths. In Mathematics and Computer Science II: Algorithms, Trees, Combinatorics and Probabilities (Chauvin, B.et al., eds), Springer, pp. 223–245.Google Scholar
[29]Wästlund, J. (2009) An easy proof of the ζ(2) limit in the random assignment problem. Electron. Comm. Probab. 14 261–269.Google Scholar
[30]Wright, E. M. (1977) The number of connected sparsely edged graphs. J. Graph Theory 1 317–330.CrossRefGoogle Scholar