Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T14:19:01.620Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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