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On the total length of the random minimal directed spanning tree

Part of: Geometric probability and stochastic geometry Limit theorems Graph theory

Published online by Cambridge University Press:  01 July 2016

Mathew D. Penrose  and
Andrew R. Wade
Mathew D. Penrose*
Affiliation:
University of Bath
Andrew R. Wade*
Affiliation:
University of Bath
*
∗ Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.
∗ Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.
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Abstract

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In Bhatt and Roy's minimal directed spanning tree construction for a random, partially ordered set of points in the unit square, all edges must respect the ‘coordinatewise’ partial order and there must be a directed path from each vertex to a minimal element. We study the asymptotic behaviour of the total length of this graph with power-weighted edges. The limiting distribution is given by the sum of a normal component away from the boundary plus a contribution introduced by the boundary effects, which can be characterized by a fixed-point equation, and is reminiscent of limits arising in the probabilistic analysis of certain algorithms. As the exponent of the power weighting increases, the distribution undergoes a phase transition from the normal contribution being dominant to the boundary effects being dominant. In the critical case in which the weight is simple Euclidean length, both effects contribute significantly to the limit law.

Keywords

Spanning treenearest-neighbour graphweak convergencefixed-point equationphase transitionfragmentation process

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60F05: Central limit and other weak theorems
Secondary: 05C80: Random graphs 05C05: Trees
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 38 , Issue 2 , June 2006 , pp. 336 - 372
Copyright
Copyright © Applied Probability Trust 2006 

Footnotes

Presented at the ICMS Workshop on Spatial Stochastic Modelling with Applications to Communications Networks (Edinburgh, June 2004).

References

Bai, Z.-D., Lee, S. and Penrose, M. D. (2006). Rooted edges in a minimal directed spanning tree. Adv. Appl. Prob. 38, 1–30.CrossRefGoogle Scholar
Barndorff-Nielsen, O. and Sobel, M. (1966). On the distribution of the number of admissible points in a vector random sample. Theory Prob. Appl. 11, 249–269.CrossRefGoogle Scholar
Bertoin, J. and Gnedin, A. (2004). Asymptotic laws for nonconservative selfsimilar fragmentations. Electron. J. Prob. 9, 575–593.CrossRefGoogle Scholar
Berger, N. et al. (2003). Degree distribution of the FKP model. In Automata, Languages and Programming (Proc. 30th Internat. Colloquium, ICALP 2003, Eindhoven; Lecture Notes Comp. Sci. 2719), eds Baeten, J. C. M., Lenstra, J. K., Parrow, J. and Woeginger, G. J., Springer, Berlin, pp. 725–738.CrossRefGoogle Scholar
Bhatt, A. G. and Roy, R. (2004). On a random directed spanning tree. Adv. Appl. Prob. 36, 19–42.CrossRefGoogle Scholar
Hoare, C. A. R. (1961). Algorithm 64: Quicksort. Commun. Assoc. Comput. Mach. 4, 321.Google Scholar
Hwang, H.-K. (1998). Asymptotics of divide-and-conquer recurrences: Batcher's sorting algorithm and a minimum Euclidean matching heuristic. Algorithmica 22, 529–546.CrossRefGoogle Scholar
Kesten, H. and Lee, S. (1996). The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Prob. 6, 495–527.CrossRefGoogle Scholar
Kingman, J. F. C. (1993). Poisson Processes (Oxford Stud. Prob. 3). Clarendon Press, Oxford.Google Scholar
Neininger, R. and Rüschendorf, L. (2004). A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Prob. 14, 378–418.CrossRefGoogle Scholar
Penrose, M. (2003). Random Geometric Graphs (Oxford Stud. Prob. 5). Oxford University Press.CrossRefGoogle Scholar
Penrose, M. D. (2005). Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Prob. 33, 1945–1945.CrossRefGoogle Scholar
Penrose, M. D. and Wade, A. R. (2004). On the total length of the random minimal directed spanning tree. Preprint. Available at http://arxiv.org/abs/math.PR/0409201.Google Scholar
Penrose, M. D. and Wade, A. R. (2004). Random minimal directed spanning trees and Dickman-type distributions. Adv. Appl. Prob. 36, 691–714.CrossRefGoogle Scholar
Penrose, M. D. and Wade, A. R. (2006). Limit theory for the random on-line nearest-neighbour graph. Preprint. Available at http://arxiv.org/abs/math.PR/0603561.Google Scholar
Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Prob. 11, 1005–1041.CrossRefGoogle Scholar
Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Prob. 13, 277–303.CrossRefGoogle Scholar
Rodriguez-Iturbe, I. and Rinaldo, A. (1997). Fractal River Basins: Chance and Self-Organization. Cambridge University Press.Google Scholar
Rösler, U. (1992). A fixed point theorem for distributions. Stoch. Process. Appl. 42, 195–214.CrossRefGoogle Scholar
Rösler, U. and Rüschendorf, L. (2001). The contraction method for recursive algorithms. Algorithmica 29, 3–33.CrossRefGoogle Scholar
Seppäläinen, T. and Yukich, J. E. (2001). Large deviation principles for Euclidean functionals and other nearly additive processes. Prob. Theory Relat. Fields 120, 309–345.CrossRefGoogle Scholar
Steele, J. M. (1997). Probability Theory and Combinatorial Optimization. Society for Industrial and Applied Mathematics, Philadelphia, PA.CrossRefGoogle Scholar
Yukich, J. E. (1998). Probability Theory of Classical Euclidean Optimization Problems (Lecture Notes Math. 1675). Springer, Berlin.CrossRefGoogle Scholar