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On the total length of the random minimal directed spanning tree

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

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).

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