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Size and Weight of Shortest Path Trees with Exponential Link Weights

Published online by Cambridge University Press:  15 August 2006

REMCO VAN DER HOFSTAD ,
GERARD HOOGHIEMSTRA  and
PIET VAN MIEGHEM
REMCO VAN DER HOFSTAD
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (e-mail: rhofstad@win.tue.nl)
GERARD HOOGHIEMSTRA
Affiliation:
Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, PO Box 5031, 2600 GA Delft, The Netherlands (e-mail: G.Hooghiemstra@ewi.tudelft.nl and P.VanMieghem@ewi.tudelft.nl)
PIET VAN MIEGHEM
Affiliation:
Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, PO Box 5031, 2600 GA Delft, The Netherlands (e-mail: G.Hooghiemstra@ewi.tudelft.nl and P.VanMieghem@ewi.tudelft.nl)
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Abstract

We derive the distribution of the number of links and the average weight for the shortest path tree (SPT) rooted at an arbitrary node to $m$ uniformly chosen nodes in the complete graph of size $N$ with i.i.d. exponential link weights. We rely on the fact that the full shortest path tree to all destinations (ie, $m=N-1$) is a uniform recursive tree to derive a recursion for the generating function of the number of links of the SPT, and solve this recursion exactly.

The explicit form of the generating function allows us to compute the expectation and variance of the size of the subtree for all $m$. We also obtain exact expressions for the average weight of the subtree.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 15 , Issue 6 , November 2006 , pp. 903 - 926
Copyright
2006 Cambridge University Press

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