Poisson approximations for functionals of random trees
Proceedings of the seventh international conference on Random structures and algorithms
Scaling of multicast trees: comments on the Chuang-Sirbu scaling law
Proceedings of the conference on Applications, technologies, architectures, and protocols for computer communication
On power-law relationships of the Internet topology
Proceedings of the conference on Applications, technologies, architectures, and protocols for computer communication
On the efficiency of multicast
IEEE/ACM Transactions on Networking (TON)
On the covariance of the level sizes in random recursive trees
Random Structures & Algorithms
One, Two and Three Times log n/n for Paths in a Complete Graph with Random Weights
Combinatorics, Probability and Computing
On the Value of a Random Minimum Weight Steiner Tree
Combinatorica
The weight of the shortest path tree
Random Structures & Algorithms
The longest minimum-weight path in a complete graph
Combinatorics, Probability and Computing
Sampling networks by the union of m shortest path trees
Computer Networks: The International Journal of Computer and Telecommunications Networking
The blind passenger and the assignment problem
Combinatorics, Probability and Computing
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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.