The Longest Minimum-Weight Path in a Complete Graph

@article{AddarioBerry2009TheLM,
  title={The Longest Minimum-Weight Path in a Complete Graph},
  author={Louigi Addario-Berry and Nicolas Broutin and G{\'a}bor Lugosi},
  journal={Combinatorics, Probability and Computing},
  year={2009},
  volume={19},
  pages={1 - 19}
}
We consider the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d. exponentially distributed random variables. We show that the longest of these minimum-weight paths has about α* log n edges, where α* ≈ 3.5911 is the unique solution of the equation α log α − α = 1. This answers a question posed by Janson [8]. 
Diameter of the Stochastic Mean-Field Model of Distance
TLDR
This work considers the complete graph 𝜅n on n vertices with exponential mean n edge lengths and shows that max i,j∈[n] Cij − 3 logn converges in distribution to some limiting random variable that can be identified via a maximization procedure on a limiting infinite random structure.
All-Pairs Shortest Paths in O(n²) Time with High Probability
TLDR
The algorithm is a variant of the dynamic all-pairs shortest paths algorithm of Demetrescu and Italiano and relies on a proof that the number of \emph{locally shortest paths} in such randomly weighted graphs is $O(n^2)$, in expectation and with high probability.
All-Pairs Shortest Paths in O(n2) Time with High Probability
TLDR
The algorithm is a variant of the dynamic all-pairs shortest paths algorithm of Demetrescu and Italiano and relies on a proof that the number of \emph{locally shortest paths} in such randomly weighted graphs is $O(n^2)$, in expectation and with high probability.
Shortest-Weight Paths in Random Regular Graphs
TLDR
This paper establishes a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices.
Random Shortest Paths: Non-Euclidean Instances for Metric Optimization Problems
TLDR
This work proves structural properties of the random shortest path metrics generated in this way and applies these findings to analyze the approximation ratios of heuristics for matching, the traveling salesman problem (TSP), and the $$k$$k-median problem, as well as the running-time of the 2-opt heuristic for the TSP.
First passage percolation on random graphs with finite mean degrees
TLDR
This study continues the program initiated in [5] of showing that log n is the correct scaling for the hopcount under i.i.d. edge disorder, and proves convergence in distribution of an appropriately centered version.
Random Shortest Paths: Non-euclidean Instances for Metric Optimization Problems
Probabilistic analysis for metric optimization problems has mostly been conducted on random Euclidean instances, but little is known about metric instances drawn from distributions other than the
Random Graphs and Complex Networks
  • R. Hofstad
  • Computer Science
    Cambridge Series in Statistical and Probabilistic Mathematics
  • 2016
TLDR
This chapter explains why many real-world networks are small worlds and have large fluctuations in their degrees, and why Probability theory offers a highly effective way to deal with the complexity of networks, and leads us to consider random graphs.
Why are certain polynomials hard?: A look at non-commutative, parameterized and homomorphism polynomials
TLDR
This thesis tries to answer the question why specific polynomials have no small suspected arithmetic circuits and introduces a new framework for arithmetic circuits, similar to fixed parameter tractability in the boolean setting.
Random trees, graphs and recursive partitions
Je presente dans ce memoire mes travaux sur les limites d'echelle de grandes structures aleatoires. Il s'agit de decrire les structures combinatoires dans la limite des grandes tailles en prenant un
...
1
2
...

References

SHOWING 1-10 OF 24 REFERENCES
The weight of the shortest path tree
The minimal weight of the shortest path tree in a complete graph with independent and exponential (mean 1) random link weights is shown to converge to a Gaussian distribution. We prove a conditional
The Weight and Hopcount of the Shortest Path in the Complete Graph with Exponential Weights
TLDR
The joint distribution of the pair (HN, WN) is considered and derive, after proper scaling, the joint limiting distribution and it is shown that HN and WN, properly scaled, are asymptotically independent.
One, Two and Three Times log n/n for Paths in a Complete Graph with Random Weights
  • S. Janson
  • Mathematics, Computer Science
    Combinatorics, Probability and Computing
  • 1999
TLDR
It is shown that the minimal weights of paths between two points in a complete graph with random weights on the edges is log n/n for two given points, and that the maximum if one point is fixed and the other varies is 2 logn/n.
Size and Weight of Shortest Path Trees with Exponential Link Weights
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 $N$
A lower bound for the critical probability in a certain percolation process
Consider a lattice L in the Cartesian plane consisting of all points (x, y) such that either x or y is an integer. Points with integer coordinates (positive, negative, or zero) are called vertices
Branching processes in the analysis of the heights of trees
  • L. Devroye
  • Computer Science, Mathematics
    Acta Informatica
  • 2004
TLDR
It is shown how the theory of branching processes can be applied in the analysis of the expected height of random trees under various models of randomization.
Note on the Heights of Random Recursive Trees and Random m-ary Search Trees
  • B. Pittel
  • Mathematics
    Random Struct. Algorithms
  • 1994
TLDR
A process of growing a random recursive tree Tn is studied and the sequence is shown to be a sequence of “snapshots” of a Crump–Mode branching process, which provides a short proof of Devroye's limit law for the height of a random m‐ary search tree.
The Probabilistic Method
TLDR
A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
Correlation inequalities on some partially ordered sets
We prove that increasing function on a finite distributive lattice are positively correlated by positive measures satisfying a suitable convexity property. Applications to Ising ferromagnets in an
A Remark on Stirling’s Formula
We shall prove Stirling’s formula by showing that for n=1, 2,… $$n! = \sqrt {2\pi } {n^{n + 1/2}}{e^{ - n}} \cdot {e^{{r_n}}}$$ (1) where rn satisfies the double inequality
...
1
2
3
...