# Beardwood–Halton–Hammersley theorem for stationary ergodic sequences: A counterexample

@article{Arlotto2016BeardwoodHaltonHammersleyTF, title={Beardwood–Halton–Hammersley theorem for stationary ergodic sequences: A counterexample}, author={Alessandro Arlotto and J. Michael Steele}, journal={Annals of Applied Probability}, year={2016}, volume={26}, pages={2141-2168} }

We construct a stationary ergodic process X1;X2;::: such that each Xt has the uniform distribution on the unit square and the length Ln of the shortest path through the points X1;X2;:::;Xn is not asymptotic to a constant times the square root of n. In other words, we show that the Beardwood, Halton and Hammersley theorem does not extend from the case of independent uniformly distributed random variables to the case of stationary ergodic sequences with uniform marginal distributions.

## 7 Citations

PROBABILITY DISTRIBUTION OF THE LENGTH OF THE SHORTEST TOUR BETWEEN A FEW RANDOM POINTS : A SIMULATION STUDY

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Inspired by an application in the field of on-demand public transportation, we perform a Monte Carlo simulation study on the probability distribution of the length of Traveling-Salesman-Problem (TSP)…

PROBABILITY DISTRIBUTION OF THE LENGTH OF THE SHORTEST TOUR BETWEEN A FEW RANDOM POINTS: A SIMULATION STUDY

- Mathematics, Computer Science2018 Winter Simulation Conference (WSC)
- 2018

It is shown that, under certain assumptions on the shape of the region and the probability distribution of locations, the length of the TSP tour is well-approximated by a normal distribution, even for as few as five locations.

Traveling in randomly embedded random graphs

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A threshold for a pair of points to be connected by a geodesic of length arbitrarily close to their Euclidean distance is shown, and the minimum length Traveling Salesperson Tour is analyzed, extending the Beardwood-Halton-Hammersley theorem to this setting.

Bounds for the traveling salesman paths of two-dimensional modular lattices

- Mathematics, Computer ScienceJ. Comb. Optim.
- 2017

Tight upper and lower bounds are presented for the traveling salesman path through the points of two-dimensional modular lattices based on earlier work on shortest vectors in lattices as well as on the strong convergence of Jacobi–Perron type algorithms.

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This paper presents an overview of the literature on continuous approximation models in the context of distribution management. It first describes the seminal contributions of Beardwood, Halton and…

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The world needs around 150 Pg of negative carbon emissions to mitigate climate change. Global soils may provide a stable, sizeable reservoir to help achieve this goal by sequestering atmospheric…

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The world needs around 150 Pg of negative carbon emissions to mitigate climate change. Global soils may provide a stable, sizeable reservoir to help achieve this goal by sequestering atmospheric…

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