Abstract:
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This work deals with convergence theorems and bounds on the
cost of several layout measures for lattice graphs, random
lattice graphs and sparse random geometric graphs. For full
square lattices, we give optimal layouts for the problems
still open. Our convergence theorems can be viewed as an
analogue of the Beardwood, Halton and Hammersley theorem for
the Euclidian TSP on random points in the $d$-dimensional
cube. As the considered layout measures are
non-subadditive, we use percolation theory to obtain our
results on random lattices and random geometric graphs. In
particular, we deal with the subcritical regimes on these
class of graphs. |