76 research outputs found
Degree-constrained spanners for multidimensional grids
AbstractA spanning subgraph S = (V, Eβ²) of a connected simple graph G = (V, E) is a f (x) -spanner if for any pair of nodes u and v, ds(u, v) β©½ f (dG(u, v)) where dG and ds are the usual distance functions in graphs G and S, respectively. The delay of the f (x) -spanner is f(x) β x. We construct four spanners with maximum degree 4 for infinite d-dimensional grids with delays 2d β 4, 2βd2β + 2[(d β 2)/4] + 2, 2β(d β 6)/8β + 4βd + 1)/4β+ 6, and β(βd/2β + 1)/ (1 + 1)rl + 2β d2β + 21 + 2. All of these constructions can be modified to produce spanners of finite (d-dimensional grids with essentially the same delay. We also construct a (5d + 4 + x) -spanner with maximum degree 3 for infinite d-dimensional grids. This construction can be used to produce spanners of finite d-dimensional grids where all dimensions are even with the same delay. We prove an Ξ©(d) lower bound for the delay of maximum degree 3 or 4 spanners of finite or infinite d-dimensional grids. For the particular cases of infinite 3- and 4-dimensional grids, we construct (6 + x) -spanners and (14 + x) -spanners, respectively. The former can be modified to construct a (6 + x) -spanner of a finite 3-dimensional grid where all dimensions are even or where all dimensions are odd and a (8 + x) -spanner of a finite 3-dimensional grid otherwise. The latter yields (14 + x) -spanners of finite 4-dimensional grids where all dimensions are even
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
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