Degree-constrained spanners for multidimensional grids

Abstract

A spanning subgraph S = ( V, E′) of a connected simple graph G = ( V, E) is a f (x) -spanner if for any pair of nodes u and v, d s ( u, v) ⩽ f ( d G ( u, v)) where d G and d s are the usual distance functions in graphs G and S, respectively. The delay of the f (x) -spanner is f(x) − x. We construct four spanners with maximum degree 4 for infinite d-dimensional grids with delays 2 d − 4, 2⌈ d 2 ⌉ + 2[(d − 2)/4] + 2, 2⌈(d − 6)/8⌉ + 4⌈d + 1)/4⌉+ 6, and ⌈(⌈d/2⌉ + 1)/ (1 + 1)rl + 2⌈ d 2 ⌉ + 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 (5 d + 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 Ω(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.