Degree-constrained spanners for multidimensional grids

AbstractA 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, ds(u, v) ⩽ 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) − x. We construct four spanners with maximum degree 4 for infinite d-dimensional grids with delays 2d − 4, 2⌈d2⌉ + 2[(d − 2)/4] + 2, 2⌈(d − 6)/8⌉ + 4⌈d + 1)/4⌉+ 6, and ⌈(⌈d/2⌉ + 1)/ (1 + 1)rl + 2⌈ d2⌉ + 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 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

Spanners for Geometric Intersection Graphs

Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.Comment: 16 pages, 5 figures, Late

Sparse geometric graphs with small dilation

Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For any k, we also construct planar n-point sets for which any geometric graph with n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position