Additive Spanners and Distance Oracles in Quadratic Time

Let G be an unweighted, undirected graph. An additive k-spanner of G is a subgraph H that approximates all distances between pairs of nodes up to an additive error of +k, that is, it satisfies dH(u, v) ≤ dG(u, v) + k for all nodes u, v, where d is the shortest path distance. We give a deterministic algorithm that constructs an additive O(1)-spanner with O ( n4/3 ) edges in O ( n2 ) time. This should be compared with the randomized Monte Carlo algorithm by Woodruff [ICALP 2010] giving an additive 6-spanner with O ( n4/3 log3 n ) edges in expected time O ( n2 log2 n ) . An (α, β)-approximate distance oracle for G is a data structure that supports the following distance queries between pairs of nodes in G. Given two nodes u, v it can in constant time compute a distance estimate d̃ that satisfies d ≤ d̃ ≤ αd + β where d is the distance between u and v in G. Sommer [ICALP 2016] gave a randomized Monte Carlo (2, 1)-distance oracle of size O ( n5/3 poly logn ) in expected time O ( n2 poly logn ) . As an application of the additive O(1)-spanner we improve the construction by Sommer [ICALP 2016] and give a Las Vegas (2, 1)distance oracle of size O ( n5/3 ) in time O ( n2 ) . This also implies an algorithm that in O ( n2 ) time gives approximate distance for all pairs of nodes in G improving on the O ( n2 logn ) algorithm by Baswana and Kavitha [SICOMP 2010]. 1998 ACM Subject Classification G.2.2 Graph Theory