Distance Oracles for Stretch Less Than 2

We present distance oracles for weighted undirected graphs that return distances of stretch less than 2. For the realistic case of sparse graphs, our distance oracles exhibit a smooth three-way trade-off between space, stretch and query time --- a phenomenon that does not occur in dense graphs. In particular, for any positive integer t and for any 1 ≤ α ≤ n, our distance oracle is of size O(m + n2/α) and returns distances of stretch at most (1 + 2/t+1) in time O((αμ)t), where μ = 2m/n is the average degree of the graph. The query time can be further reduced to O((α + μ)t) at the expense of a small additive stretch.