Distance Oracles beyond the Thorup-Zwick Bound

We give the first improvement to the space/approximation trade-off of distance oracles since the seminal result of Thorup and Zwick [STOC´01]. For unweighted graphs, our distance oracle has size $O(n^{5/3}) = O(n^{1.66\cdots})$ and, when queried about vertices at distance $d$, returns a path of length $2d+1$. For weighted graphs with $m=n^2/\alpha$ edges, our distance oracle has size $O(n^2 / \sqrt[3]{\alpha})$ and returns a factor 2 approximation. Based on a plausible conjecture about the hardness of set intersection queries, we show that a 2-approximate distance oracle requires space $\tOmega(n^2 / \sqrt{\alpha})$. For unweighted graphs, this implies a $\tOmega(n^{1.5})$ space lower bound to achieve approximation $2d+1$.