Approximate Distance Oracles with Improved Query Time

Given an undirected graph $G$ with $m$ edges, $n$ vertices, and non-negative edge weights, and given an integer $k\geq 2$, we show that a $(2k-1)$-approximate distance oracle for $G$ of size $O(kn^{1 + 1/k})$ and with $O(\log k)$ query time can be constructed in $O(\min{kmn^{1/k},\sqrt km + kn^{1 + c/\sqrt k}})$ time for some constant $c$. This improves the $O(k)$ query time of Thorup and Zwick. Furthermore, for any $0 < \epsilon \leq 1$, we give an oracle of size $O(kn^{1 + 1/k})$ that answers $((2 + \epsilon)k)$-approximate distance queries in $O(1/\epsilon)$ time. At the cost of a $k$-factor in size, this improves the $128k$ approximation achieved by the constant query time oracle of Mendel and Naor and approaches the best possible tradeoff between size and stretch, implied by a widely believed girth conjecture of Erd\H{o}s. We can match the $O(n^{1 + 1/k})$ size bound of Mendel and Naor for any constant $\epsilon > 0$ and $k = O(\log n/\log\log n)$.