Path-Reporting Distance Oracles with Logarithmic Stretch and Linear Size

Given an undirected graph G = (V, E, 𝐰) on n vertices with positive edge weights, a distance oracle is a space-efficient data structure that answers pairwise distance queries in fast runtime. The quality of a distance oracle is measured by three parameters: space, query time, and stretch. In a landmark paper by [Thorup and Zwick, 2001], they showed that for any integer parameter k ≥ 1, there exists a distance oracle with size O(kn^{1+1/k}), O(k) query time, and (2k-1)-stretch error on the approximate distances. After that, there has been a line of subsequent improvements which culminated in the optimal trade-off of O(n^{1+1/k}) space, O(1) query time, and (2k-1)-stretch [Chechik, 2015]. However, these line of constructions did not require that the distance oracle is capable of printing an actual path besides an approximate distance estimate, and there has been a performance gap between path-reporting distance oracles and ones that are not path-reporting. It is known that the earliest construction by [Thorup and Zwick, 2001] is path-reporting, but the parameters are worse by a factor of k. In a later construction by [Wulff-Nilsen, 2013], the query time was improved from O(k) to O(log k). Better trade-offs were discovered in [Elkin and Pettie, 2015] where the authors broke the O(kn^{1+1/k}) space barrier and achieved O(n^{1+1/k}log k) space with O(log k) query time, but their stretch was blown up to a polynomial O(k^{log_{4/3}7}); they also gave an alternative choice of O(n^{1+1/k}) space which is optimal, and O(k)-stretch which is also optimal up to a constant factor, but their query time rose exponentially to O(n^ε). In a recent work [Elkin and Shabat, 2023], the authors obtained significant improvements of O(n^{1+1/k}log k) space, O(k)-stretch, and O(log log k) query time, or O(n^{1+1/k}) space, O(klog k)-stretch, and O(log log k) query time. All the above constructions of path-reporting distance oracles share a common barrier; that is, they could not achieve optimal space O(n^{1+1/k}) and stretch O(k) simultaneously within logarithmic query time; for example, in the natural regime where k = ⌈log n⌉, previous distance oracles had to pay an extra factor of log log n either in the space or stretch. As our result, we bypass this barrier by a new construction of path-reporting distance oracles with O(n^{1+1/k}) space and O(k)-stretch and O(log log k) query time.