Path-Reporting Distance Oracles with Linear Size

Given an undirected weighted graph, an (approximate) distance oracle is a data structure that can (approximately) answer distance queries. A {\em Path-Reporting Distance Oracle}, or {\em PRDO}, is a distance oracle that must also return a path between the queried vertices. Given a graph on $n$ vertices and an integer parameter $k\ge 1$, Thorup and Zwick \cite{TZ01} showed a PRDO with stretch $2k-1$, size $O(k\cdot n^{1+1/k})$ and query time $O(k)$ (for the query time of PRDOs, we omit the time needed to report the path itself). Subsequent works \cite{MN06,C14,C15} improved the size to $O(n^{1+1/k})$ and the query time to $O(1)$. However, these improvements produce distance oracles which are not path-reporting. Several other works \cite{ENW16,EP15} focused on small size PRDO for general graphs, but all known results on distance oracles with linear size suffer from polynomial stretch, polynomial query time, or not being path-reporting. In this paper we devise the first linear size PRDO with poly-logarithmic stretch and low query time $O(\log\log n)$. More generally, for any integer $k\ge 1$, we obtain a PRDO with stretch at most $O(k^{4.82})$, size $O(n^{1+1/k})$, and query time $O(\log k)$. In addition, we can make the size of our PRDO as small as $n+o(n)$, at the cost of increasing the query time to poly-logarithmic. For unweighted graphs, we improve the stretch to $O(k^2)$. We also consider {\em pairwise PRDO}, which is a PRDO that is only required to answer queries from a given set of pairs ${\cal P}$. An exact PRDO of size $O(n+|{\cal P}|^2)$ and constant query time was provided in \cite{EP15}. In this work we dramatically improve the size, at the cost of slightly increasing the stretch. Specifically, given any $\epsilon>0$, we devise a pairwise PRDO with stretch $1+\epsilon$, constant query time, and near optimal size $n^{o(1)}\cdot (n+|{\cal P}|)$.