Path-Reporting Distance Oracles with Linear Size

TL;DR

The first linear size PRDO with poly-logarithmic stretch and low query time $O(\log\log n) is devised, and the size is dramatically improved, at the cost of slightly increasing the stretch.

Abstract

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 $ε>0$, we devise a pairwise PRDO with stretch $1+ε$, constant query time, and near optimal size $n^{o(1)}\cdot (n+|{\cal P}|)$.

Figures (2)

• Figure 1: An illustration of the graph $\mathcal{H}_\kappa[p,l]$. The internal vertices of $\ddot{B}[p,l]$ are replaced by copies of $\mathcal{H}_{\kappa-1}[p',l]$, where $p'$ is the number of edges-labels in $\ddot{B}[p,l]$. An edge of $\ddot{B}[p,l]$ that had label $a$, and is from an even layer to an odd layer, is replaced by an edge that connects the $\pi(a)$-th input ports of the corresponding copies of $\mathcal{H}_{\kappa-1}[p',l]$ (input port are represented in the figure by a square shape). If the edge is from an odd layer to an even layer, the same happens for the output ports of these copies (represented by circular shape).
• Figure 2: Given an edge $e=(v_1,v_2)$ (colored orange in the figure), we consider the BFS trees $T^i_1$ and $T^{\delta-1-i}_2$ rooted at $v_1$ and $v_2$ respectively, up to distance $i$ and $\delta-1-i$ respectively. There are no cycles within these two trees, because of the girth guarantee. By regularity, we know that each vertex in these trees, except the leaves, has exactly $p$ children. Every path of length $\delta$ that passes through $e$, such as the blue path in the figure, is determined by a leaf of $T^i_1$ and a leaf of $T^{\delta-1-i}_2$.

Theorems & Definitions (45)

• Definition 1: Path-Reporting Pairwise Spanner
• Remark 2
• Theorem 3: ES23
• Theorem 4: ES23
• Definition 5
• Definition 6: Path-Reporting Prioritized Spanner
• Theorem 7: Theorem 6 in ES23
• Theorem 8: ES23
• Definition 9
• Theorem 10: TZ01WN13