Path-Reporting Distance Oracles for Vertex-Labeled Graphs

2026-04-29

AI Summary

Existing distance oracles based on vertex-labeling spanners commonly suffer from large stretch factors and an inability to report actual paths. This work proposes the first vertex-labeling spanner-based distance oracle that supports path reporting, integrating hierarchical cover sets, tree embeddings, and sparse adjacency representations to simultaneously achieve efficient query performance, path reconstructability, and high approximation accuracy. The main contributions are two novel oracles: the first achieves a stretch of $(4k-5)(1+\varepsilon)$ while enabling path reporting; the second attains the theoretically optimal stretch of $2k-1$, with query time $O(\ell^{1/k}\log n)$ and space complexity $O(k n \ell^{1/k})$.

Abstract

Let $G=(V,E)$ be a weighted undirected graph, with $n$ vertices. A distance oracle is a data structure that can quickly answer distance queries, with some stretch factor. A seminal work of \cite{TZ01}, given an integer $k\ge 1$, provides such an oracle with stretch $2k-1$, query time $O(k)$, and size $O(k\cdot n^{1+1/k})$. Furthermore, this oracle can also report a path in $G$ corresponding to the returned distance. In this paper we focus on vertex-labeled graphs, in which each vertex is given a label from a set $L$ of size $\ell$. A {\em vertex-label distance oracle} answers queries of the form $(v,λ)$, where $v\in V$ and $λ\in L$, by reporting (an approximation to) the distance from $v$ to the closest vertex of label $λ$. Following \cite{HLWY11}, it was shown in \cite{C12} that for any integer $k> 1$, there exists a vertex-label distance oracle with stretch $4k-5$, query time $O(k)$, and size $O(k\cdot n\cdot \ell^{1/k})$. This state-of-the-art result suffers from two main drawbacks: The stretch is roughly a factor of 2 larger than in \cite{TZ01}, and it is not path-reporting. We address these concerns in this work, and provide the following results: First, we devise a {\em path-reporting} vertex-label distance oracle, at the cost of a slight increase in stretch and size. For any constant $0<ε<1$, our oracle has stretch $(4k-5)\cdot(1+ε)$, query time $O(k)$, and size $O(n^{1+o(1)}\cdot \ell^{1/k})$. Second, we show how to improve the stretch to the optimal $2k-1$, at the cost of mildly increasing the query time. Specifically, we devise a vertex-label distance oracle with stretch $2k-1$, query time $O(\ell^{1/k}\cdot\log n)$, and size $O(k\cdot n\cdot \ell^{1/k})$. \end{itemize}

Problem

• vertex-labeled graphs
• distance oracles
• path reporting
• stretch factor
• nearest labeled vertex

Innovation

• path-reporting
• vertex-labeled graphs
• distance oracles
• stretch factor
• approximate shortest paths