Shortest Beer Path Queries in Outerplanar Graphs

TL;DR

This paper introduces an efficient data structure for answering shortest beer path queries in outerplanar graphs, enabling rapid retrieval of path weights and paths with optimal preprocessing.

Contribution

It presents a linear-time preprocessing method for outerplanar beer graphs that supports fast shortest beer path queries, improving over previous approaches.

Findings

Preprocessing time is O(n) for outerplanar beer graphs.

Query time for shortest beer path weight is O(α(n)).

Query time for retrieving the shortest beer path is proportional to its length.

Abstract

A \emph{beer graph} is an undirected graph $G$, in which each edge has a positive weight and some vertices have a beer store. A \emph{beer path} between two vertices $u$ and $v$ in $G$ is any path in $G$ between $u$ and $v$ that visits at least one beer store. We show that any outerplanar beer graph $G$ with $n$ vertices can be preprocessed in $O(n)$ time into a data structure of size $O(n)$, such that for any two query vertices $u$ and $v$, (i) the weight of the shortest beer path between $u$ and $v$ can be reported in $O(\alpha(n))$ time (where $\alpha(n)$ is the inverse Ackermann function), and (ii) the shortest beer path between $u$ and $v$ can be reported in $O(L)$ time, where $L$ is the number of vertices on this path. Both results are optimal, even when $G$ is a beer tree (i.e., a beer graph whose underlying graph is a tree).