Shortest Beer Path Queries in Outerplanar Graphs

Abstract

A beer graph is an undirected graph G, in which each edge has a positive weight and some vertices have a beer store. A 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. Note that the running time for (ii) does not depend on the number of vertices of G. Both results are optimal, even when G is a beer tree (i.e., a beer graph whose underlying graph is a tree).