Shortest Beer Path Queries in Outerplanar Graphs

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(α (n)) time (where α (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).