Shortest Beer Path Queries based on Graph Decomposition
Tesshu Hanaka, Hirotaka Ono, Kunihiko Sadakane, Kosuke Sugiyama

TL;DR
This paper introduces new graph indexing structures based on decomposition techniques to efficiently answer shortest beer path queries in general graphs, extending previous work on special graph classes.
Contribution
It proposes novel indexing data structures for beer path queries in general graphs using tree and triconnected component decompositions, achieving optimal performance for series-parallel graphs.
Findings
Indexes with size O(m+nr^2) based on triconnected decomposition.
Query time for beer distance is O(α(m)).
Achieves optimal performance for series-parallel graphs.
Abstract
Given a directed edge-weighted graph with beer vertices , a beer path between two vertices and is a path between and that visits at least one beer vertex in , and the beer distance between two vertices is the shortest length of beer paths. We consider \emph{indexing problems} on beer paths, that is, a graph is given a priori, and we construct some data structures (called indexes) for the graph. Then later, we are given two vertices, and we find the beer distance or beer path between them using the data structure. For such a scheme, efficient algorithms using indexes for the beer distance and beer path queries have been proposed for outerplanar graphs and interval graphs. For example, Bacic et al. (2021) present indexes with size for outerplanar graphs and an algorithm using them that answers the beer distance between given two vertices…
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