Shortest Beer Path Queries in Interval Graphs
Rathish Das, Meng He, Eitan Kondratovsky, J. Ian Munro, Anurag Murty, Naredla, Kaiyu Wu

TL;DR
This paper introduces efficient data structures for answering shortest beer path and beer distance queries in interval graphs, optimizing space and query time with theoretical bounds and trade-offs.
Contribution
It presents novel space-efficient representations for interval graphs supporting fast beer path and distance queries, including tight lower bounds.
Findings
Representation of unweighted interval graphs using $2n \log n + O(n) + O(|B|\log n)$ bits.
Answering beer distance queries in $O(\log^\varepsilon n)$ time.
Answering shortest beer path queries in $O(\log^\varepsilon n + d)$ time.
Abstract
Our interest is in paths between pairs of vertices that go through at least one of a subset of the vertices known as beer vertices. Such a path is called a beer path, and the beer distance between two vertices is the length of the shortest beer path. We show that we can represent unweighted interval graphs using bits where is the number of beer vertices. This data structure answers beer distance queries in time for any constant and shortest beer path queries in time, where is the beer distance between the two nodes. We also show that proper interval graphs may be represented using bits to support beer distance queries in time for any and shortest beer path queries in time. All of these results also have time-space…
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