eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-11-28
35:1
35:17
10.4230/LIPIcs.ISAAC.2023.35
article
Shortest Beer Path Queries in Digraphs with Bounded Treewidth
Gudmundsson, Joachim
1
Sha, Yuan
1
The University of Sydney, Australia
A beer digraph G is a real-valued weighted directed graph where some of the vertices have beer stores. A beer path from a vertex u to a vertex v in G is a path in G from u to v that visits at least one beer store.
In this paper we consider the online shortest beer path query in beer digraphs with bounded treewidth t. Assume that a tree decomposition of treewidth t on a beer digraph with n vertices is given. We show that after O(t³n) time preprocessing on the beer digraph, (i) a beer distance query can be answered in O(t³α(n)) time, where α(n) is the inverse Ackermann function, and (ii) a shortest beer path can be reported in O(t³α(n)L) time, where L is the number of edges on the path. In the process we show an improved O(t³α(n)L) time shortest path query algorithm, compared with the currently best O(t⁴α(n)L) time algorithm [Chaudhuri & Zaroliagis, 2000].
We also consider queries in a dynamic setting where the weight of an edge in G can change over time. We show two data structures. Assume t is constant and let β be any constant in (0,1). The first data structure uses O(n) preprocessing time, answers a beer distance query in O(α(n)) time and reports a shortest beer path in O(α(n) L) time. It can be updated in O(n^β) time after an edge weight change. The second data structure has O(n) preprocessing time, answers a beer distance query in O(log n) time, reports a shortest beer path in O(log n + L) time, and can be updated in O(log n) time after an edge weight change.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol283-isaac2023/LIPIcs.ISAAC.2023.35/LIPIcs.ISAAC.2023.35.pdf
Graph algorithms
Shortest Path
Data structures
Bounded treewidth