Abstract
We give a fully dynamic single-source shortest paths data structure for planar weighted digraphs with O˜(n4/5) worst-case update time and O(log2n) query time. Here, a single update can either change the graph by inserting or deleting an edge, or reset the source s of interest. All known non-trivial planarity-exploiting exact dynamic single-source shortest paths algorithms to date had polynomial query time. We then extend our approach, obtaining a data structure that can maintain a planar digraph under edge insertions and deletions, and is capable of returning the identifier of the strongly connected component of any query vertex. The worst-case update and query time bounds are the same as for our single-source distance oracle. To the best of our knowledge, this is the first fully dynamic strong-connectivity algorithm achieving both sublinear update time and polylogarithmic query time for an important class of digraphs.
Highlights
The dynamic shortest paths problem seeks for a data structure maintaining a graph under updates and supporting shortest path queries.1 Depending on the set of supported updates, we call such a graph data structure fully dynamic if both edge insertions and deletions are allowed, incremental if only edge insertions are supported, or decremental if only edge deletions are allowed
The lack of progress on obtaining an exact fully dynamic single-source shortest paths algorithms with O(n3− ) initialization time, O(m1− ) amortized update time and O(n1− ) query time at the same time can be explained by a matching lower bound conditional on the APSP conjecture [73]
In this paper we show the first exact dynamic single-source shortest paths algorithm for planar graphs with strongly sublinear update time and polylogarithmic query time
Summary
The dynamic shortest paths problem seeks for a data structure maintaining a graph under updates and supporting shortest path queries. Depending on the set of supported updates, we call such a graph data structure fully dynamic if both edge insertions and deletions are allowed, incremental if only edge insertions (or edge weight decreases) are supported, or decremental if only edge deletions (or weight increases) are allowed. The lack of progress on obtaining an exact fully dynamic single-source shortest paths algorithms with O(n3− ) initialization time, O(m1− ) amortized update time and O(n1− ) query time at the same time can be explained by a matching lower bound conditional on the (static) APSP conjecture [73] Breaking this barrier even in the partially dynamic setting for undirected unweighted graphs would be a large breakthrough [43]. Klein and Subramamian were the first to give a planarity-exploiting dynamic shortest paths algorithm [66] – their data structure worked for undirected graphs, was fully dynamic, (1+ )-approximate and had O(n2/3) update and query time bounds. In this paper we show the first exact dynamic single-source shortest paths algorithm for planar graphs with strongly sublinear update time and polylogarithmic query time. To the best of our knowledge, we obtain the first fully dynamic strongly connected components algorithm to achieve sublinear update-query time product for any important class of digraphs. For fully dynamic planar graphs polylogarithmic worst-case update bounds are known to be achievable even deterministically [34]
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