# Decremental Single-Source Reachability in Planar Digraphs

## TL;DR

This paper introduces efficient decremental algorithms for single-source reachability and strongly connected components in directed planar graphs, significantly improving update times over previous solutions and being among the first with polylogarithmic bounds for such problems.

## Contribution

It presents the first almost optimal decremental algorithms with polylogarithmic update times for reachability and strongly connected components in directed planar graphs.

## Key findings

- Total update time is $O(n ext{log}^2 n ext{log} ext{log} n)$
- Achieves polylogarithmic update times for non-trivial reachability problems
- First such algorithms with these bounds for directed planar graphs

## Abstract

In this paper we show a new algorithm for the decremental single-source reachability problem in directed planar graphs. It processes any sequence of edge deletions in $O(n\log^2{n}\log\log{n})$ total time and explicitly maintains the set of vertices reachable from a fixed source vertex. Hence, if all edges are eventually deleted, the amortized time of processing each edge deletion is only $O(\log^2 n \log \log n)$, which improves upon a previously known $O(\sqrt{n})$ solution. We also show an algorithm for decremental maintenance of strongly connected components in directed planar graphs with the same total update time. These results constitute the first almost optimal (up to polylogarithmic factors) algorithms for both problems. To the best of our knowledge, these are the first dynamic algorithms with polylogarithmic update times on general directed planar graphs for non-trivial reachability-type problems, for which only polynomial bounds are known in general graphs.