Good r-Divisions Imply Optimal Amortized Decremental Biconnectivity

We present a data structure that, given a graph G of n vertices and m edges, and a suitable pair of nested r-divisions of G, preprocesses G in O(m+n) time and handles any series of edge-deletions in O(m) total time while answering queries to pairwise biconnectivity in worst-case O(1) time. In case the vertices are not biconnected, the data structure can return a cutvertex separating them in worst-case O(1) time. As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs

Faster Shortest Paths in Dense Distance Graphs, with Applications

We show how to combine two techniques for efficiently computing shortest paths in directed planar graphs. The first is the linear-time shortest-path algorithm of Henzinger, Klein, Subramanian, and Rao [STOC'94]. The second is Fakcharoenphol and Rao's algorithm [FOCS'01] for emulating Dijkstra's algorithm on the dense distance graph (DDG). A DDG is defined for a decomposition of a planar graph $G$ into regions of at most $r$ vertices each, for some parameter $r < n$. The vertex set of the DDG is the set of $\Theta(n/\sqrt r)$ vertices of $G$ that belong to more than one region (boundary vertices). The DDG has $\Theta(n)$ arcs, such that distances in the DDG are equal to the distances in $G$. Fakcharoenphol and Rao's implementation of Dijkstra's algorithm on the DDG (nicknamed FR-Dijkstra) runs in $O(n\log(n) r^{-1/2} \log r)$ time, and is a key component in many state-of-the-art planar graph algorithms for shortest paths, minimum cuts, and maximum flows. By combining these two techniques we remove the $\log n$ dependency in the running time of the shortest-path algorithm, making it $O(n r^{-1/2} \log^2r)$. This work is part of a research agenda that aims to develop new techniques that would lead to faster, possibly linear-time, algorithms for problems such as minimum-cut, maximum-flow, and shortest paths with negative arc lengths. As immediate applications, we show how to compute maximum flow in directed weighted planar graphs in $O(n \log p)$ time, where $p$ is the minimum number of edges on any path from the source to the sink. We also show how to compute any part of the DDG that corresponds to a region with $r$ vertices and $k$ boundary vertices in $O(r \log k)$ time, which is faster than has been previously known for small values of $k$

Decremental Single-Source Reachability in Planar Digraphs

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

Structured recursive separator decompositions for planar graphs in linear time (Extended Abstract)

Given a triangulated planar graph G on n vertices and an integer r &lt; n, an r-division of G with few holes is a decomposition of G into O(n/r) regions of size at most r such that each region contains at most a constant number of faces that are not faces of G (also called holes), and such that, for each region, the total number of vertices on these faces is O( √ r). We provide an algorithm for computing r-divisions with few holes in linear time. In fact, our algorithm computes a structure, called decomposition tree, which represents a recursive decomposition of G that includes r-divisions for essentially all values of r. In particular, given an exponentially increasing sequence r = (r1, r2, ...), our algorithm can produce a recursive r-division with few holes in linear time. r-divisions with few holes have been used in efficient algorithms to compute shortest paths, minimum cuts, and maximum flows. Our linear-time algorithm improves upon the decomposition algorithm used in the state-of-the-art algorithm for minimum st-cut (Italiano, Nussbaum, Sankowski, and Wulff-Nilsen, STOC 2011), removing one of the bottlenecks in the overall running time of their algorithm (analogously for minimum cut in planar and bounded-genus graphs)