Improved Algorithms for Decremental Single-Source Reachability on Directed Graphs

Recently we presented the first algorithm for maintaining the set of nodes reachable from a source node in a directed graph that is modified by edge deletions with $o(mn)$ total update time, where $m$ is the number of edges and $n$ is the number of nodes in the graph [Henzinger et al. STOC 2014]. The algorithm is a combination of several different algorithms, each for a different $m$ vs. $n$ trade-off. For the case of $m = \Theta(n^{1.5})$ the running time is $O(n^{2.47})$, just barely below $mn = \Theta(n^{2.5})$. In this paper we simplify the previous algorithm using new algorithmic ideas and achieve an improved running time of $\tilde O(\min(m^{7/6} n^{2/3}, m^{3/4} n^{5/4 + o(1)}, m^{2/3} n^{4/3+o(1)} + m^{3/7} n^{12/7+o(1)}))$. This gives, e.g., $O(n^{2.36})$ for the notorious case $m = \Theta(n^{1.5})$. We obtain the same upper bounds for the problem of maintaining the strongly connected components of a directed graph undergoing edge deletions. Our algorithms are correct with high probabililty against an oblivious adversary.Comment: This paper was presented at the International Colloquium on Automata, Languages and Programming (ICALP) 2015. A full version combining the findings of this paper and its predecessor [Henzinger et al. STOC 2014] is available at arXiv:1504.0795

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

Faster Fully Dynamic Transitive Closure in Practice

The fully dynamic transitive closure problem asks to maintain reachability information in a directed graph between arbitrary pairs of vertices, while the graph undergoes a sequence of edge insertions and deletions. The problem has been thoroughly investigated in theory and many specialized algorithms for solving it have been proposed in the last decades. In two large studies [Frigioni ea, 2001; Krommidas and Zaroliagis, 2008], a number of these algorithms have been evaluated experimentally against simple, static algorithms for graph traversal, showing the competitiveness and even superiority of the simple algorithms in practice, except for very dense random graphs or very high ratios of queries. A major drawback of those studies is that only small and mostly randomly generated graphs are considered. In this paper, we engineer new algorithms to maintain all-pairs reachability information which are simple and space-efficient. Moreover, we perform an extensive experimental evaluation on both generated and real-world instances that are several orders of magnitude larger than those in the previous studies. Our results indicate that our new algorithms outperform all state-of-the-art algorithms on all types of input considerably in practice

Deterministic Fully Dynamic SSSP and More

We present the first non-trivial fully dynamic algorithm maintaining exact single-source distances in unweighted graphs. This resolves an open problem stated by Sankowski [COCOON 2005] and van den Brand and Nanongkai [FOCS 2019]. Previous fully dynamic single-source distances data structures were all approximate, but so far, non-trivial dynamic algorithms for the exact setting could only be ruled out for polynomially weighted graphs (Abboud and Vassilevska Williams, [FOCS 2014]). The exact unweighted case remained the main case for which neither a subquadratic dynamic algorithm nor a quadratic lower bound was known. Our dynamic algorithm works on directed graphs, is deterministic, and can report a single-source shortest paths tree in subquadratic time as well. Thus we also obtain the first deterministic fully dynamic data structure for reachability (transitive closure) with subquadratic update and query time. This answers an open problem of van den Brand, Nanongkai, and Saranurak [FOCS 2019]. Finally, using the same framework we obtain the first fully dynamic data structure maintaining all-pairs $(1+\epsilon)$-approximate distances within non-trivial sub-$n^\omega$ worst-case update time while supporting optimal-time approximate shortest path reporting at the same time. This data structure is also deterministic and therefore implies the first known non-trivial deterministic worst-case bound for recomputing the transitive closure of a digraph.Comment: Extended abstract to appear in FOCS 202

On Fully Dynamic Strongly Connected Components

We consider maintaining strongly connected components (SCCs) of a directed graph subject to edge insertions and deletions. For this problem, we show a randomized algebraic data structure with conditionally tight O(n^1.529) worst-case update time. The only previously described subquadratic update bound for this problem [Karczmarz, Mukherjee, and Sankowski, STOC\u2722] holds exclusively in the amortized sense. For the less general dynamic strong connectivity problem, where one is only interested in maintaining whether the graph is strongly connected, we give an efficient deterministic black-box reduction to (arbitrary-pair) dynamic reachability. Consequently, for dynamic strong connectivity we match the best-known O(n^1.407) worst-case upper bound for dynamic reachability [van den Brand, Nanongkai, and Saranurak FOCS\u2719]. This is also conditionally optimal and improves upon the previous O(n^1.529) bound. Our reduction also yields the first fully dynamic algorithms for maintaining the minimum strong connectivity augmentation of a digraph

Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler

In the decremental $(1+\epsilon)$-approximate Single-Source Shortest Path (SSSP) problem, we are given a graph $G=(V,E)$ with $n = |V|, m = |E|$, undergoing edge deletions, and a distinguished source $s \in V$, and we are asked to process edge deletions efficiently and answer queries for distance estimates $\widetilde{\mathbf{dist}}_G(s,v)$ for each $v \in V$, at any stage, such that $\mathbf{dist}_G(s,v) \leq \widetilde{\mathbf{dist}}_G(s,v) \leq (1+ \epsilon)\mathbf{dist}_G(s,v)$. In the decremental $(1+\epsilon)$-approximate All-Pairs Shortest Path (APSP) problem, we are asked to answer queries for distance estimates $\widetilde{\mathbf{dist}}_G(u,v)$ for every $u,v \in V$. In this article, we consider the problems for undirected, unweighted graphs. We present a new \emph{deterministic} algorithm for the decremental $(1+\epsilon)$-approximate SSSP problem that takes total update time $O(mn^{0.5 + o(1)})$. Our algorithm improves on the currently best algorithm for dense graphs by Chechik and Bernstein [STOC 2016] with total update time $\tilde{O}(n^2)$ and the best existing algorithm for sparse graphs with running time $\tilde{O}(n^{1.25}\sqrt{m})$ [SODA 2017] whenever $m = O(n^{1.5 - o(1)})$. In order to obtain this new algorithm, we develop several new techniques including improved decremental cover data structures for graphs, a more efficient notion of the heavy/light decomposition framework introduced by Chechik and Bernstein and the first clustering technique to maintain a dynamic \emph{sparse} emulator in the deterministic setting. As a by-product, we also obtain a new simple deterministic algorithm for the decremental $(1+\epsilon)$-approximate APSP problem with near-optimal total running time $\tilde{O}(mn /\epsilon)$ matching the time complexity of the sophisticated but rather involved algorithm by Henzinger, Forster and Nanongkai [FOCS 2013].Comment: Appeared in SODA'2