Incremental 22-Edge-Connectivity in Directed Graphs

In this paper, we initiate the study of the dynamic maintenance of $2$-edge-connectivity relationships in directed graphs. We present an algorithm that can update the $2$-edge-connected blocks of a directed graph with $n$ vertices through a sequence of $m$ edge insertions in a total of $O(mn)$ time. After each insertion, we can answer the following queries in asymptotically optimal time: (i) Test in constant time if two query vertices $v$ and $w$ are $2$-edge-connected. Moreover, if $v$ and $w$ are not $2$-edge-connected, we can produce in constant time a "witness" of this property, by exhibiting an edge that is contained in all paths from $v$ to $w$ or in all paths from $w$ to $v$. (ii) Report in $O(n)$ time all the $2$-edge-connected blocks of $G$. To the best of our knowledge, this is the first dynamic algorithm for $2$-connectivity problems on directed graphs, and it matches the best known bounds for simpler problems, such as incremental transitive closure.Comment: Full version of paper presented at ICALP 201

Strong Connectivity in Directed Graphs under Failures, with Application

In this paper, we investigate some basic connectivity problems in directed graphs (digraphs). Let $G$ be a digraph with $m$ edges and $n$ vertices, and let $G\setminus e$ be the digraph obtained after deleting edge $e$ from $G$. As a first result, we show how to compute in $O(m+n)$ worst-case time: $(i)$ The total number of strongly connected components in $G\setminus e$, for all edges $e$ in $G$. $(ii)$ The size of the largest and of the smallest strongly connected components in $G\setminus e$, for all edges $e$ in $G$. Let $G$ be strongly connected. We say that edge $e$ separates two vertices $x$ and $y$, if $x$ and $y$ are no longer strongly connected in $G\setminus e$. As a second set of results, we show how to build in $O(m+n)$ time $O(n)$-space data structures that can answer in optimal time the following basic connectivity queries on digraphs: $(i)$ Report in $O(n)$ worst-case time all the strongly connected components of $G\setminus e$, for a query edge $e$. $(ii)$ Test whether an edge separates two query vertices in $O(1)$ worst-case time. $(iii)$ Report all edges that separate two query vertices in optimal worst-case time, i.e., in time $O(k)$, where $k$ is the number of separating edges. (For $k=0$, the time is $O(1)$). All of the above results extend to vertex failures. All our bounds are tight and are obtained with a common algorithmic framework, based on a novel compact representation of the decompositions induced by the $1$-connectivity (i.e., $1$-edge and $1$-vertex) cuts in digraphs, which might be of independent interest. With the help of our data structures we can design efficient algorithms for several other connectivity problems on digraphs and we can also obtain in linear time a strongly connected spanning subgraph of $G$ with $O(n)$ edges that maintains the $1$-connectivity cuts of $G$ and the decompositions induced by those cuts.Comment: An extended abstract of this work appeared in the SODA 201

Approximating the Smallest Spanning Subgraph for 2-Edge-Connectivity in Directed Graphs

Let $G$ be a strongly connected directed graph. We consider the following three problems, where we wish to compute the smallest strongly connected spanning subgraph of $G$ that maintains respectively: the $2$-edge-connected blocks of $G$ (\textsf{2EC-B}); the $2$-edge-connected components of $G$ (\textsf{2EC-C}); both the $2$-edge-connected blocks and the $2$-edge-connected components of $G$ (\textsf{2EC-B-C}). All three problems are NP-hard, and thus we are interested in efficient approximation algorithms. For \textsf{2EC-C} we can obtain a $3/2$-approximation by combining previously known results. For \textsf{2EC-B} and \textsf{2EC-B-C}, we present new $4$-approximation algorithms that run in linear time. We also propose various heuristics to improve the size of the computed subgraphs in practice, and conduct a thorough experimental study to assess their merits in practical scenarios

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

Data Structures and Algorithms for Disjoint Set Union Problems

This paper surveys algorithmic techniques and data structures that have been proposed to solve the set union problem and its variants. Their discovery required a new set of algorithmic tools that have proven useful in other areas. Special attention is devoted to recent extensions of the original set union problem, and some effort is made to provide a unifying theoretical framework for this growing body of algorithms

On-line algorithms for polynomially solvable satisfiability problems

AbstractGiven a propositional Horn formula, we show how to maintain on-line information about its satisfiability during the insertion of new clauses. A data structure is presented which answers each satisfiability question in O(1) time and inserts a new clause of length q in O(q) amortized time. This significantly outperforms previously known solutions of the same problem. This result is extended also to a particular class of non-Horn formulae already considered in the literature, for which the space bound is improved. Other operations are considered, such as testing whether a given hypothesis is consistent with a satisfying interpretation of the given formula and determining a truth assignment which satisfies a given formula. The on-line time and space complexity of these operations is also analyzed