1,550 research outputs found

### Incremental $2$-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

### Dynamic Algorithms for the Massively Parallel Computation Model

The Massive Parallel Computing (MPC) model gained popularity during the last
decade and it is now seen as the standard model for processing large scale
data. One significant shortcoming of the model is that it assumes to work on
static datasets while, in practice, real-world datasets evolve continuously. To
overcome this issue, in this paper we initiate the study of dynamic algorithms
in the MPC model.
We first discuss the main requirements for a dynamic parallel model and we
show how to adapt the classic MPC model to capture them. Then we analyze the
connection between classic dynamic algorithms and dynamic algorithms in the MPC
model. Finally, we provide new efficient dynamic MPC algorithms for a variety
of fundamental graph problems, including connectivity, minimum spanning tree
and matching.Comment: Accepted to the 31st ACM Symposium on Parallelism in Algorithms and
Architectures (SPAA 2019

### Dynamic Dominators and Low-High Orders in DAGs

We consider practical algorithms for maintaining the dominator tree and a low-high order in directed acyclic graphs (DAGs) subject to dynamic operations. Let G be a directed graph with a distinguished start vertex s. The dominator tree D of G is a tree rooted at s, such that a vertex v is an ancestor of a vertex w if and only if all paths from s to w in G include v. The dominator tree is a central tool in program optimization and code generation, and has many applications in other diverse areas including constraint programming, circuit testing, biology, and in algorithms for graph connectivity problems. A low-high order of G is a preorder of D that certifies the correctness of D, and has further applications in connectivity and path-determination problems.
We first provide a practical and carefully engineered version of a recent algorithm [ICALP 2017] for maintaining the dominator tree of a DAG through a sequence of edge deletions. The algorithm runs in O(mn) total time and O(m) space, where n is the number of vertices and m is the number of edges before any deletion. In addition, we present a new algorithm that maintains a low-high order of a DAG under edge deletions within the same bounds. Both results extend to the case of reducible graphs (a class that includes DAGs). Furthermore, we present a fully dynamic algorithm for maintaining the dominator tree of a DAG under an intermixed sequence of edge insertions and deletions. Although it does not maintain the O(mn) worst-case bound of the decremental algorithm, our experiments highlight that the fully dynamic algorithm performs very well in practice. Finally, we study the practical efficiency of all our algorithms by conducting an extensive experimental study on real-world and synthetic graphs

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