655 research outputs found
Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time
We present faster algorithms for computing the 2-edge and 2-vertex strongly
connected components of a directed graph, which are straightforward
generalizations of strongly connected components. While in undirected graphs
the 2-edge and 2-vertex connected components can be found in linear time, in
directed graphs only rather simple -time algorithms were known. We use
a hierarchical sparsification technique to obtain algorithms that run in time
. For 2-edge strongly connected components our algorithm gives the
first running time improvement in 20 years. Additionally we present an -time algorithm for 2-edge strongly connected components, and thus
improve over the running time also when . Our approach
extends to k-edge and k-vertex strongly connected components for any constant k
with a running time of for edges and for vertices
2-Vertex Connectivity in Directed Graphs
We complement our study of 2-connectivity in directed graphs, by considering
the computation of the following 2-vertex-connectivity relations: We say that
two vertices v and w are 2-vertex-connected if there are two internally
vertex-disjoint paths from v to w and two internally vertex-disjoint paths from
w to v. We also say that v and w are vertex-resilient if the removal of any
vertex different from v and w leaves v and w in the same strongly connected
component. We show how to compute the above relations in linear time so that we
can report in constant time if two vertices are 2-vertex-connected or if they
are vertex-resilient. We also show how to compute in linear time a sparse
certificate for these relations, i.e., a subgraph of the input graph that has
O(n) edges and maintains the same 2-vertex-connectivity and vertex-resilience
relations as the input graph, where n is the number of vertices.Comment: arXiv admin note: substantial text overlap with arXiv:1407.304
Approximating the Smallest Spanning Subgraph for 2-Edge-Connectivity in Directed Graphs
Let be a strongly connected directed graph. We consider the following
three problems, where we wish to compute the smallest strongly connected
spanning subgraph of that maintains respectively: the -edge-connected
blocks of (\textsf{2EC-B}); the -edge-connected components of
(\textsf{2EC-C}); both the -edge-connected blocks and the -edge-connected
components of (\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 -approximation by combining previously known results. For
\textsf{2EC-B} and \textsf{2EC-B-C}, we present new -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
An optimal maximal independent setalgorithm for bounded-independence graphs
We present a novel distributed algorithm for the maximal independent set problem (This is an extended journal version of Schneider and Wattenhofer in Twenty-seventh annual ACM SIGACT-SIGOPS symposium on principles of distributed computing, 2008). On bounded-independence graphs our deterministic algorithm finishes in O(log* n) time, n being the number of nodes. In light of Linial's Ω(log* n) lower bound our algorithm is asymptotically optimal. Furthermore, it solves the connected dominating set problem for unit disk graphs in O(log* n) time, exponentially faster than the state-of-the-art algorithm. With a new extension our algorithm also computes a δ+1 coloring and a maximal matching in O(log* n) time, where δ is the maximum degree of the grap
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