12,553 research outputs found
Fast Dynamic Graph Algorithms for Parameterized Problems
Fully dynamic graph is a data structure that (1) supports edge insertions and
deletions and (2) answers problem specific queries. The time complexity of (1)
and (2) are referred to as the update time and the query time respectively.
There are many researches on dynamic graphs whose update time and query time
are , that is, sublinear in the graph size. However, almost all such
researches are for problems in P. In this paper, we investigate dynamic graphs
for NP-hard problems exploiting the notion of fixed parameter tractability
(FPT).
We give dynamic graphs for Vertex Cover and Cluster Vertex Deletion
parameterized by the solution size . These dynamic graphs achieve almost the
best possible update time and the query time
, where is the time complexity of any static
graph algorithm for the problems. We obtain these results by dynamically
maintaining an approximate solution which can be used to construct a small
problem kernel. Exploiting the dynamic graph for Cluster Vertex Deletion, as a
corollary, we obtain a quasilinear-time (polynomial) kernelization algorithm
for Cluster Vertex Deletion. Until now, only quadratic time kernelization
algorithms are known for this problem.
We also give a dynamic graph for Chromatic Number parameterized by the
solution size of Cluster Vertex Deletion, and a dynamic graph for
bounded-degree Feedback Vertex Set parameterized by the solution size. Assuming
the parameter is a constant, each dynamic graph can be updated in
time and can compute a solution in time. These results are obtained by
another approach.Comment: SWAT 2014 to appea
On Efficiently Detecting Overlapping Communities over Distributed Dynamic Graphs
Modern networks are of huge sizes as well as high dynamics, which challenges
the efficiency of community detection algorithms. In this paper, we study the
problem of overlapping community detection on distributed and dynamic graphs.
Given a distributed, undirected and unweighted graph, the goal is to detect
overlapping communities incrementally as the graph is dynamically changing. We
propose an efficient algorithm, called \textit{randomized Speaker-Listener
Label Propagation Algorithm} (rSLPA), based on the \textit{Speaker-Listener
Label Propagation Algorithm} (SLPA) by relaxing the probability distribution of
label propagation. Besides detecting high-quality communities, rSLPA can
incrementally update the detected communities after a batch of edge insertion
and deletion operations. To the best of our knowledge, rSLPA is the first
algorithm that can incrementally capture the same communities as those obtained
by applying the detection algorithm from the scratch on the updated graph.
Extensive experiments are conducted on both synthetic and real-world datasets,
and the results show that our algorithm can achieve high accuracy and
efficiency at the same time.Comment: A short version of this paper will be published as ICDE'2018 poste
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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