292 research outputs found
Strong Connectivity in Directed Graphs under Failures, with Application
In this paper, we investigate some basic connectivity problems in directed
graphs (digraphs). Let be a digraph with edges and vertices, and
let be the digraph obtained after deleting edge from . As
a first result, we show how to compute in worst-case time: The
total number of strongly connected components in , for all edges
in . The size of the largest and of the smallest strongly
connected components in , for all edges in .
Let be strongly connected. We say that edge separates two vertices
and , if and are no longer strongly connected in .
As a second set of results, we show how to build in time -space
data structures that can answer in optimal time the following basic
connectivity queries on digraphs: Report in worst-case time all
the strongly connected components of , for a query edge .
Test whether an edge separates two query vertices in worst-case
time. Report all edges that separate two query vertices in optimal
worst-case time, i.e., in time , where is the number of separating
edges. (For , the time is ).
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 -connectivity (i.e.,
-edge and -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 with
edges that maintains the -connectivity cuts of and the decompositions
induced by those cuts.Comment: An extended abstract of this work appeared in the SODA 201
Incremental -Edge-Connectivity in Directed Graphs
In this paper, we initiate the study of the dynamic maintenance of
-edge-connectivity relationships in directed graphs. We present an algorithm
that can update the -edge-connected blocks of a directed graph with
vertices through a sequence of edge insertions in a total of time.
After each insertion, we can answer the following queries in asymptotically
optimal time: (i) Test in constant time if two query vertices and are
-edge-connected. Moreover, if and are not -edge-connected, we can
produce in constant time a "witness" of this property, by exhibiting an edge
that is contained in all paths from to or in all paths from to .
(ii) Report in time all the -edge-connected blocks of . To the
best of our knowledge, this is the first dynamic algorithm for -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
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
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
Incremental Low-High Orders of Directed Graphs and Applications
A flow graph G = (V, E, s) is 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 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 d of D that certifies the correctness of D, and has further applications in connectivity and path-determination problems.
In this paper we consider how to maintain efficiently a low-high order of a flow graph incrementally under edge insertions. We present algorithms that run in O(mn) total time for a sequence of edge insertions in a flow graph with n vertices, where m is the total number of edges after all insertions. These immediately provide the first incremental certifying algorithms for maintaining the dominator tree in O(mn) total time, and also imply incremental algorithms for other problems. Hence, we provide a substantial improvement over the O(m^2) straightforward algorithms, which recompute the solution from scratch after each edge insertion. Furthermore, we provide efficient implementations of our algorithms and conduct an extensive experimental study on real-world graphs taken from a variety of application areas. The experimental results show that our algorithms perform very well in practice
Static Scheduling for Barrier MIMD Architectures
Barrier MIMDs are asynchronous Multiple Instruction stream Multiple Data stream architectures capable of parallel execution of variable-execution-time instructions and arbitrary control flow (e.g., w h ile loops and calls); however, they differ from conventional MIMDs in that the need for run-time synchronization is significantly reduced. Whenever a group of processors within a barrier MIMD encounters a synchronization point (barrier), static timing constraints become precise, hence, conceptual synchronizations between the processors often can be statically resolved with zero cost — as in a SIMD or VLIW and using similar compiler technology. Unlike these machines, however, as execution continues past the synchronization point the accuracy within which the compiler can track the relative timing between processors is reduced. Where this imprecision becomes too large, the compiler simply inserts a synchronization barrier to insure that timing imprecision at that point is zero, and again employs static, implicit synchronization. This paper describes new scheduling and barrier placement algorithms for barrier MIMDs that are based loosely on the list scheduling approach employed for VLIWs [Elli85]. In addition, the experimental results from scheduling more than 3500 synthetic benchmark programs for a parameterized barrier MIMD machine are presented
Dominators in Directed Graphs: A Survey of Recent Results, Applications, and Open Problems
The computation of dominators is a central tool in program optimization and code generation, and it has applications in other diverse areas includingconstraint programming, circuit testing, and biology. In this paper we survey recent results, applications, and open problems related to the notion of dominators in directed graphs,including dominator verification and certification, computing independent spanning trees, and connectivity and path-determination problems in directed graphs
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