677 research outputs found
The Minimum Wiener Connector
The Wiener index of a graph is the sum of all pairwise shortest-path
distances between its vertices. In this paper we study the novel problem of
finding a minimum Wiener connector: given a connected graph and a set
of query vertices, find a subgraph of that connects all
query vertices and has minimum Wiener index.
We show that The Minimum Wiener Connector admits a polynomial-time (albeit
impractical) exact algorithm for the special case where the number of query
vertices is bounded. We show that in general the problem is NP-hard, and has no
PTAS unless . Our main contribution is a
constant-factor approximation algorithm running in time
.
A thorough experimentation on a large variety of real-world graphs confirms
that our method returns smaller and denser solutions than other methods, and
does so by adding to the query set a small number of important vertices
(i.e., vertices with high centrality).Comment: Published in Proceedings of the 2015 ACM SIGMOD International
Conference on Management of Dat
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
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
Implicit Decomposition for Write-Efficient Connectivity Algorithms
The future of main memory appears to lie in the direction of new technologies
that provide strong capacity-to-performance ratios, but have write operations
that are much more expensive than reads in terms of latency, bandwidth, and
energy. Motivated by this trend, we propose sequential and parallel algorithms
to solve graph connectivity problems using significantly fewer writes than
conventional algorithms. Our primary algorithmic tool is the construction of an
-sized "implicit decomposition" of a bounded-degree graph on
nodes, which combined with read-only access to enables fast answers to
connectivity and biconnectivity queries on . The construction breaks the
linear-write "barrier", resulting in costs that are asymptotically lower than
conventional algorithms while adding only a modest cost to querying time. For
general non-sparse graphs on edges, we also provide the first writes
and operations parallel algorithms for connectivity and biconnectivity.
These algorithms provide insight into how applications can efficiently process
computations on large graphs in systems with read-write asymmetry
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
MIMO Detection for High-Order QAM Based on a Gaussian Tree Approximation
This paper proposes a new detection algorithm for MIMO communication systems
employing high order QAM constellations. The factor graph that corresponds to
this problem is very loopy; in fact, it is a complete graph. Hence, a
straightforward application of the Belief Propagation (BP) algorithm yields
very poor results. Our algorithm is based on an optimal tree approximation of
the Gaussian density of the unconstrained linear system. The finite-set
constraint is then applied to obtain a loop-free discrete distribution. It is
shown that even though the approximation is not directly applied to the exact
discrete distribution, applying the BP algorithm to the loop-free factor graph
outperforms current methods in terms of both performance and complexity. The
improved performance of the proposed algorithm is demonstrated on the problem
of MIMO detection
Robust Inference of Trees
This paper is concerned with the reliable inference of optimal
tree-approximations to the dependency structure of an unknown distribution
generating data. The traditional approach to the problem measures the
dependency strength between random variables by the index called mutual
information. In this paper reliability is achieved by Walley's imprecise
Dirichlet model, which generalizes Bayesian learning with Dirichlet priors.
Adopting the imprecise Dirichlet model results in posterior interval
expectation for mutual information, and in a set of plausible trees consistent
with the data. Reliable inference about the actual tree is achieved by focusing
on the substructure common to all the plausible trees. We develop an exact
algorithm that infers the substructure in time O(m^4), m being the number of
random variables. The new algorithm is applied to a set of data sampled from a
known distribution. The method is shown to reliably infer edges of the actual
tree even when the data are very scarce, unlike the traditional approach.
Finally, we provide lower and upper credibility limits for mutual information
under the imprecise Dirichlet model. These enable the previous developments to
be extended to a full inferential method for trees.Comment: 26 pages, 7 figure
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