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43
The complexity of theoremproving procedures
 In STOC
, 1971
"... It is shown that any recognition problem solved by a polynomial timebounded nondeterministic Turing machine can be “reduced ” to the problem of determining whether a given propositional formula is a tautology. Here “reduced ” means, roughly speaking, that the first problem can be solved determinist ..."
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Cited by 772 (4 self)
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It is shown that any recognition problem solved by a polynomial timebounded nondeterministic Turing machine can be “reduced ” to the problem of determining whether a given propositional formula is a tautology. Here “reduced ” means, roughly speaking, that the first problem can be solved deterministically in polynomial time provided an oracle is available for solving the second. From this notion of reducible, polynomial degrees of difficulty are defined, and it is shown that the problem of determining tautologyhood has the same polynomial degree as the problem of determining whether the first of two given graphs is isomorphic to a subgraph of the second. Other examples are discussed. A method of measuring the complexity of proof procedures for the predicate calculus is introduced and discussed. Throughout this paper, a set of strings 1 means a set of strings on some fixed, large, finite alphabet Σ. This alphabet is large enough to include symbols for all sets described here. All Turing machines are deterministic recognition devices, unless the contrary is explicitly stated. 1 Tautologies and Polynomial ReReducibility. Let us fix a formalism for the propositional calculus in which formulas are written as strings on Σ. Since we will require infinitely many proposition symbols (atoms), each such symbol will consist of a member of Σ followed by a number in binary notation to distinguish that symbol. Thus a formula of length n can
An algorithm for drawing general undirected graphs
 Information Processing Letters
, 1989
"... Graphs (networks) are very common data structures which are handled in computers. Diagrams are widely used to represent the graph structures visually in many information systems. In order to automatically draw the diagrams which are, for example, state graphs, dataflow graphs, Petri nets, and entit ..."
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Cited by 455 (2 self)
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Graphs (networks) are very common data structures which are handled in computers. Diagrams are widely used to represent the graph structures visually in many information systems. In order to automatically draw the diagrams which are, for example, state graphs, dataflow graphs, Petri nets, and entityrelationship diagrams, basic graph drawing algorithms are required.
Algorithmics and Applications of Tree and Graph Searching
 In Symposium on Principles of Database Systems
, 2002
"... Modern search engines answer keywordbased queries extremely efficiently. The impressive speed is due to clever inverted index structures, caching, a domainindependent knowledge of strings, and thousands of machines. Several research efforts have attempted to generalize keyword search to keytree an ..."
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Cited by 108 (8 self)
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Modern search engines answer keywordbased queries extremely efficiently. The impressive speed is due to clever inverted index structures, caching, a domainindependent knowledge of strings, and thousands of machines. Several research efforts have attempted to generalize keyword search to keytree and keygraph searching, because trees and graphs have many applications in nextgeneration database systems. This paper surveys both algorithms and applications, giving some emphasis to our own work.
An improved algorithm for matching large graphs
 In: 3rd IAPRTC15 Workshop on Graphbased Representations in Pattern Recognition, Cuen
, 2001
"... In this paper an improved version of a graph matching algorithm is presented, which is able to efficiently solve the graph isomorphism and graphsubgraph isomorphism problems on Attributed Relational Graphs. This version is particularly suited to work with very large graphs, since its memory require ..."
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Cited by 68 (2 self)
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In this paper an improved version of a graph matching algorithm is presented, which is able to efficiently solve the graph isomorphism and graphsubgraph isomorphism problems on Attributed Relational Graphs. This version is particularly suited to work with very large graphs, since its memory requirements are quite smaller than those of other algorithms of the same kind. After a detailed description of the algorithm, an experimental comparison is made against both the previous version (developed by the same authors) and the Ullmann’s algorithm. 1.
Resisting Structural Reidentification in Anonymized Social Networks
, 2008
"... We identify privacy risks associated with releasing network data sets and provide an algorithm that mitigates those risks. A network consists of entities connected by links representing relations such as friendship, communication, or shared activity. Maintaining privacy when publishing networked dat ..."
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Cited by 60 (7 self)
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We identify privacy risks associated with releasing network data sets and provide an algorithm that mitigates those risks. A network consists of entities connected by links representing relations such as friendship, communication, or shared activity. Maintaining privacy when publishing networked data is uniquely challenging because an individual’s network context can be used to identify them even if other identifying information is removed. In this paper, we quantify the privacy risks associated with three classes of attacks on the privacy of individuals in networks, based on the knowledge used by the adversary. We show that the risks of these attacks vary greatly based on network structure and size. We propose a novel approach to anonymizing network data that models aggregate network structure and then allows samples to be drawn from that model. The approach guarantees anonymity for network entities while preserving the ability to estimate a wide variety of network measures with relatively little bias.
Anonymizing Social Networks
 VLDB 2008
, 2008
"... Advances in technology have made it possible to collect data about individuals and the connections between them, such as email correspondence and friendships. Agencies and researchers who have collected such social network data often have a compelling interest in allowing others to analyze the data. ..."
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Cited by 45 (3 self)
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Advances in technology have made it possible to collect data about individuals and the connections between them, such as email correspondence and friendships. Agencies and researchers who have collected such social network data often have a compelling interest in allowing others to analyze the data. However, in many cases the data describes relationships that are private (e.g., email correspondence) and sharing the data in full can result in unacceptable disclosures. In this paper, we present a framework for assessing the privacy risk of sharing anonymized network data. This includes a model of adversary knowledge, for which we consider several variants and make connections to known graph theoretical results. On several realworld social networks, we show that simple anonymization techniques are inadequate, resulting in substantial breaches of privacy for even modestly informed adversaries. We propose a novel anonymization technique based on perturbing the network and demonstrate empirically that it leads to substantial reduction of the privacy threat. We also analyze the effect that anonymizing the network has on the utility of the data for social network analysis.
Autonomous Deployment and Repair of a Sensor Network Using an Unmanned Aerial Vehicle
 in IEEE International Conference on Robotics and Automation
, 2004
"... We describe a sensor network deployment method using autonomous flying robots. Such networks are suitable for tasks such as largescale environmental monitoring or for command and control in emergency situations. We describe in detail the algorithms used for deployment and for measuring network conn ..."
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Cited by 43 (8 self)
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We describe a sensor network deployment method using autonomous flying robots. Such networks are suitable for tasks such as largescale environmental monitoring or for command and control in emergency situations. We describe in detail the algorithms used for deployment and for measuring network connectivity and provide experimental data we collected from field trials. A particular focus is on determining gaps in connectivity of the deployed network and generating a plan for a second, repair, pass to complete the connectivity. This project is the result of a collaboration between three robotics labs (CSIRO, USC, and Dartmouth.) I.
A performance comparison of five algorithms for graph isomorphism
 in Proceedings of the 3rd IAPR TC15 Workshop on Graphbased Representations in Pattern Recognition
, 2001
"... Despite the significant number of isomorphism algorithms presented in the literature, till now no efforts have been done for characterizing their performance. Consequently, it is not clear how the behavior of those algorithms varies as the type and the size of the graphs to be matched varies in case ..."
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Cited by 34 (2 self)
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Despite the significant number of isomorphism algorithms presented in the literature, till now no efforts have been done for characterizing their performance. Consequently, it is not clear how the behavior of those algorithms varies as the type and the size of the graphs to be matched varies in case of real applications. In this paper we present a benchmarking activity for characterizing the performance of a bunch of algorithms for exact graph isomorphism. To this purpose we use a large database containing 10,000 couples of isomorphic graphs with different topologies (regular graphs, randomly connected graphs, bounded valence graph), enriched with suitably modified versions of them for simulating distortions occurring in real cases. The size of the considered graphs ranges from a few nodes to about 1000 nodes. 1.
Efficient Aggregation for Graph Summarization
"... Graphs are widely used to model real world objects and their relationships, and large graph datasets are common in many application domains. To understand the underlying characteristics of large graphs, graph summarization techniques are critical. However, existing graph summarization methods are mo ..."
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Cited by 33 (3 self)
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Graphs are widely used to model real world objects and their relationships, and large graph datasets are common in many application domains. To understand the underlying characteristics of large graphs, graph summarization techniques are critical. However, existing graph summarization methods are mostly statistical (studying statistics such as degree distributions, hopplots and clustering coefficients). These statistical methods are very useful, but the resolutions of the summaries are hard to control. In this paper, we introduce two databasestyle operations to summarize graphs. Like the OLAPstyle aggregation methods that allow users to drilldown or rollup to control the resolution of summarization, our methods provide an analogous functionality for large graph datasets. The first operation, called SNAP, produces a summary graph by grouping nodes based on userselected node attributes and relationships. The second operation, called kSNAP, further allows users to control the resolutions of summaries and provides the “drilldown ” and “rollup ” abilities to navigate through summaries with different resolutions. We propose an efficient algorithm to evaluate the SNAP operation. In addition, we prove that the kSNAP computation is NPcomplete. We propose two heuristic methods to approximate the kSNAP results. Through extensive experiments on a variety of real and synthetic datasets, we demonstrate the effectiveness and efficiency of the proposed methods.
A Lagrangian Relaxation Network for Graph Matching
 IEEE Trans. Neural Networks
, 1996
"... A Lagrangian relaxation network for graph matching is presented. The problem is formulated as follows: given graphs G and g, find a permutation matrix M that brings the two sets of vertices into correspondence. Permutation matrix constraints are formulated in the framework of deterministic annealing ..."
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Cited by 26 (7 self)
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A Lagrangian relaxation network for graph matching is presented. The problem is formulated as follows: given graphs G and g, find a permutation matrix M that brings the two sets of vertices into correspondence. Permutation matrix constraints are formulated in the framework of deterministic annealing. Our approach is in the same spirit as a Lagrangian decomposition approach in that the row and column constraints are satisfied separately with a Lagrange multiplier used to equate the two "solutions." Due to the unavoidable symmetries in graph isomorphism (resulting in multiple global minima), we add a symmetrybreaking selfamplification term in order to obtain a permutation matrix. With the application of a fixpoint preserving algebraic transformation to both the distance measure and selfamplification terms, we obtain a Lagrangian relaxation network. The network performs minimization with respect to the Lagrange parameters and maximization with respect to the permutation matrix variable...