Fast processing of graph queries on a large database of small and medium-sized data graphs

We propose a new way of indexing a large database of small and medium-sized graphs and processing exact subgraph matching (or subgraph isomorphism) and approximate (full) graph matching queries. Rather than decomposing a graph into smaller units (e.g., paths, trees, graphs) for indexing purposes, we represent each graph in the database by its graph signature, which is essentially a multiset. We construct a disk-based index on all the signatures via bulk loading. During query processing, a query graph is also mapped into its signature, and this signature is searched using the index by performing multiset operations. To improve the precision of exact subgraph matching, we develop a new scheme using the concept of line graphs. Through extensive evaluation on real and synthetic graph datasets, we demonstrate that our approach provides a scalable and efficient disk-based solution for a large database of small and medium-sized graphs.     

Cache-oblivious index for approximate string matching

This paper revisits the problem of indexing a text for approximate string matching. Specifically, given a text T of length n and a positive integer k, we want to construct an index of T such that for any input pattern P, we can find all its k-error matches in T efficiently. This problem is well-studied in the internal-memory setting. Here, we extend some of these recent results to external-memory solutions, which are also cache-oblivious. Our first index occupies O((nlog^kn)/B) disk pages and finds all k-error matches with O((|P|+occ)/B+log^knloglog_Bn) I/Os, where B denotes the number of words in a disk page. To the best of our knowledge, this index is the first external-memory data structure that does not require Ω(|P|+occ+poly(logn)) I/Os. The second index reduces the space to O((nlogn)/B) disk pages, and the I/O complexity is O((|P|+occ)/B+log^k(k+1)nloglogn).     

A distance measure for large graphs based on prime graphs

Graphs are universal modeling tools. They are used to represent objects and their relationships in almost all domains: they are used to represent DNA, images, videos, social networks, XML documents, etc. When objects are represented by graphs, the problem of their comparison is a problem of comparing graphs. Comparing objects is a key task in our daily life. It is the core of a search engine, the backbone of a mining tool, etc. Nowadays, comparing objects faces the challenge of the large amount of data that this task must deal with. Moreover, when graphs are used to model these objects, it is known that graph comparison is very complex and computationally hard especially for large graphs. So, research on simplifying graph comparison gainedan interest and several solutions are proposed. In this paper, we explore and evaluate a new solution for the comparison of large graphs. Our approach relies on a compact encoding of graphs called prime graphs. Prime graphs are smaller and simpler than the original ones but they retain the structure and properties of the encoded graphs. We propose to approximate the similarity between two graphs by comparing the corresponding prime graphs. Simulations results show that this approach is effective for large graphs.     

A subquadratic algorithm for approximate limited expression matching

In this paper we present an efficient subquadratic-time algorithm for matching strings and limited expressions in large texts. Limited expressions are a subset of regular expressions that appear often in practice. The generalization from simple strings to limited expressions has a negligible affect on the speed of our algorithm, yet allows much more flexibility. Our algorithm is similar in spirit to that of Masek and Paterson [MP], but it is much faster in practice. Our experiments show a factor of four to five speedup against the algorithms of Sellers [Se] and Ukkonen [Uk1] independent of the sizes of the input strings. Experiments also reveal our algorithm to be faster, in most cases, than a recent improvement by Chang and Lampe [CL2], especially for small alphabet sizes for which it is two to three times faster.The research of U. Manber was supported in part by a Presidential Young Investigator Award DCR-8451397, with matching funds from AT&T, and by NSF Grant CCR-9001619. G. Myers research was supported in part by NIH Grant LM04960, NSF Grant CCR-9001619, and the Aspen Center for Physics.     

HEGM: A hierarchical elastic graph matching for hand gesture recognition

A hierarchical scheme for elastic graph matching applied to hand gesture recognition is proposed. The proposed algorithm exploits the relative discriminatory capabilities of visual features scattered on the images, assigning the corresponding weights to each feature. A boosting algorithm is used to determine the structure of the hierarchy of a given graph. The graph is expressed by annotating the nodes of interest over the target object to form a bunch graph. Three annotation techniques, manual, semi-automatic, and automatic annotation are used to determine the position of the nodes. The scheme and the annotation approaches are applied to explore the hand gesture recognition performance. A number of filter banks are applied to hand gestures images to investigate the effect of using different feature representation approaches. Experimental results show that the hierarchical elastic graph matching (HEGM) approach classified the hand posture with a gesture recognition accuracy of 99.85% when visual features were extracted by utilizing the Histogram of Oriented Gradient (HOG) representation. The results also provide the performance measures from the aspect of recognition accuracy to matching benefits, node positions correlation and consistency on three annotation approaches, showing that the semi-automatic annotation method is more efficient and accurate than the other two methods.     

Fast parallel algorithms for graph similarity and matching

This paper addresses the problem of global graph alignment on supercomputer-class clusters. We define the alignment of two graphs, as a mapping of each vertex in the first graph to a unique vertex in the second graph so as to optimize a given similarity-based cost function.¹ Using a state of the art serial algorithm for the computation of vertex similarity scores called Network Similarity Decomposition (NSD), we derive corresponding parallel formulations. Coupling this parallel similarity algorithm with a parallel auction-based bipartite matching technique, we obtain a highly efficient and scalable graph matching pipeline. We validate the performance of our integrated approach on a large parallel platform and on diverse graph instances (including Protein Interaction, Wikipedia and Web networks). Experimental results demonstrate that our algorithms scale to large machine configurations (thousands of cores) and problem instances, enabling the alignment of networks of sizes two orders of magnitude larger than reported in the current literature.     

The labeled perfect matching in bipartite graphs

In this paper, we deal with both the complexity and the approximability of the labeled perfect matching problem in bipartite graphs. Given a simple graph G=(V,E) with |V|=2n vertices such that E contains a perfect matching (of size n), together with a color (or label) function , the labeled perfect matching problem consists in finding a perfect matching on G that uses a minimum or a maximum number of colors.     

Inexact graph matching for model-based recognition: Evaluation and comparison of optimization algorithms

A method for segmentation and recognition of image structures based on graph homomorphisms is presented in this paper. It is a model-based recognition method where the input image is over-segmented and the obtained regions are represented by an attributed relational graph (ARG). This graph is then matched against a model graph thus accomplishing the model-based recognition task. This type of problem calls for inexact graph matching through a homomorphism between the graphs since no bijective correspondence can be expected, because of the over-segmentation of the image with respect to the model. The search for the best homomorphism is carried out by optimizing an objective function based on similarities between object and relational attributes defined on the graphs. The following optimization procedures are compared and discussed: deterministic tree search, for which new algorithms are detailed, genetic algorithms and estimation of distribution algorithms. In order to assess the performance of these algorithms using real data, experimental results on supervised classification of facial features using face images from public databases are presented.     

Recognizing hand gestures using the weighted elastic graph matching (WEGM) method

This paper proposes a weighted scheme for elastic graph matching hand posture recognition. Visual features scattered on the elastic graph are assigned corresponding weights according to their relative ability to discriminate between gestures. The weights' values are determined using adaptive boosting. A dictionary representing the variability of each gesture class is expressed in the form of a bunch graph. The positions of the nodes in the bunch graph are determined using three techniques: manually, semi-automatically, and automatically. Experimental results also show that the semi-automatic annotation method is efficient and accurate in terms of three performance measures; assignment cost, accuracy, and transformation error. In terms of the recognition accuracy, our results show that the hierarchical weighting on features has more significant discriminative power than the classic method (uniform weighting). The hierarchical elastic graph matching (WEGM) approach was used to classify a lexicon of ten hand postures, and it was found that the poses were recognized with a recognition accuracy of 97.08% on average. Using the weighted scheme, computing cycles can be decreased by only computing the features for those nodes whose weight is relatively high and ignoring the remaining nodes. It was found that only 30% of the nodes need to be computed to obtain a recognition accuracy of over 90%.     

Matching preclusion and conditional matching preclusion for pancake and burnt pancake graphs

The matching preclusion number of a graph with an even number of vertices is the minimum number of edges whose deletion destroys all perfect matchings in the graph. The optimal matching preclusion sets are often precisely those which are induced by a single vertex of minimum degree. To look for obstruction sets beyond these, the conditional matching preclusion number was introduced, which is defined similarly with the additional restriction that the resulting graph has no isolated vertices. In this paper we find the matching preclusion and conditional matching preclusion numbers and classify all optimal sets for the pancake graphs and burnt pancake graphs.     

Improving the bit-parallel NFA of Baeza-Yates and Navarro for approximate string matching

We propose a new variant of the bit-parallel NFA of Baeza-Yates and Navarro (BPD) for approximate string matching [R. Baeza-Yates, G. Navarro, Faster approximate string matching, Algorithmica 23 (1999) 127-158]. BPD is one of the most practical approximate string matching algorithms under moderate pattern lengths and error levels [G. Myers, A fast bit-vector algorithm for approximate string matching based on dynamic programming, J. ACM 46 (3) 1989 395-415; G. Navarro, M. Raffinot, Flexible Pattern Matching in Strings—Practical On-line Search Algorithms for Texts and Biological Sequences, Cambridge University Press, Cambridge, UK, 2002]. Given a length-m pattern and an error threshold k, the original BPD requires (m−k)(k+2) bits of space to represent an NFA with (m−k)(k+1) states. In this paper we remove redundancy from the original NFA representation. Our variant requires (m−k)(k+1) bits of space, which is optimal in the sense that exactly one bit per state is used. The space efficiency is achieved by using an alternative, but equally or even more efficient, simulation algorithm for the bit-parallel NFA. We also present experimental results to compare our modified NFA against the original BPD and its main competitors. Our new variant is more efficient than the original BPD, and it hence takes over/extends the role of the original BPD as one of the most practical approximate string matching algorithms under moderate values of k and m.     

A pictorial index mechanism for model-based matching

We are currently developing unified query processing strategies for image databases. To perform this task, model-based representations of images by content are being used, as well as a hierarchical generalization of a relatively new object-recognition technique called data-driven indexed hypotheses. As the name implies, it is index-based, from which its efficiency derives. Earlier approaches to data-driven model-based object recognition techniques were not capable of handling complex image data containing overlapping, partially visible, and touching objects due to the limitations of the features used for building models. Recently, a few data-driven techniques capable of handling complex image data have been proposed. In these techniques, as in traditional databases, iconic index structures are employed to store the image and shape representation in such a way that searching for a given shape or image feature can be conducted efficiently. Some of these techniques handle the insertion and deletion of shapes and/or image representations very efficiently and with very little influence on the overall system performance. However, the main disadvantage of all previous data-driven implementations is that they are main memory based. In the present paper, we describe a secondary memory implementation of data-driven indexed hypotheses along with some performance studies we have conducted.     

Approximate string matching for music analysis

Approximate pattern matching algorithms have become an important tool in computer assisted music analysis and information retrieval. The number of different problem formulations has greatly expanded in recent years, not least because of the subjective nature of measuring musical similarity. From an algorithmic perspective, the complexity of each problem depends crucially on the exact definition of the difference between two strings. We present an overview of advances in approximate string matching in this field focusing on new measures of approximation.     

An online composite graphics recognition approach based on matching of spatial relation graphs

A spatial relation graph (SRG) and its partial matching method are proposed for online composite graphics representation and recognition. The SRG-based approach emphasizes three characteristics of online graphics recognition: partial, structural, and independent of stroke order and stroke number. A constrained partial permutation strategy is also proposed to reduce the computational cost of matching two SRGs, which is originally an NP-complete problem as is graph isomorphism. Experimental results show that our proposed SRG-based approach is both efficient and effective for online composite graphics recognition in our sketch-based graphics input system - SmartSketchpad.Received: 13 March 2003, Accepted: 13 March 2004, Published online: 1 June 2004     

Symbol recognition in electrical diagrams using probabilistic graph matching

In this paper a method is proposed to recognize symbols in electrical diagrams based on probabilistic matching. The skeletons of the symbols are represented by graphs. After finding the pose of the graph (orientation, translation, scale) by a bounded search for a minimum error transformation, the observed graph is matched to the class models and the likelihood of the match is calculated. Results are given for computer-generated symbols and hand drawn symbols with and without a template. Error rates range from <1% to 8%.     

Generalized median graph computation by means of graph embedding in vector spaces

The median graph has been presented as a useful tool to represent a set of graphs. Nevertheless its computation is very complex and the existing algorithms are restricted to use limited amount of data. In this paper we propose a new approach for the computation of the median graph based on graph embedding. Graphs are embedded into a vector space and the median is computed in the vector domain. We have designed a procedure based on the weighted mean of a pair of graphs to go from the vector domain back to the graph domain in order to obtain a final approximation of the median graph. Experiments on three different databases containing large graphs show that we succeed to compute good approximations of the median graph. We have also applied the median graph to perform some basic classification tasks achieving reasonable good results. These experiments on real data open the door to the application of the median graph to a number of more complex machine learning algorithms where a representative of a set of graphs is needed.     

A sparse nonnegative matrix factorization technique for graph matching problems

Graph matching problem that incorporates pairwise constraints can be cast as an Integer Quadratic Programming (IQP). Since it is NP-hard, approximate methods are required. In this paper, a new approximate method based on nonnegative matrix factorization with sparse constraints is presented. Firstly, the graph matching is formulated as an optimization problem with nonnegative and sparse constraints, followed by an efficient algorithm to solve this constrained problem. Then, we show the strong relationship between the sparsity of the relaxation solution and its effectiveness for graph matching based on our model. A key benefit of our method is that the solution is sparse and thus can approximately impose the one-to-one mapping constraints in the optimization process naturally. Therefore, our method can approximate the original IQP problem more closely than other approximate methods. Extensive and comparative experimental results on both synthetic and real-world data demonstrate the effectiveness of our graph matching method.