fast algorithms for computing tree lcs the lcs of two rooted ordered and labeled trees f and g is the largest forest that can be obtained from both trees by deleting nodes we present algorithms for computing tree lcs which exploit the sparsity inherent to the tree lcs problem assuming g is smaller than f our first algorithm runs in time o email protected height f height g lglg g where r is the number of pairs email protected f email protected g such that v and w have the same label our second algorithm runs in time o email protected lglg g where l is the size of the lcs of f and g for this algorithm we present a novel three dimensional alignment graph our third algorithm is intended for the constrained variant of the problem in which only nodes with zero or one children can be deleted for this case we obtain an o rhlglg g time algorithm where h height f height g

eigenfaces vs fisherfaces recognition using class specific linear projection we develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression taking a pattern classification approach we consider each pixel in an image as a coordinate in a high dimensional space we take advantage of the observation that the images of a particular face under varying illumination but fixed pose lie in a 3d linear subspace of the high dimensional image space if the face is a lambertian surface without shadowing however since faces are not truly lambertian surfaces and do indeed produce self shadowing images will deviate from this linear subspace rather than explicitly modeling this deviation we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation our projection method is based on fisher s linear discriminant and produces well separated classes in a low dimensional subspace even under severe variation in lighting and facial expressions the eigenface technique another method based on linearly projecting the image space to a low dimensional subspace has similar computational requirements yet extensive experimental results demonstrate that the proposed fisherface method has error rates that are lower than those of the eigenface technique for tests on the harvard and yale face databases

kernelization and parameterized algorithms for covering a tree by a set of stars or paths two restricted versions of the subforest isomorphism problem the covering a tree by a set of stars ctss and the covering a tree by a set of paths ctsp problems are studied both problems are np complete the problems are closely related to a number of well studied problems including the problems subgraph isomorphism tree editing and graph packing it is shown that the problems ctss and ctsp are fixed parameter tractable thorough development of parameterized algorithms and kernelization algorithms for these problems are presented

algorithms for approximate graph matching abstract labeled graphs are graphs in which each node and edge has a label the distance between two labeled graphs is considered to be the weighted sum of the costs of edit operations insert delete and relabel the nodes and edges to transform one graph to the other the paper considers two variants of the approximate graph matching problem agm given a pattern graph p and a data graph d 1 1 what is the distance between p and d 2 2 what is the minimum distance between p and d when subgraphs can be freely removed from d we first show that no efficient algorithm can solve either variant of the agm unless p np then we present several heuristic algorithms leading to approximate solutions the heuristics are based on probabilistic hill climbing and maximum flow techniques our experimental results involving the comparison of real and simulated data demonstrate the good performance of the algorithms presented

graph characterization via ihara coefficients the novel contributions of this paper are twofold first we demonstrate how to characterize unweighted graphs in a permutation invariant manner using the polynomial coefficients from the ihara zeta function i e the ihara coefficients second we generalize the definition of the ihara coefficients to edge weighted graphs for an unweighted graph the ihara zeta function is the reciprocal of a quasi characteristic polynomial of the adjacency matrix of the associated oriented line graph since the ihara zeta function has poles that give rise to infinities the most convenient numerically stable representation is to work with the coefficients of the quasi characteristic polynomial moreover the polynomial coefficients are invariant to vertex order permutations and also convey information concerning the cycle structure of the graph to generalize the representation to edge weighted graphs we make use of the reduced bartholdi zeta function we prove that the computation of the ihara coefficients for unweighted graphs is a special case of our proposed method for unit edge weights we also present a spectral analysis of the ihara coefficients and indicate their advantages over other graph spectral methods we apply the proposed graph characterization method to capturing graph class structure and clustering graphs experimental results reveal that the ihara coefficients are more effective than methods based on laplacian spectra

tensor based aam with continuous variation estimation application to variation robust face recognition the active appearance model aam is a well known model that can represent a non rigid object effectively however the fitting result is often unsatisfactory when an input image deviates from the training images due to its fixed shape and appearance model to obtain more robust aam fitting we propose a tensor based aam that can handle a variety of subjects poses expressions and illuminations in the tensor algebra framework which consists of an image tensor and a model tensor the image tensor estimates image variations such as pose expression and illumination of the input image using two different variation estimation techniques discrete and continuous variation estimation the model tensor generates variation specific aam basis vectors from the estimated image variations which leads to more accurate fitting results to validate the usefulness of the tensor based aam we performed variation robust face recognition using the tensor based aam fitting results to do we propose indirect aam feature transformation experimental results show that tensor based aam with continuous variation estimation outperforms that with discrete variation estimation and conventional aam in terms of the average fitting error and the face recognition rate

finding similar consensus between trees an algorithm and a distance hierarchy abstract the problem of finding this similar consensus also known as the largest approximately common substructures of two trees arises in many pattern recognition applications this paper presents a dynamic programming algorithm to solve the problem based on the distance measure originated from tanaka and tanaka the algorithm runs as fast as the best known algorithm for comparing two trees using tanaka s distance measure when the allowed distance between the common substructures is a constant independent of the input trees in addition we establish a hierarchy among tanaka s distance measure and three other edit based distance measures published in the literature

diffusion wavelet embedding a multi resolution approach for graph embedding in vector space abstract in this article we propose a multiscale method of embedding a graph into a vector space using diffusion wavelets at each scale we extract a detail subspace and a corresponding lower scale approximation subspace to represent the graph representative features are then extracted at each scale to provide a scale space description of the graph the lower scale is constructed using a super node merging strategy based on nearest neighbor or maximum participation and the new adjacency matrix is generated using vertex identification this approach allows the comparison of graphs where the important structural differences may be present at varying scales additionally this method can improve the differentiating power of the embedded vectors and this property reduces the possibility of cospectrality typical in spectral methods substantially the experimental results show that augmenting the features of abstract levels to the graph features increases the graph classification accuracies in different datasets

bayesian graph edit distance this paper describes a novel framework for comparing and matching corrupted relational graphs the paper develops the idea of edit distance originally introduced for graph matching by sanfeliu and fu 1983 we show how the levenshtein distance 1966 can be used to model the probability distribution for structural errors in the graph matching problem this probability distribution is used to locate matches using map label updates we compare the resulting graph matching algorithm with that recently reported by wilson and hancock the use of edit distance offers an elegant alternative to the exhaustive compilation of label dictionaries moreover the method is polynomial rather than exponential in its worst case complexity we support our approach with an experimental study on synthetic data and illustrate its effectiveness on an uncalibrated stereo correspondence problem this demonstrates experimentally that the gain in efficiency is not at the expense of quality of match

enhancing bilinear subspace learning by element rearrangement the success of bilinear subspace learning heavily depends on reducing correlations among features along rows and columns of the data matrices in this work we study the problem of rearranging elements within a matrix in order to maximize these correlations so that information redundancy in matrix data can be more extensively removed by existing bilinear subspace learning algorithms an efficient iterative algorithm is proposed to tackle this essentially integer programming problem in each step the matrix structure is refined with a constrained earth mover s distance procedure that incrementally rearranges matrices to become more similar to their low rank approximations which have high correlation among features along rows and columns in addition we present two extensions of the algorithm for conducting supervised bilinear subspace learning experiments in both unsupervised and supervised bilinear subspace learning demonstrate the effectiveness of our proposed algorithms in improving data compression performance and classification accuracy

extensions of convexity models the notion of convexity model is introduced to provide a general frame for statistical theories of physical interest this frame encompasses in particular the classical and the quantum cases in a convexity model the states of the physical system and the convex structure they form play a basic role observables and related quantities are then naturally defined the notion of extensions of a convexity model is studied it appears physically relevant to cope with several needs paradigmatically with the one of viewing the physical system as a part of a compound system we focus attention on quantum like extensions of both the usual classical and quantum convexity models as well as on classical like extensions of the quantum model the behavior of state overlapping and state superposition under model extension is briefly examined

on the similarity metric and the distance metric similarity and dissimilarity measures are widely used in many research areas and applications when a dissimilarity measure is used it is normally required to be a distance metric however when a similarity measure is used there is no formal requirement in this article we have three contributions first we give a formal definition of similarity metric second we show the relationship between similarity metric and distance metric third we present general solutions to normalize a given similarity metric or distance metric

graph embedding using tree edit union in this paper we address the problem of how to learn a structural prototype that can be used to represent the variations present in a set of trees the prototype serves as a pattern space representation for the set of trees to do this we construct a super tree to span the union of the set of trees this is a chicken and egg problem since before the structure can be estimated correspondences between the nodes of the super tree and the nodes of the sample tree must be to hand we demonstrate how to simultaneously estimate the structure of the super tree and recover the required correspondences by minimizing the sum of the tree edit distances over pairs of trees subject to edge consistency constraints each node of the super tree corresponds to a dimension of the pattern space and for each tree we construct a pattern vector in which the elements of the weights corresponding to each of the dimensions of the super tree we perform pattern analysis on the set of trees by performing principal components analysis on the vectors the method is illustrated on a shape analysis problem involving shock trees extracted from the skeletons of 2d objects

average complexity of the jiang wang zhang pairwise tree alignment algorithm and of a rna secondary structure alignment algorithm we prove that the average complexity of the pairwise ordered tree alignment algorithm of jiang wang and zhang is in o nm where n and m stand for the sizes of the two trees respectively we show that the same result holds for the average complexity of pairwise comparison of rna secondary structures using a set of biologically relevant operations

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

generalized transition probability mobility and symmetries in the framework of mackey s description of a physical system thegeneralized transition probability as defined in an earlier paper is shown to be non decreasing while the system evolves and invariant when the evolution is reversible it is also in variant under a natural action of the space time symmetry group

error correcting graph matching on the influence of the underlying cost function investigates the influence of the cost function on the optimal match between two graphs it is shown that for a given cost function there are an infinite number of other cost functions that lead for any given pair of graphs to the same optimal error correcting matching furthermore it is shown that well known concepts from graph theory such as graph isomorphism subgraph isomorphism and maximum common subgraph are special cases of optimal error correcting graph matching under particular cost functions

structural matching by discrete relaxation this paper describes a bayesian framework for performing relational graph matching by discrete relaxation our basic aim is to draw on this framework to provide a comparative evaluation of a number of contrasting approaches to relational matching broadly speaking there are two main aspects to this study firstly we focus on the issue of how relational inexactness may be quantified we illustrate that several popular relational distance measures can be recovered as specific limiting cases of the bayesian consistency measure the second aspect of our comparison concerns the way in which structural inexactness is controlled we investigate three different realizations of the matching process which draw on contrasting control models the main conclusion of our study is that the active process of graph editing outperforms the alternatives in terms of its ability to effectively control a large population of contaminating clutter

hierarchical inference of unicast network topologies based on end to end measurements in this paper we address the problem of topology discovery in unicast logical tree networks using end to end measurements without any cooperation from the internal routers topology estimation can be formulated as hierarchical clustering of the leaf nodes based on pairwise correlations as similarity metrics unlike previous work that first assumes the network topology is a binary tree and then tries to generalize to a nonbinary tree we provide a framework that directly deals with general logical tree topologies a hierarchical algorithm to estimate the topology is developed in a recursive manner by finding the best partitions of the leaf nodes level by level our simulations show that the algorithm is more robust than binary tree based methods

computing the aspect graph for line drawings of polyhedral objects an algorithm for computing the aspect graph for polyhedral objects is described the aspects graph is a representation of three dimensional objects by a set of two dimensional views the set of viewpoints on the gaussian sphere is partitioned into regions such that in each region the qualitative structure of the line drawing remains the same at the boundaries between adjacent regions are the accidental viewpoints where the structure for the line drawing changes it is shown that for polyhedral objects there are two fundamental visual events 1 the projections of an edge and a vertex coincide and 2 the projections of three nonadjacent edges intersect at a point the geometry of the object is reflected in the locus of the accidental viewpoints the algorithm computes the partition together with a representative view for each region of the partition

graph embedding and extensions a general framework for dimensionality reduction a large family of algorithms supervised or unsupervised stemming from statistics or geometry theory has been designed to provide different solutions to the problem of dimensionality reduction despite the different motivations of these algorithms we present in this paper a general formulation known as graph embedding to unify them within a common framework in graph embedding each algorithm can be considered as the direct graph embedding or its linear kernel tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided furthermore the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms by utilizing this framework as a tool we propose a new supervised dimensionality reduction algorithm called marginal fisher analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class while the penalty graph connects the marginal points and characterizes the interclass separability we show that mfa effectively overcomes the limitations of the traditional linear discriminant analysis algorithm due to data distribution assumptions and available projection directions real face recognition experiments show the superiority of our proposed mfa in comparison to lda also for corresponding kernel and tensor extensions

on the pattern recognition of noisy subsequence trees we consider the problem of recognizing ordered labeled trees by processing their noisy subsequence trees which are patched up noisy portions of their fragments we assume that h a finite dictionary of ordered labeled trees is given x is an unknown element of h and u is any arbitrary subsequence tree of x we consider the problem of estimating x by processing y which is a noisy version of u the solution which we present is to our knowledge the first reported solution to the problem we solve the problem by sequentially comparing y with every element x of h the basis of comparison being a new dissimilarity measure between two trees which implicitly captures the properties of the corrupting mechanism that noisily garbles u into y the algorithm which incorporates this constraint has been used to test our pattern recognition system and the experimental results obtained demonstrate good accuracy

computing similarity between rna structures the primary structure of a ribonucleic acid rna molecule is a sequence of nucleotides bases over the four letter alphabet a c g u the secondary or tertiary structure of an rna is a set of base pairs nucleotide pairs which forms bonds between au and cg for secondary structures these bonds have been traditionally assumed to be one to one and non crossing this paper considers a notion of similarity between two rna molecule structures taking into account the primary the secondary and the tertiary structures we show that for tertiary structures it is max snp hard for both minimization and maximization versions we show a stronger result for the maximization version where it cannot be approximated within ratio 2logn in polynomial time unless npdtime 2polylogn we then present an algorithm that can be used for practical application our algorithm will produce an optimal solution for the case where at least one of the rna involved is of a secondary structure we also show an approximation algorithm

hypergraph based image retrieval for graph based representation in this paper we introduce a novel method for graph indexing we propose a hypergraph based model for graph data sets by allowing cluster overlapping more precisely in this representation one graph can be assigned to more than one cluster using the concept of the graph median and a given threshold the proposed algorithm detects automatically the number of classes in the graph database we consider clusters as hyperedges in our hypergraph model and we index the graph set by the hyperedge centroids this model is interesting to traverse the data set and efficient to retrieve graphs

an algebraic view of the relation between largest common subtrees and smallest common supertrees the relationship between two important problems in tree pattern matching the largest common subtree and the smallest common supertree problems is established by means of simple constructions which allow one to obtain a largest common subtree of two trees from a smallest common supertree of them and vice versa these constructions are the same for isomorphic homeomorphic topological and minor embeddings they take only time linear in the size of the trees and they turn out to have a clear algebraic meaning

constrained tree editing the distance between two ordered labeled trees is considered to be the minimum sum of the weights associated with the edit operations insertion deletion and substitution required to transform one tree to another the problem of computing this distance and the optimal transformation using no edit constraints has been studied in the literature 3 4 7 9 111 in this paper we consider the problem of transforming one tree t1 to another tree t2 using any arbitrary edit constraint involving the number and type of edit operations to be performed an algorithm to compute this constrained distance is presented if for a tree t span t is defined as the min depth t leaves t the time and space complexities of this algorithm are

finding common structured patterns in linear graphs a linear graph is a graph whose vertices are linearly ordered this linear ordering allows pairs of disjoint edges to be either preceding nesting or crossing given a family of linear graphs and a non empty subset email protected we are interested in the maximum common structured pattern mcsp problem find a maximum size edge disjoint graph with edge pairs all comparable by one of the relations in r that occurs as a subgraph in each of the linear graphs of the family the mcsp problem generalizes many structure comparison and structure prediction problems that arise in computational molecular biology we give tight hardness results for the mcsp problem for structured patterns and structured patterns furthermore we prove that the problem is approximable within ratios i email protected k for structured patterns ii k 1 2 for structured patterns and iii o klogk for structured patterns where k is the size of the optimal solution and k email protected i 1 k1 i is the kth harmonic number also we provide combinatorial results concerning different types of structured patterns that are of independent interest in their own right

what is the set of images of an object under all possible illumination conditions the appearance of an object depends on both the viewpoint from which it is observed and the light sources by which it is illuminated if the appearance of two objects is never identical for any pose or lighting conditions then in theory the objects can always be distinguished or recognized the question arises what is the set of images of an object under all lighting conditions and pose in this paper we consider only the set of images of an object under variable illumination including multiple extended light sources and shadows we prove that the set of n pixel images of a convex object with a lambertian reflectance function illuminated by an arbitrary number of point light sources at infinity forms a convex polyhedral cone in r n and that the dimension of this illumination cone equals the number of distinct surface normals furthermore the illumination cone can be constructed from as few as three images in addition the set of n pixel images of an object of any shape and with a more general reflectance function seen under all possible illumination conditions still forms a convex cone in r n extensions of these results to color images are presented these results immediately suggest certain approaches to object recognition throughout we present results demonstrating the illumination cone representation

computing exact aspect graphs of curved objects algebraic surfaces this article presents an algorithm for computing the exact aspect graph of an opaque solid bounded by a smooth algebraic surface orthographic projection is assumed the algorithm is based on a catalog of visual events available from singularity theory it uses curve tracing cell decomposition homotopy continuation and ray tracing to construct the regions of the view sphere delineated by visual event curves the algorithm has been fully implemented and examples are presented

a physics based analysis of image appearance models linear and multilinear models pca 3dmm aam asm and multilinear tensors of object shape appearance have been very popular in computer vision in this paper we analyze the applicability of these heuristic models from the fundamental physical laws of object motion and image formation we prove that under suitable conditions the image appearance space can be closely approximated to be multilinear with the illumination and texture subspaces being trilinearly combined with the direct sum of the motion and deformation subspaces this result provides a physics based understanding of many of the successes and limitations of the linear and multilinear approaches existing in the computer vision literature and also identifies some of the conditions under which they are valid it provides an analytical representation of the image space in terms of different physical factors that affect the image formation process numerical analysis of the accuracy of the physics based models is performed and tracking results on real data are presented

identifying consensus of trees through alignment abstract in this paper we present a dynamic programming algorithm for identifying the similar consensus sc or the largest approximately common substructures of two ordered labeled trees based on the alignment distance we consider a substructure of a tree t to be a connected subgraph of t given two trees t 1 t 2 and an integer d the sc problem is to find a substructure u 1 of t 1 and a substructure u 2 of t 2 such that u 1 is within distance d of u 2 and where there does not exist any other substructure v 1 of t 1 and v 2 of t 2 such that v 1 and v 2 satisfy the distance constraint and the sum of the sizes of v 1 and v 2 is greater than the sum of the sizes of u 1 and u 2 the proposed algorithm solves the sc problem in time o d 2 t 1 t 2 g 1 g 2 2 where t i i 1 2 is the size of tree t i and g i is the maximum degree of t i this is as fast as the best known algorithm for aligning two ordered labeled trees when the distance allowed in the common substructures is a constant independent of the input trees to demonstrate the utility of our algorithm we discuss its application to characterizing the conformation space of rna sequences pertaining to the poliovirus human rhinovirus and coxsackievirus where the rna structures are represented by ordered labeled trees

metrics on the set of states of a w algebra abstract for any q 1 a metric d q is defined on the set of states of a w algebra it is shown that d 1 is the jauch misra gibson metric whereas d 2 is the bures cantoni metric all these metrics are uniformly equivalent limit properties and alternative expressions are given

an image understanding system using attributed symbolic representation and inexact graph matching this paper presents a powerful image understanding system that utilizes a semantic syntactic or attributed synibolic representation scheme in the form of attributed relational graphs arg s for comprehending the global information contents of images nodes in the arg represent the global image features while the relations between those features are represented by attributed branches between their corresponding nodes the extraction of arg representation from images is achieved by a multilayer graph transducer scheme this scheme is basically a rule based system that uses a combination of model driven and data driven concepts in performing a hierarchical symbolic mapping of the image information content from the spatial domain representation into a global representation further analysis and inter pretation of the imagery data is performed on the extracted arg representation a distance measure between images is defined in terms of the distance between their respective arg representations the distance between two arg s and the inexact matching of their respective components are calculated by an efficient dynamic programming technique the system handles noise distortion and ambiguity in real world images by two means namely through modeling and embedding them into the transducer s mapping rules as well as through the appropriate cost of error transformation for the inexact matching of the arg image representation two illustrative experiments are presented to demonstrate some capabilities of the proposed system experiment i deals with locating objects in multiobject scenes while experiment ii is concerned with target detection in sar images

learning shape classes using a mixture of tree unions this paper poses the problem of tree clustering as that of fitting a mixture of tree unions to a set of sample trees the tree unions are structures from which the individual data samples belonging to a cluster can be obtained by edit operations the distribution of observed tree nodes in each cluster sample is assumed to be governed by a bernoulli distribution the clustering method is designed to operate when the correspondences between nodes are unknown and must be inferred as part of the learning process we adopt a minimum description length approach to the problem of fitting the mixture model to data we make maximum likelihood estimates of the bernoulli parameters the tree unions and the mixing proportions are sought so as to minimize the description length criterion this is the sum of the negative logarithm of the bernoulli distribution and a message length criterion that encodes both the complexity of the union trees and the number of mixture components we locate node correspondences by minimizing the edit distance with the current tree unions and show that the edit distance is linked to the description length criterion the method can be applied to both unweighted and weighted trees we illustrate the utility of the resulting algorithm on the problem of classifying 20 shapes using a shock graph representation

simultaneous tensor decomposition and completion using factor priors the success of research on matrix completion is evident in a variety of real world applications tensor completion which is a high order extension of matrix completion has also generated a great deal of research interest in recent years given a tensor with incomplete entries existing methods use either factorization or completion schemes to recover the missing parts however as the number of missing entries increases factorization schemes may overfit the model because of incorrectly predefined ranks while completion schemes may fail to interpret the model factors in this paper we introduce a novel concept complete the missing entries and simultaneously capture the underlying model structure to this end we propose a method called simultaneous tensor decomposition and completion stdc that combines a rank minimization technique with tucker model decomposition moreover as the model structure is implicitly included in the tucker model we use factor priors which are usually known a priori in real world tensor objects to characterize the underlying joint manifold drawn from the model factors by exploiting this auxiliary information our method leverages two classic schemes and accurately estimates the model factors and missing entries we conducted experiments to empirically verify the convergence of our algorithm on synthetic data and evaluate its effectiveness on various kinds of real world data the results demonstrate the efficacy of the proposed method and its potential usage in tensor based applications it also outperforms state of the art methods on multilinear model analysis and visual data completion tasks

consistency driven alternating optimization for multigraph matching a unified approach the problem of graph matching gm in general is nondeterministic polynomial complete and many approximate pairwise matching techniques have been proposed for a general setting in real applications it typically requires to find the consistent matching across a batch of graphs sequentially performing pairwise matching is prone to error propagation along the pairwise matching sequence and the sequences generated in different pairwise matching orders can lead to contradictory solutions motivated by devising a robust and consistent multiple gm model we propose a unified alternating optimization framework for multi gm in addition we define and use two metrics related to graphwise and pairwise consistencies the former is used to find an appropriate reference graph which induces a set of basis variables and launches the iteration procedure the latter defines the order in which the considered graphs in the iterations are manipulated we show two embodiments under the proposed framework that can cope with the nonfactorized and factorized affinity matrix respectively our multi gm model has two major characters 1 the affinity information across multiple graphs are explored in each iteration by fixing part of the matching variables via a consistency driven mechanism and 2 the framework is flexible to incorporate various existing pairwise gm solvers in an out of box fashion and also can proceed with the output of other multi gm methods the experimental results on both synthetic data and real images empirically show that the proposed framework performs competitively with the state of the art

forest alignment with affine gaps and anchors applied in rna structure comparison we present two enhancements to jiang s tree alignment algorithm motivated by experience with its use for rna structure alignment one enhancement is the introduction of an affine gap model which can be accommodated with a runtime increase by a constant factor the second enhancement is a speed up of the alignment algorithm when certain nodes in the trees are pre aligned by a so called anchoring both enhancements are included in a new implementation of the tool rnaforester we evaluate the new algorithm with two applications related to rna secondary structure analysis based on our experience we suggest a new formulation of the tree alignment model based on regular tree languages and rewrite rules

recent advances in graph based pattern recognition with applications in document analysis graphs are a powerful and popular representation formalism in pattern recognition particularly in the field of document analysis they have found widespread application from the formal point of view however graphs are quite limited in the sense that the majority of mathematical operations needed to build common algorithms such as classifiers or clustering schemes are not defined consequently we observe a severe lack of algorithmic procedures that can directly be applied to graphs there exists recent work however aimed at overcoming these limitations the present paper first provides a review of the use of graph representations in document analysis then we discuss a number of novel approaches suitable for making tools from statistical pattern recognition available to graphs these novel approaches include graph kernels and graph embedding with several experiments using different data sets from the field of document analysis we show that the new methods have great potential to outperform traditional procedures applied to graph representations

a long trip in the charming world of graphs for pattern recognition this paper is a historical overview of graph based methodologies in pattern recognition in the last 40 years history is interpreted with the aim of recognizing the rationale inspiring the papers published in these years so as to roughly classify them despite the extent of scientific production in this field it is possible to identify three historical periods each having its own connotation common to most of the corresponding papers which are called here as the pure the impure and extreme periods highlightsthe paper is a historical survey of graph based methodologies in pattern recognition the scientific evolution of graph based methods over their history is discussed the paper analyses the trend of reusing vector space methods for graphs three distinctive periods are recognized wrt how the vector graph gap is filled reserve a ticket for a historical trip in this charming world and discover it soon

an algorithm for finding the largest approximately common substructures of two trees ordered labeled trees are trees in which each node has a label and the left to right order of its children if it has any is fixed such trees have many applications in vision pattern recognition molecular biology and natural language processing we consider a substructure of an ordered labeled tree t to be a connected subgraph of t given two ordered labeled trees t sub 1 and t sub 2 and an integer d the largest approximately common substructure problem is to find a substructure u sub 1 of t sub 1 and a substructure u sub 2 of t sub 2 such that u sub 1 is within edit distance d of u sub 2 and where there does not exist any other substructure v sub 1 of t sub 1 and v sub 2 of t sub 2 such that v sub 1 and v sub 2 satisfy the distance constraint and the sum of the sizes of v sub 1 and v sub 2 is greater than the sum of the sizes of u sub 1 and u sub 2 we present a dynamic programming algorithm to solve this problem which runs as fast as the fastest known algorithm for computing the edit distance of two trees when the distance allowed in the common substructures is a constant independent of the input trees to demonstrate the utility of our algorithm we discuss its application to discovering motifs in multiple rna secondary structures which are ordered labeled trees

finding approximate patterns in undirected acyclic graphs abstract we consider an approximate pattern matching problem for undirected acyclic graphs specifically let p be a pattern graph d a data graph and t an integer we present an algorithm to locate a subgraph in d whose distance from p is at most t the distance measure used here is the degree 2 metric published previously the time complexity of our algorithm is o n p n d d d log d where n p and n d are the number of nodes in p and d respectively d min d p d d d p and d d are the maximum degree of p and d respectively central to our algorithm is a procedure for finding approximate patterns in rooted unordered trees and freely allowing cuts we discuss two applications of the algorithms in chemical information search and website management on the internet

fuzzy multilevel graph embedding structural pattern recognition approaches offer the most expressive convenient powerful but computational expensive representations of underlying relational information to benefit from mature less expensive and efficient state of the art machine learning models of statistical pattern recognition they must be mapped to a low dimensional vector space our method of explicit graph embedding bridges the gap between structural and statistical pattern recognition we extract the topological structural and attribute information from a graph and encode numeric details by fuzzy histograms and symbolic details by crisp histograms the histograms are concatenated to achieve a simple and straightforward embedding of graph into a low dimensional numeric feature vector experimentation on standard public graph datasets shows that our method outperforms the state of the art methods of graph embedding for richly attributed graphs highlights we propose an explicit graph embedding method we perform multilevel analysis of graph to extract global topological structural and attribute information we use homogeneity of subgraphs in graph for extracting topological structural details we encode numeric information by fuzzy histograms and symbolic information by crisp histograms our method outperforms graph embedding methods for richly attributed graphs

on cantoni s generalized transition probability we obtain simple expressions of the generalized transition probability proposed by v cantoni for both classical and quantum mechanics we compare the result with the ordinary quantum mechanical transition probability

face recognition using laplacianfaces we propose an appearance based face recognition method called the laplacianface approach by using locality preserving projections lpp the face images are mapped into a face subspace for analysis different from principal component analysis pca and linear discriminant analysis lda which effectively see only the euclidean structure of face space lpp finds an embedding that preserves local information and obtains a face subspace that best detects the essential face manifold structure the laplacianfaces are the optimal linear approximations to the eigenfunctions of the laplace beltrami operator on the face manifold in this way the unwanted variations resulting from changes in lighting facial expression and pose may be eliminated or reduced theoretical analysis shows that pca lda and lpp can be obtained from different graph models we compare the proposed laplacianface approach with eigenface and fisherface methods on three different face data sets experimental results suggest that the proposed laplacianface approach provides a better representation and achieves lower error rates in face recognition

first order gaussian graphs for efficient structure classification abstract first order random graphs as introduced by wong are a promising tool for structure based classification their complexity however hampers their practical application we describe an extension to first order random graphs which uses continuous gaussian distributions to model the densities of all random elements in a random graph these first order gaussian graphs foggs are shown to have several nice properties which allow for fast and efficient clustering and classification specifically we show how the entropy of a fogg may be computed directly from the gaussian parameters of its random elements this allows for fast and memoryless computation of the objective function used in the clustering procedure used for learning a graphical model of a class we give a comparative evaluation between foggs and several traditional statistical classifiers on our example problem selected from the area of document analysis our first order gaussian graph classifier significantly outperforms statistical feature based classifiers the fogg classifier achieves a classification accuracy of approximately 98 while the best statistical classifiers only manage approximately 91

a constrained edit distance algorithm between semi ordered trees in this paper we propose a formal definition of a new class of trees called semi ordered trees and a polynomial dynamic programming algorithm to compute a constrained edit distance between such trees the core of the method relies on a similar approach to compare unordered kaizhong zhang a constrained edit distance between unordered labeled trees algorithmica 15 1996 205 222 and ordered trees kaizhong zhang algorithms for the constrained editing distance between ordered labeled trees and related problems pattern recognition 28 3 1995 463 474 the method is currently applied to evaluate the similarity between architectures of apple trees vincent segura aida ouangraoua pascal ferraro evelyne costes comparison of tree architecture using tree edit distances application to two year old apple tree euphytica 161 2007 155 164

the bell phenomenon in classical frameworks the experimentally tested occurrence of probabilities that do not find a representation within the usual kolmogorov probability theory is mathematically formalized through the notion of the bell phenomenon for probability measures and for observables reference is made to a physically natural definition of observables that includes both the classical and the quantum usual versions it is shown that even a classical framework can host the bell phenomenon provided fuzzy observables are called into play

polynomial time metrics for attributed trees we address the problem of comparing attributed trees and propose four novel distance measures centered around the notion of a maximal similarity common subtree the proposed measures are general and defined on trees endowed with either symbolic or continuous valued attributes and can be applied to rooted as well as unrooted trees we prove that our measures satisfy the metric constraints and provide a polynomial time algorithm to compute them this is a remarkable and attractive property since the computation of traditional edit distance based metrics is in general np complete at least in the unordered case we experimentally validate the usefulness of our metrics on shape matching tasks and compare them with an approximation of edit distance

multilinear graph embedding representation and regularization for images given a set of images finding a compact and discriminative representation is still a big challenge especially when multiple latent factors are hidden in the way of data generation to represent multifactor images although multilinear models are widely used to parameterize the data most methods are based on high order singular value decomposition hosvd which preserves global statistics but interprets local variations inadequately to this end we propose a novel method called multilinear graph embedding mge as well as its kernelization mkge to leverage the manifold learning techniques into multilinear models our method theoretically links the linear nonlinear and multilinear dimensionality reduction we also show that the supervised mge encodes informative image priors for image regularization provided that an image is represented as a high order tensor from our experiments on face and gait recognition the superior performance demonstrates that mge better represents multifactor images than classic methods including hosvd and its variants in addition the significant improvement in image or tensor completion validates the potential of mge for image regularization

a new algorithm for computing similarity between rna structures abstract the primary structure of a ribonucleic acid rna molecule is a sequence of nucleotides bases over the four letter alphabet a c g u the secondary or tertiary structure of an rna is a set of base pairs nucleotide pairs which form bonds between a u and c g for secondary structures these bonds have been traditionally assumed to be one to one and non crossing we consider the edit distance between two rna structures this is a notion of similarity introduced in proceedings of the tenth symposium on combinatorial pattern matching lecture notes in computer science vol 1645 springer berlin 1999 p 281 between two rna molecule structures taking into account the primary the secondary and the tertiary structures in general this problem is np hard for tertiary structures in this paper we consider this notion under some constraints we present an algorithm and then show how to use this algorithm for practical applications

rna secondary structure comparison exact analysis of the zhang shasha tree edit algorithm we are interested in rna secondary structure comparison using an approach which consists to represent these structures by labeled ordered trees following the problem considered this tree representation can be rough considering only the structural patterns or refined until an exact coding of the structure is obtained after some preliminary definitions and the description of the zhang shasha siam j comput 18 6 1989 1245 tree edit algorithm which is on the one hand the reference when dealing with ordered labeled trees comparison and on the other hand the starting point of our work this article will present an exact analysis of its complexity the purpose of this work is also to lead us to a better comprehension of the parameters of this algorithm in order to be able to modify it more easily without changing its time complexity to take into account biological constraints that occur when comparing rna secondary structures

indexing hierarchical structures using graph spectra hierarchical image structures are abundant in computer vision and have been used to encode part structure scale spaces and a variety of multiresolution features in this paper we describe a framework for indexing such representations that embeds the topological structure of a directed acyclic graph dag into a low dimensional vector space based on a novel spectral characterization of a dag this topological signature allows us to efficiently retrieve a promising set of candidates from a database of models using a simple nearest neighbor search we establish the insensitivity of the signature to minor perturbation of graph structure due to noise occlusion or node split merge to accommodate large scale occlusion the dag rooted at each nonleaf node of the query votes for model objects that share that part effectively accumulating local evidence in a model dag s topological subspaces we demonstrate the approach with a series of indexing experiments in the domain of view based 3d object recognition using shock graphs

pattern vectors from algebraic graph theory graph structures have proven computationally cumbersome for pattern analysis the reason for this is that before graphs can be converted to pattern vectors correspondences must be established between the nodes of structures which are potentially of different size to overcome this problem in this paper we turn to the spectral decomposition of the laplacian matrix we show how the elements of the spectral matrix for the laplacian can be used to construct symmetric polynomials that are permutation invariants the coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner we extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a hermitian property matrix that can be viewed as a complex analogue of the laplacian to embed the graphs in a pattern space we explore whether the vectors of invariants can be embedded in a low dimensional space using a number of alternative strategies including principal components analysis pca multidimensional scaling mds and locality preserving projection lpp experimentally we demonstrate that the embeddings result in well defined graph clusters our experiments with the spectral representation involve both synthetic and real world data the experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure the real world experiments show that the method can be used to locate clusters of graphs

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

alignment of trees an alternative to tree edit abstract in this paper we propose the alignment of trees as a measure of the similarity between two labeled trees both ordered and unordered trees are considered an algorithm is designed for ordered trees the time complexity of this algorithm is o t 1 t 2 deg t 1 deg t 2 2 where t 1 is the number of nodes in t 1 and deg t 1 is the degree of t 1 i 1 2 the algorithm is faster than the best known algorithm for tree edit when deg t 1 and deg t 2 are smaller than the depths of t 1 and t 2 for unordered trees we show that the alignment problem can be solved in polynomial time if the trees have a bounded degree and becomes max snp hard if one of the trees is allowed to have an arbitrary degree in contrast the edit problem for unordered trees is max snp hard even if both trees have a bounded degree zhang and jiang 1994 finally multiple alignment of trees is discussed

illumination planning for object recognition using parametric eigenspaces presents a novel approach to the problem of illumination planning for robust object recognition in structured environments given a set of objects the goal is to determine the illumination for which the objects are most distinguishable in appearance from each other correlation is used as a measure of similarity between objects for each object a large number of images is automatically obtained by varying the pose and the illumination direction images of all objects together constitute the planning image set the planning set is compressed using the karhunen loeve transform to obtain a low dimensional subspace called the eigenspace for each illumination direction objects are represented as parametrized manifolds in the eigenspace the minimum distance between the manifolds of two objects represents the similarity between the objects in the correlation sense the optimal source direction is therefore the one that maximizes the shortest distance between the object manifolds several experiments have been conducted using real objects the results produced by the illumination planner have been used to enhance the performance of an object recognition system

graph edit distance from spectral seriation this paper is concerned with computing graph edit distance one of the criticisms that can be leveled at existing methods for computing graph edit distance is that they lack some of the formality and rigor of the computation of string edit distance hence our aim is to convert graphs to string sequences so that string matching techniques can be used to do this we use a graph spectral seriation method to convert the adjacency matrix into a string or sequence order we show how the serial ordering can be established using the leading eigenvector of the graph adjacency matrix we pose the problem of graph matching as a maximum a posteriori probability map alignment of the seriation sequences for pairs of graphs this treatment leads to an expression in which the edit cost is the negative logarithm of the a posteriori sequence alignment probability we compute the edit distance by finding the sequence of string edit operations which minimizes the cost of the path traversing the edit lattice the edit costs are determined by the components of the leading eigenvectors of the adjacency matrix and by the edge densities of the graphs being matched we demonstrate the utility of the edit distance on a number of graph clustering problems

structural matching in computer vision using probabilistic relaxation in this paper we develop the theory of probabilistic relaxation for matching features extracted from 2d images derive as limiting cases the various heuristic formulae used by researchers in matching problems and state the conditions under which they apply we successfully apply our theory to the problem of matching and recognizing aerial road network images based on road network models and to the problem of edge matching in a stereo pair for this purpose each line network is represented by an attributed relational graph where each node is a straight line segment characterized by certain attributes and related with every other node via a set of binary relations

an introduction to kernel based learning algorithms this paper provides an introduction to support vector machines kernel fisher discriminant analysis and kernel principal component analysis as examples for successful kernel based learning methods we first give a short background about vapnik chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and unsupervised scenarios including practical and algorithmic considerations we illustrate the usefulness of kernel algorithms by discussing applications such as optical character recognition and dna analysis

a survey on tree edit distance and related problems we survey the problem of comparing labeled trees based on simple local operations of deleting inserting and relabeling nodes these operations lead to the tree edit distance alignment distance and inclusion problem for each problem we review the results available and present in detail one or more of the central algorithms for solving the problem

simple fast algorithms for the editing distance between trees and related problems ordered labeled trees are trees in which the left to right order among siblings is significant the distance between two ordered trees is considered to be the weighted number of edit operations insert delete and modify to transform one tree to another the problem of approximate tree matching is also considered specifically algorithms are designed to answer the following kinds of questions 1 what is the distance between two trees 2 what is the minimum distance between t 1 and t 2 when zero or more subtrees can be removed from t 2 3 let the pruning of a tree at node n mean removing all the descendants of node n the analogous question for prunings as for subtrees is answered a dynamic programming algorithm is presented to solve the three questions in sequential time o t 1 times t 2 times min textit depth t 1 textit leaves t 1 times min textit depth t 2 textit leaves t 2 and space o t 1 times t 2 compared with o t 1 times t 2 t

effect algebras and statistical physical theories the dichotomic physical quantities of a physical system can be naturally hosted in a mathematical structure called effect algebra of which orthomodular posets and boolean algebras are particular examples we examine how effect algebras arise inside statistical physical theories and conversely we study to what extent an effect algebra can be taken as a primitive structure on which a satisfactory statistical physical model equipped with a convex set of states can be constructed

image categorization graph edit distance edge direction histogram this paper presents a novel algorithm for computing graph edit distance ged in image categorization this algorithm is purely structural i e it needs only connectivity structure of the graph and does not draw on node or edge attributes there are two major contributions 1 introducing edge direction histogram edh to characterize shape features of images it is shown that ged can be employed as distance of edhs this algorithm is completely independent on cost function which is difficult to be defined exactly 2 computing distance of edhs with earth mover distance emd which takes neighborhood bins into account so as to compute distance of edhs correctly a set of experiments demonstrate that the newly presented algorithm is available for classifying and clustering images and is immune to the planar rotation of images compared with ged from spectral seriation our algorithm can capture the structure change of graphs better and consume 12 79 time used by the former one the average classification rate is 5 and average clustering rate is 25 higher than the spectral seriation method

the geometry of view space of opaque objects bounded by smooth surfaces abstract a view of a smooth opaque object is a line drawing consisting of contour fragments which form closed loops terminate or form t junctions given a compact object we have a global decomposition of the space of camera positions into cells from which topologically equivalent views are obtained to apply certain image based object recognition techniques based on characteristic views of an object one needs to know this cellular decomposition of view space the boundaries of such cells of stable views are formed by the view bifurcation set from which transitional degenerate views are seen we derive a finite list of geometrical models of the view bifurcation set the view bifurcation set of any particular object is a combination of such model surfaces from this list to match a characteristic view with an image one has to find the visible fragments of the apparent contour in the image and we briefly describe an observation which might lead to an algorithm for finding such contour fragments

the enumerative geometry of projective algebraic surfaces and the complexity of aspect graphs the aspect graph is a popular viewer centered representation that enumerates all the topologically distinct views of an object building the aspect graph requires partitioning viewpoint space in view equivalent cells by a certain number of visual event surfaces if the object is piecewise smooth algebraic then all visual event surfaces are either made of lines having specified contacts with the object or made of lines supporting the points of contacts of planes having specified contacts with the object in this paper we present a general framework for studying the enumerative properties of line and plane systems the context is that of enumerative geometry and intersection theory in particular we give exact results for the degrees of all visual event surfaces coming up in the construction of aspect graphs of piecewise smooth algebraic bodies we conclude by giving a bound on the number of topologically distinet views of such objects

quantal effects and maxent convex operational models coms are considered as great extrapolations to larger settings of any statistical theory in this article we generalize the maximum entropy principle maxent of jaynes to any com after expressing maxent in a geometrical and lattice theoretical setting we are able to cast it for any com this scope amplification opens the door to a new systematization of the principle and sheds light into its geometrical structure

numerical similarity and dissimilarity measures between two trees quantifying the measure of similarity between two trees is a problem of intrinsic importance in the study of algorithms and data structures and has applications in computational molecular biology structural syntactic pattern recognition and in data management in this paper we define and formulate an abstract measure of comparison spl omega t sub 1 t sub 2 between two trees t sub 1 and t sub 2 presented in terms of a set of elementary intersymbol measures spl omega and two abstract operators spl oplus and spl otimes by appropriately choosing the concrete values for these two operators and for spl omega this measure can be used to define various quantities including 1 the edit distance between two trees 2 the size of their largest common subtree 3 prob t sub 2 t sub 1 the probability of receiving t sub 2 given that t sub 1 was transmitted across a channel causing independent substitution and deletion errors and 4 the a posteriori probability of t sub 1 being the transmitted tree given that t sub 2 is the received tree containing independent substitution insertion and deletion errors the recursive properties of spl omega t sub 1 t sub 2 have been derived and a single generic iterative dynamic programming scheme to compute all the above quantities has been developed the time and space complexities of the algorithm have been analyzed and the implications of our results in both theoretical and applied fields has been discussed

algorithms for the constrained editing distance between ordered labeled trees and related problems abstract this paper considers the problem of computing a constrained editing distance between ordered labeled trees this distance metric can be applied to pattern recognition syntactic tree comparison classification tree comparison and other applications the problem of approximate ordered tree matching is also considered we present optimal algorithms for solving these problems in sequential time o t1 t2

generalized lcs the longest common subsequence lcs is a well studied problem having a wide range of implementations its motivation is in comparing strings it has long been of interest to devise a similar measure for comparing higher dimensional objects and more complex structures in this paper we study the longest common substructure of two matrices and show that this problem is np hard we also study the longest common subforest problem for multiple trees including a constrained version as well we show np hardness for k 2 unordered trees in the constrained lcs we also give polynomial time algorithms for ordered trees and prove a lower bound for any decomposition strategy for k trees

computing exact aspect graphs of curved objects solid of revolution this paper introduces a new approach to computing the exact orthographic aspect graph of curved objects curves corresponding to various visual events partition the view sphere into regions where the image structure is stable a catalog of these events for piecewise smooth objects is available from singularity theory for a solid of revolution whose generator is an algebraic curve each visual event is characterized by a system of polynomial equations whose roots can be computed by continuation methods within each region the stable image structure is characterized by a variation of cylindrical algebraic decomposition and ray tracing this approach has been implemented and several examples are presented

identifying approximately common substructures in trees based on a restricted edit distance in this paper we present a dynamic programming algorithm for identifying the largest approximately common substructures lacss of two ordered labeled trees we consider a substructure of a tree t to be a connected subgraph of t given two trees t1 t2 and an integer d the lacs problem is to find a substructure u1 of t1 and a substructure u2 of t2 such that u1 is within distance d of u2 and where there does not exist any other substructure v1 of t1 and v2 of t2 such that v1 and v2 satisfy the distance restriction and the sum of the sizes of v1 and v2 is greater than the sum of the sizes of u1 and u2 the distance measure considered in the paper is a restricted edit distance originated from selkow the lacs problem is motivated by the studies of program and document comparison the proposed algorithm solves the problem in time o d2 t1 t2 which is as fast as the best known algorithm for calculating selkow s distance of two trees when the distance allowed in the common substructures is a constant independent of the input trees

a spectral approach to learning structural variations in graphs this paper shows how to construct a linear deformable model for graph structure by performing principal components analysis pca on the vectorised adjacency matrix we commence by using correspondence information to place the nodes of each of a set of graphs in a standard reference order using the correspondences order we convert the adjacency matrices to long vectors and compute the long vector covariance matrix by projecting the vectorised adjacency matrices onto the leading eigenvectors of the covariance matrix we embed the graphs in a pattern space we illustrate the utility of the resulting method for shape analysis

spectral embedding of graphs in this paper we explore how to embed symbolic relational graphs with unweighted edges in a pattern space we adopt a graph spectral approach we use the leading eigenvectors of the graph adjacency matrix to define eigenmodes of the adjacency matrix for each eigenmode we compute vectors of spectral properties these include the eigenmode perimeter eigenmode volume cheeger number inter mode adjacency matrices and intermode edge distance we embed these vectors in a pattern space using two contrasting approaches the first of these involves performing principal or independent components analysis on the covariance matrix for the spectral pattern vectors the second approach involves performing multidimensional scaling on the l2 norm for pairs of pattern vectors we illustrate the utility of the embedding methods on neighbourhood graphs representing the arrangement of corner features in 2d images of 3d polyhedral objects two problems are investigated the first of these is the clustering of graphs representing distinct objects viewed from different directions the second is the identification of characteristic views of single objects these two studies reveal that both embedding methods result in well structured view spaces for graph data extracted from 2d views of 3d objects

a new algorithm for aligning nested arc annotated sequences under arbitrary weight schemes in this paper we propose a new algorithm for the alignment of nested arc annotated sequences having applications in the comparison of rna secondary structures without pseudo knots we use a general edit distance model between arc annotated sequences that considers classical sequences of edit operations and structural edit operations on arcs in this model the general edit distance problem under a non constrained weight scheme is np hard recently a hierarchy of arc annotated sequence alignment problems that highlights less general but tractable problems was introduced we refine this hierarchy of alignment problems and extend the class of tractable alignment problems up to date the alignment problem we solve is the most general one that is known to be tractable in the considered edit distance model and under arbitrary weight schemes this algorithm is efficient as its asymptotic time and space complexities are the same as the complexities of the best previously published algorithm