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

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

roughness in cayley graphs in this paper rough approximations of cayley graphs are studied and rough edge cayley graphs are introduced furthermore a new algebraic definition for pseudo cayley graphs containing cayley graphs is proposed and a rough approximation is expanded to pseudo cayley graphs in addition rough vertex pseudo cayley graphs and rough pseudo cayley graphs are introduced some theorems are provided from which properties such as connectivity and optimal connectivity are derived this approach opens new research fields such as data networks

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

distance patterns in structural similarity similarity of edge labeled graphs is considered in the sense of minimum squared distance between corresponding values vertex correspondences are established by isomorphisms if both graphs are of equal size and by subisomorphisms if one graph has fewer vertices than the other best fit isomorphisms and subisomorphisms amount to solutions of quadratic assignment problems and are computed exactly as well as approximately by minimum cost flow linear assignment relaxations and related graph algorithms

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

on the structure of generalized rough sets in this paper we consider some fundamental properties of generalized rough sets induced by binary relations on algebras and show that 1 any reflexive binary relation determines a topology 2 if q is a reflexive and symmetric relation on a set x then o email protected x q a a is a topology such that a is open if and only if it is closed 3 conversely for every topological space x o satisfying the condition that a is open if and only if it is closed there exists a reflexive and symmetric relation r such that o email protected x r a a 4 let q be an equivalence relation on x for any pseudo w closed subset a of x q a is an w closed set if and only if w x x x email protected a for any email protected x moreover we consider properties of generalized rough sets

extensions and intentions in the rough set theory abstract the approach to rough set theory proposed in this paper is based on the mutual correspondence of the concepts of extension and intension it is different from the well known approaches in the literature in that the upper approximations and the lower approximations of unknown sets are considered as certain families of known sets this approach makes it possible to formulate necessary and sufficient conditions for the existence of operations on rough sets which are analogous to classical operations on sets the basic results presented in this paper based on certain ideas of the second author were formulated by the first author in his doctoral dissertation prepared under the supervision of the third author

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

rough set theory applied to fuzzy ideal theory we use covers of the universal set to define approximation operators on the power set of the given set in section 1 we determine basic properties of the upper approximation operator and show how it can be used to give algebraic structural properties of certain subsets we define a particular cover on the set of ideals of a commutative ring with identity in such a way that both the concepts of the fuzzy prime spectrum of a ring and rough set theory can simultaneously be brought to bear on the study of fuzzy ideals of a ring

on the structure of rough prime primary ideals and rough fuzzy prime primary ideals in commutative rings this paper is a continuation of ideas presented by davvaz roughness in rings inform sci 164 2004 147 163 roughness based on fuzzy ideals inform sci 176 2006 2417 2437 we introduce the notions of rough prime primary ideals and rough fuzzy prime primary ideals in a ring and give some properties of such ideals also we discuss the relations between the upper and lower rough prime primary ideals and the upper and lower approximations of their homomorphism images

approximations and reducts with covering generalized rough sets the covering generalized rough sets are an improvement of traditional rough set model to deal with more complex practical problems which the traditional one cannot handle it is well known that any generalization of traditional rough set theory should first have practical applied background and two important theoretical issues must be addressed the first one is to present reasonable definitions of set approximations and the second one is to develop reasonable algorithms for attributes reduct the existing covering generalized rough sets however mainly pay attention to constructing approximation operators the ideas of constructing lower approximations are similar but the ideas of constructing upper approximations are different and they all seem to be unreasonable furthermore less effort has been put on the discussion of the applied background and the attributes reduct of covering generalized rough sets in this paper we concentrate our discussion on the above two issues we first discuss the applied background of covering generalized rough sets by proposing three kinds of datasets which the traditional rough sets cannot handle and improve the definition of upper approximation for covering generalized rough sets to make it more reasonable than the existing ones then we study the attributes reduct with covering generalized rough sets and present an algorithm by using discernibility matrix to compute all the attributes reducts with covering generalized rough sets with these discussions we can set up a basic foundation of the covering generalized rough set theory and broaden its applications

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

roughness of a fuzzy set an integration between the theories of fuzzy sets and rough sets has been attempted by providing a measure of roughness of a fuzzy set several properties of this new measure are established some of the possible applications for handling uncertainties in the field of pattern recognition are mentioned

existence and multiplicity of solutions for asymptotically linear noncooperative elliptic systems abstract using the minimax methods in critical point theory and a generalized landesman lazer type condition we establish two existence results of solutions for asymptotically linear noncooperative elliptic systems at resonance besides this we obtain two solutions in the case of near resonance

relationship between generalized rough sets based on binary relation and covering rough set theory is a powerful tool for dealing with uncertainty granularity and incompleteness of knowledge in information systems this paper systematically studies a type of generalized rough sets based on covering and the relationship between this type of covering based rough sets and the generalized rough sets based on binary relation firstly we present basic concepts and properties of this kind of rough sets then we investigate the relationships between this type of generalized rough sets and other five types of covering based rough sets the major contribution in this paper is that we establish the equivalency between this type of covering based rough sets and a type of binary relation based rough sets through existing results in binary relation based rough sets we present axiomatic systems for this type of covering based lower and upper approximation operations in addition we explore the relationships among several important concepts such as minimal description reduction representative covering exact covering and unary covering in covering based rough sets investigation of this type of covering based will benefit to our understanding of other types of rough sets based on covering and binary relation

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

a tree matching algorithm based on node splitting and merging an algorithm that computes the best matching of two trees is described the degree of mismatch i e the distance is measured in terms of the number of node splitting and merging operations required the proposed tree distance is a more appropriate measurement of structural defonnation than the tree distance measure in terms of the number of insertions deletions and substitutions of tree nodes as defined in previous studies an algorithm that uses a divide and conquer strategy is presented the analysis shows that the time complexity is o nm2 where n and al are the number of nodes of the two trees respectively the algorithm has been implemented on a vax 11 780

distance transform for images represented by quadtrees the concept of distance used in binary array representations of images is adapted to a quadtree representation the chessboard distance metric is shown to be particularly suitable for the quadtree a chessboard distance transform for a quadtree is defined as the minimum distance in the plane from each black node to the border of a white node an algorithm is presented which computes this transform by only examining the black node s adjacent and abutting neighbors and their progeny however unlike prior work with quadtrees computation of the distance transform requires a capability of finding neighbors in the diagonal direction rather than merely in the horizontal and vertical directions the algorithm s average execution time is proportional to the number of leaf nodes in the quadtree

a short note on algebraic t rough sets our aim in this paper is to introduce the concept of a set valued homomorphism for groups which is a generalization of ordinary homomorphism then using the definitions of lower inverse and upper inverse we introduce the definition of uniform set valued homomorphism and prove that every set valued homomorphism is uniform finally the concept of probabilistic t rough set is defined

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

some properties of generalized rough sets some algebraic and topological properties of generalized rough sets associated with generalized lower and upper approximations are studied generalized definable sets give rise to two topological spaces degree of accuracy dag for generalized rough sets induces two types of equivalence relations one is defined on x whereas the other is defined on p y both of these relations strictly manage an order in classes of x and p y it has been shown that dag induces two types of fuzzy subsets one of the set x and the other of p y finally some properties of these fuzzy sets have been discussed

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

roughness in mv algebras in this paper by considering the notion of an mv algebra we consider a relationship between rough sets and mv algebra theory we introduce the notion of rough ideal with respect to an ideal of an mv algebra which is an extended notion of ideal in an mv algebra and we give some properties of the lower and the upper approximations in an mv algebra

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

the further investigation of covering based rough sets uncertainty characterization similarity measure and generalized models the notion of rough sets was originally proposed by pawlak in pawlak s rough set theory the equivalence relation or partition plays an important role however the equivalence relation or partition is restrictive for many applications because it can only deal with complete information systems this limits the theory s application to a certain extent therefore covering based rough sets are derived by replacing the partitions of a universe with its coverings this paper focuses on the further investigation of covering based rough sets firstly we discuss the uncertainty of covering in the covering approximation space and show that it can be characterized by rough entropy and the granulation of covering secondly since it is necessary to measure the similarity between covering rough sets in practical applications such as pattern recognition image processing and fuzzy reasoning we present an approach which measures these similarities using a triangular norm we show that in a covering approximation space a triangular norm can induce an inclusion degree and that the similarity measure between covering rough sets can be given according to this triangular norm and inclusion degree thirdly two generalized covering based rough set models are proposed and we employ practical examples to illustrate their applications finally relationships between the proposed covering based rough set models and the existing rough set models are also made

bifurcation from infinity and multiple solutions for some discrete sturm liouville problems in this paper we study nonlinear discrete boundary value problems of the form d p t 1 dy t 1 q t y t l email protected y t f t y t h t email protected t a 1 1y a a 1 email protected a 0 a 2 1y b 1 a 2 email protected b 1 0 where l k is an eigenvalue of the associated linear problem l is a parameter and f satisfies the sublinear growth condition f t s a s a b email protected a 1 b 1 email protected r for some email protected email protected 1 and a email protected 0 we study the existence and multiplicity of solutions for the above mentioned problems by establishing some a priori bounds together with leray schauder continuation and bifurcation arguments

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

roughness in modules the theory of rough sets was proposed by pawlak in 1982 the relations between rough sets and algebraic systems have been already considered by many mathematicians important algebraic structures are groups rings and modules rough groups and rough rings have been investigated by biswas and nanda kuroki and wang and davvaz in this paper we consider an r module as a universal set and we introduce the notion of rough submodule with respect to a submodule of an r module which is an extended notion of a submodule in an r module we also give some properties of the lower and the upper approximations in an r module

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

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

on fuzzy rough sets based on tolerance relations dubois and prade 1990 1 introduced the notion of fuzzy rough sets as a fuzzy generalization of rough sets which was originally proposed by pawlak 1982 8 later radzikowska and kerre introduced the so called i t fuzzy rough sets where i is an implication and t is a triangular norm in the present paper by using a pair of implications i j we define the so called i j fuzzy rough sets which generalize the concept of fuzzy rough sets in the sense of radzikowska and kerre and that of mi and zhang basic properties of i j fuzzy rough sets are investigated in detail

generalized rough sets based on reflexive and transitive relations in this paper we investigate the relationship between generalized rough sets induced by reflexive and transitive relations and the topologies on the universe which is not restricted to be finite it is proved that there exists a one to one correspondence between the set of all reflexive and transitive relations and the set of all topologies which satisfy a certain kind of compactness condition

on rough approximations via ideal abstract in this paper we generalize the rough set model by defining new approximation operators in more general setting of a complete atomic boolean lattice by using an ideal an ideal on a set x is a non empty collection of subsets of x with heredity property which is also closed under finite unions we introduce the concept of lower and upper approximations via ideal in a lattice theoretical setting these decrease the upper approximation and increase the lower approximation and hence increase the accuracy properties of these approximations are studied also properties of the ordered set of the lower and upper of an element of a complete atomic boolean lattice via ideal are investigated we also study the connections between the rough approximations defined by jarvinen 10 11 and our new approximations various examples are given finally we give a new approach for defining the rough approximations w r t the induced map by using an ideal we study the connections between the rough approximations defined with respect to the induced map by using an ideal and the rough approximations defined with respect to the considered map under certain conditions of the map

the lower and upper approximations in a quotient hypermodule with respect to fuzzy sets this paper provides a continuation of ideas presented by davvaz and mahdavipour b davvaz m mahdavipour roughness in modules inform sci 176 2006 3658 3674 the notion of hypermodule is a generalization of the notion of module in this paper we consider the quotient hypermodule m a and interpret the lower and upper approximations as subsets of the quotient hypermodule m a then we introduce the concept of quotient rough sub hypermodule also using the concept of fuzzy sets we introduce and discuss the concept of fuzzy rough hypermodules and then we obtain the relation between fuzzy rough sub hypermodules and level rough sets this relation is characterized as a necessary and sufficient condition

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

generalized rough sets based on relations rough set theory has been proposed by pawlak as a tool for dealing with the vagueness and granularity in information systems the core concepts of classical rough sets are lower and upper approximations based on equivalence relations this paper studies arbitrary binary relation based generalized rough sets in this setting a binary relation can generate a lower approximation operation and an upper approximation operation but some of common properties of classical lower and upper approximation operations are no longer satisfied we investigate conditions for a relation under which these properties hold for the relation based lower and upper approximation operations

multiple solutions for an elliptic system near resonance abstract we consider an elliptic system with linear part depending on two parameters and a sublinear perturbation we obtain the existence of at least two solutions when the linear part is near resonance the system is associated with a strongly indefinite functional and the solutions are obtained through saddle point theorem and galerkin approximation

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

multiple solutions for semilinear elliptic equations near resonance at higher eigenvalues abstract using the minimax methods in critical point theory and a generalized landesman lazer type condition we obtain two solutions for a class of semilinear elliptic equations near resonance at higher eigenvalues

a new methodology for fuzzification of rough sets based on α indiscernibility abstract this paper presents a new framework for fuzzification of rough sets when the objects are indiscernible that is indiscernible up to a certain degree to this end a fuzzy proximity relation has been used basic properties of the so called fuzzified rough sets are examined following radzikowska and kerre 28 approach the technique is further generalized using t norms and implication operators two algorithms followed by an example have also been presented for practical applications of the new technique

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

a metric between unrooted and unordered trees and its bottom up computing method proposes a distance measure between unrooted and unordered trees based on the strongly structure preserving mapping sspm sspm can make correspondences between the vertices of similar substructures of given structures more strictly than previously proposed mappings the time complexity of computing the distance between trees t sub a and t sub b is o m sub bsup 3 n sub a n sub b where n sub a and n sub b are the number of vertices in trees t sub a and t sub b respectively m sub a and m sub b are the maximum degrees of a vertex in t sub a and t sub b respectively and m sub aspl les m sub b is assumed the space complexity of the method is o n sub a n sub b

measuring roughness of generalized rough sets induced by a covering in this paper we propose new lower and upper approximations and obtain some important properties in generalized rough set induced by a covering especially these properties are compared with ones of pawlak s rough sets and bonikowski s covering generalized rough sets respectively moreover we define a measure of roughness based on generalized rough sets with the new approximations and discuss some significant properties of the measure

constructive and algebraic methods of the theory of rough sets this paper reviews and compares constructive and algebraic approaches in the study of rough sets in the constructive approach one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation different classes of rough set algebras are obtained from different types of binary relations in the algebraic approach one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators various classes of rough set algebras are characterized by different sets of axioms axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators

the algebraic structures of generalized rough set theory rough set theory is an important technique for knowledge discovery in databases and its algebraic structure is part of the foundation of rough set theory in this paper we present the structures of the lower and upper approximations based on arbitrary binary relations some existing results concerning the interpretation of belief functions in rough set backgrounds are also extended based on the concepts of definable sets in rough set theory two important boolean subalgebras in the generalized rough sets are investigated an algorithm to compute atoms for these two boolean algebras is presented

roughness in n ary hypergroups this paper is aimed at studying the rough sets within the context of the commutative n ary hypergroups for this approach the presentation of a detailed study of the various types of n ary subhypergroups is of the greatest importance this study was initiated in v leoreanu fotea several types of n ary subhypergroups italian journal of pure and applied mathematics in press subsequently we shall use the findings in this study to define the upper and lower approximation of a subset with respect to an invertible n ary subhypergroup of a commutative n ary hypergroup in the final part the concept of n ary rough subhypergroup is introduced

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

multiple solutions for a class of nonlinear boundary value problems near resonance a variational approach abstract a variational approach is given to the existence of multiple solutions for some nonlinear boundary value problems near resonance at the first eigenvalue

semilinear elliptic problems near resonance with a nonprincipal eigenvalue abstract we consider the dirichlet problem for the equation u u f x u h x in a bounded domain where f has a sublinear growth and h l 2 we find suitable conditions on f and h in order to have at least two solutions for near to an eigenvalue of a typical example to which our results apply is when f x u behaves at infinity like a x u q 2 u with m a x 0 and 1 q 2

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

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

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

rough sets theory for multicriteria decision analysis the original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation it operates on a data table composed of a set u of objects actions described by a set q of attributes its basic notions are indiscernibility relation on u lower and upper approximation of either a subset or a partition of u dependence and reduction of attributes from q and decision rules derived from lower approximations and boundaries of subsets identified with decision classes the original rough set idea is failing however when preference orders of attribute domains criteria are to be taken into account precisely it cannot handle inconsistencies following from violation of the dominance principle this inconsistency is characteristic for preferential information used in multicriteria decision analysis mcda problems like sorting choice or ranking in order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary the main change is the substitution of the indiscernibility relation by a dominance relation which permits approximation of ordered sets in multicriteria sorting to approximate preference relations in multicriteria choice and ranking problems another change is necessary substitution of the data table by a pairwise comparison table where each row corresponds to a pair of objects described by binary relations on particular criteria in all those mcda problems the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model it is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax in order to workout a recommendation in one of the mcda problems we propose exploitation procedures of the set of decision rules finally some other recently obtained results are given rough approximations by means of similarity relations rough set handling of missing data comparison of the rough set model with sugeno and choquet integrals and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive

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

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

generalized lower and upper approximations in a ring in this paper the concepts of set valued homomorphism and strong set valued homomorphism of a ring are introduced and related properties are investigated the notions of generalized lower and upper approximation operators constructed by means of a set valued mapping which is a generalization of the notion of lower and upper approximation of a ring are provided we also propose the notion of generalized lower and upper approximations with respect to an ideal of a ring which is an extended notation of rough ideal introduced lately by davvaz b davvaz roughness in rings information science 164 2004 147 163 in a ring and discuss some significant properties of them

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

an median graphs properties algorithms and applications in object prototype learning and similar tasks median computation is an important technique for capturing the essential information of a given set of patterns we extend the median concept to the domain of graphs in terms of graph distance we introduce the novel concepts of set median and generalized median of a set of graphs we study properties of both types of median graphs for the more complex task of computing generalized median graphs a genetic search algorithm is developed experiments conducted on randomly generated graphs demonstrate the advantage of generalized median graphs compared to set median graphs and the ability of our genetic algorithm to find approximate generalized median graphs in reasonable time application examples with both synthetic and nonsynthetic data are shown to illustrate the practical usefulness of the concept of median graphs

generalized fuzzy rough sets determined by a triangular norm the theory of rough sets has become well established as an approach for uncertainty management in a wide variety of applications various fuzzy generalizations of rough approximations have been made over the years this paper presents a general framework for the study of t fuzzy rough approximation operators in which both the constructive and axiomatic approaches are used by using a pair of dual triangular norms in the constructive approach some definitions of the upper and lower approximation operators of fuzzy sets are proposed and analyzed by means of arbitrary fuzzy relations the connections between special fuzzy relations and the t upper and t lower approximation operators of fuzzy sets are also examined in the axiomatic approach an operator oriented characterization of rough sets is proposed that is t fuzzy approximation operators are defined by axioms different axiom sets of t upper and t lower fuzzy set theoretic operators guarantee the existence of different types of fuzzy relations producing the same operators the independence of axioms characterizing the t fuzzy rough approximation operators is examined then the minimal sets of axioms for the characterization of the t fuzzy approximation operators are presented based on information theory the entropy of the generalized fuzzy approximation space which is similar to shannon s entropy is formulated to measure uncertainty in t generalized fuzzy rough sets a notion of fuzziness is introduced some basic properties of this measure are examined for a special triangular norm t min it is proved that the measure of fuzziness of the generalized fuzzy rough set is equal to zero if and only if the set is crisp and definable

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

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 roughness measure for fuzzy sets recently an attempt of integration between the theories of fuzzy sets and rough sets has resulted in providing a roughness measure for fuzzy sets m banerjee s k pal roughness of a fuzzy set information sciences 93 1996 235 246 essentially banerjee and pal s roughness measure depends on parameters that are designed as thresholds of definiteness and possibility in membership of the objects to a fuzzy set in this paper we first remark that this measure of roughness has several undesirable properties and then propose a parameter free roughness measure for fuzzy sets based on the notion of the mass assignment of a fuzzy set several interesting properties of this new measure are examined furthermore we also discuss how the proposed approach is used to describe the rough approximation quality of a fuzzy classification

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

landesman lazer type results for second order hamilton jacobi bellman equations we study the boundary value problem f d2u du u x u f x u in u 0on where the second order differential operator f is of hamilton jacobi bellman type f is sub linear in u at infinity and rn is a regular bounded domain we extend the well known landesman lazer conditions to study various bifurcation phenomena taking place near the two principal eigenvalues associated to the differential operator we provide conditions under which the solution branches extend globally along the eigenvalue gap we also present examples illustrating the results and hypotheses

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

waveform correlation by tree matching a waveform correlation scheme is presented the scheme consists of four parts 1 the representation of waveforms by trees 2 the definition of basic operations on tree nodes and tree distance 3 a tree matching algorithm and 4 a backtracking procedure to find the best node to node correlation this correlation scheme has been implemented results show that the scheme has the capability of handling distortions that result from stretching or shrinking of intervals or from missing intervals

a new measure of uncertainty based on knowledge granulation for rough sets in rough set theory accuracy and roughness are used to characterize uncertainty of a set and approximation accuracy is employed to depict accuracy of a rough classification although these measures are effective they have some limitations when the lower upper approximation of a set under one knowledge is equal to that under another knowledge to overcome these limitations we address in this paper the issues of uncertainty of a set in an information system and approximation accuracy of a rough classification in a decision table an axiomatic definition of knowledge granulation for an information system is given under which these three measures are modified theoretical studies and experimental results show that the modified measures are effective and suitable for evaluating the roughness and accuracy of a set in an information system and the approximation accuracy of a rough classification in a decision table respectively and have a much simpler and more comprehensive form than the existing ones