Subtree Isomorphism Research Articles

Graphs with a unique maximum independent set up to automorphisms

If for every two maximum independent sets A and B in a graph G there exists an automorphism of G that maps A to B , then we call G an α -iso-unique graph. We extend several known characterizations of trees that have a unique maximum independent set obtaining characterizations of α -iso-unique trees. Such trees either have the property that the deletion of any edge does not affect the independence number, or they have the central edge, which connects two isomorphic subtrees, and only the deletion of the central edge increases the independence number. One of the characterizations results in a linear time algorithm to recognize whether a given tree is α -iso-unique. In contrast, we prove that the problem of recognizing whether an arbitrary graph is α -iso-unique is not polynomial unless P = NP. Constructions of large families of α -iso-unique graphs are given involving simplicial vertices and chordal graphs. In particular, we show that every graph is an induced subgraph of an α -iso-unique graph. Finally, α -iso-unique graphs are characterized among Cartesian products G □ H of co-prime graphs G and H such that α ( G □ H ) = α ( H ) | V ( G ) | .

Research on visualisation transmission method for business innovation strategy data based on structural characteristics

In order to overcome the problems of low transfer accuracy and user satisfaction existing in the existing data visualisation transfer methods, this paper proposes a data visualisation transfer method of business innovation strategy based on structural features. The similarity calculation method of structure feature tree is used to realise the data structure feature processing. The sub-trees of all non-isomorphic patterns of K nodes are constructed. The number of isomorphic sub-trees is calculated in the tree, and the transfer data division is completed. Based on the basic program of business intelligence, the original data of business innovation strategy is processed, converted into various types of visualisation structure data, and transmitted to relevant personnel in online form to complete the visualisation transmission of business innovation strategy data. The experimental results show that the proposed method is more accurate, up to 99.3%, and users are more satisfied.

De novo glycan structural identification from mass spectra using tree merging strategy

MotivationGlycans are large molecules with specific tree structures. Glycans play important roles in a great variety of biological processes. These roles are primarily determined by the fine details of their structures, making glycan structural identification highly desirable. Mass spectrometry (MS) has become the major technology for elucidation of glycan structures. Most de novo approaches to glycan structural identification from mass spectra fall into three categories: enumerating followed by filtering approaches, heuristic and dynamic programming-based approaches. The former suffers from its low efficiency while the latter two suffer from the possibility of missing the actual glycan structures. Thus, how to reliably and efficiently identify glycan structures from mass spectra still remains challenging. ResultsIn this study we propose an efficient and reliable approach to glycan structure identification using tree merging strategy. Briefly, for each MS peak, our approach first calculated monosaccharide composition of its corresponding fragment ion, and then built a constraint that forces these monosaccharides to be directly connected in the underlying glycan tree structure. According to these connecting constraints, we next merged constituting monosaccharides of the glycan into a complete structure step by step. During this process, the intermediate structures were represented as subtrees, which were merged iteratively until a complete tree structure was generated. Finally the generated complete structures were ranked according to their compatibility to the input mass spectra. Unlike the traditional enumerating followed by filtering strategy, our approach performed deisomorphism to remove isomorphic subtrees, and ruled out invalid structures that violates the connection constraints at each tree merging step, thus significantly increasing efficiency. In addition, all complete structures satisfying the connection constraints were enumerated without any missing structure. Over a test set of 10 N-glycan standards, our approach accomplished structural identification in minutes and gave the manually-validated structure first three highest score. We further successfully applied our approach to profiling and subsequent structure assignment of glycans released from glycoprotein mAb, which was in perfect agreement with previous studies and CE analysis.

Subtree Isomorphism Revisited

The Subtree Isomorphism problem asks whether a given tree is contained in another given tree. The problem is of fundamental importance and has been studied since the 1960s. For some variants, e.g., ordered trees , near-linear time algorithms are known, but for the general case truly subquadratic algorithms remain elusive. Our first result is a reduction from the Orthogonal Vectors problem to Subtree Isomorphism, showing that a truly subquadratic algorithm for the latter refutes the Strong Exponential Time Hypothesis (SETH). In light of this conditional lower bound, we focus on natural special cases for which no truly subquadratic algorithms are known. We classify these cases against the quadratic barrier, showing in particular that: • Even for binary, rooted trees, a truly subquadratic algorithm refutes SETH. • Even for rooted trees of depth O (log log n ), where n is the total number of vertices, a truly subquadratic algorithm refutes SETH. • For every constant d , there is a constant ε d > 0 and a randomized, truly subquadratic algorithm for degree- d rooted trees of depth at most (1+ ε d ) log d n . In particular, there is an O (min { 2.85 h , n 2 }) algorithm for binary trees of depth h . Our reductions utilize new “tree gadgets” that are likely useful for future SETH-based lower bounds for problems on trees. Our upper bounds apply a folklore result from randomized decision tree complexity.

Probabilistic frequent subtrees for efficient graph classification and retrieval

Frequent subgraphs proved to be powerful features for graph classification and prediction tasks. Their practical use is, however, limited by the computational intractability of pattern enumeration and that of graph embedding into frequent subgraph feature spaces. We propose a simple probabilistic technique that resolves both limitations. In particular, we restrict the pattern language to trees and relax the demand on the completeness of the mining algorithm, as well as on the correctness of the pattern matching operator by replacing transaction and query graphs with small random samples of their spanning trees. In this way we consider only a random subset of frequent subtrees, called probabilistic frequent subtrees, that can be enumerated efficiently. Our extensive empirical evaluation on artificial and benchmark molecular graph datasets shows that probabilistic frequent subtrees can be listed in practically feasible time and that their predictive and retrieval performance is very close even to those of complete sets of frequent subgraphs. We also present different fast techniques for computing the embedding of unseen graphs into (probabilistic frequent) subtree feature spaces. These algorithms utilize the partial order on tree patterns induced by subgraph isomorphism and, as we show empirically, require much less evaluations of subtree isomorphism than the standard brute-force algorithm. We also consider partial embeddings, i.e., when only a part of the feature vector has to be calculated. In particular, we propose a highly effective practical algorithm that significantly reduces the number of pattern matching evaluations required by the classical min-hashing algorithm approximating Jaccard-similarities.

An Efficient Retrieval Technique for Trademarks Based on the Fuzzy Inference System

The existing trademark image retrieval (TIR) approaches mostly use complex image features, the integration of multi features, a tree structure, etc. to enable highly accurate retrieval. However, there is the heavy computational burden for complex image features and maximum similarity subtree isomorphism (MSSI) measurement. This paper aims to provide an efficient solution for TIR in real-time applications, especially in measuring the similarity between multi-object trademark images. In particular, we propose a novel algorithm for tree similarity measurement based on the fuzzy inference system (FIS) to improve retrieval efficiency. Furthermore, the integration of global and local geometric descriptors is used to enable accurate retrieval. The global descriptor is computed by employing the Hu moments, while the local descriptors are generated by using a tree structure based on the five geometric features: convexity, eccentricity, compactness, circle variance, and elliptic variance. During the retrieval process, the similarity coefficient between the query and the database image is obtained from the similarity of the global and local descriptors. The proposed technique is evaluated using 1800 trademark images, including 12 different classes and 416 trademark images. Additionally, the three common indices, the precision/recall rate, the Bull’s eye score, and the average normalized modified retrieval rank (ANMRR) are used as the performance indices. The experimental results show that the proposed technique is superior to the other two competitive approaches. It shows 19.43% and 26.78% precision/recall improvement, 19.56% and 30.58% improvement in the average Bull’s eye score, and 0.167 and 0.236 improvement in the ANMRR score, respectively, for the 416 query images. It can be concluded from the experimental analysis that the proposed technique not only provides reliable retrieval results but also improves the retrieval efficiency by 151 times in the retrieval process.

Algorithmic height compression of unordered trees

By nature, tree structures frequently present similarities between their sub-parts. Making use of this redundancy, different types of tree compression techniques have been designed in the literature to reduce the complexity of tree structures. A popular and efficient way to compress a tree consists of merging its isomorphic subtrees, which produces a directed acyclic graph (DAG) equivalent to the original tree. An important property of this method is that the compressed structure (i.e. the DAG) has the same height as the original tree, thus limiting partially the possibility of compression. In this paper we address the problem of further compressing this DAG in height. The difficulty is that compression must be carried out on substructures that are not exactly isomorphic as they are strictly nested within each-other. We thus introduced a notion of quasi-isomorphism between subtrees that makes it possible to define similar patterns along any given path in a tree. We then proposed an algorithm to detect these patterns and to merge them, thus leading to compressed structures corresponding to DAGs augmented with return edges. In this way, redundant information is removed from the original tree in both width and height, thus achieving minimal structural compression. The complete compression algorithm is then illustrated on the compression of various plant-like structures.

Faster algorithms for finding and counting subgraphs

In the Subgraph Isomorphism problem we are given two graphs F and G on k and n vertices respectively as an input, and the question is whether there exists a subgraph of G isomorphic to F. We show that if the treewidth of F is at most t, then there is a randomized algorithm for the Subgraph Isomorphism problem running in time O⁎(2kn2t). Our proof is based on a novel construction of an arithmetic circuit of size at most nO(t) for a new multivariate polynomial, Homomorphism Polynomial, of degree at most k, which in turn is used to solve the Subgraph Isomorphism problem. For the counting version of the Subgraph Isomorphism problem, where the objective is to count the number of distinct subgraphs of G that are isomorphic to F, we give a deterministic algorithm running in time and space O⁎((nk/2)n2p) or (nk/2)nO(tlogk). We also give an algorithm running in time O⁎(2k(nk/2)n5p) and taking O⁎(np) space. Here p and t denote the pathwidth and the treewidth of F, respectively. Our work improves on the previous results on Subgraph Isomorphism, it also extends and unifies most of the known results on sub-path and sub-tree isomorphisms.

Geometry-Based Image Retrieval in Binary Image Databases

In this paper, a geometry-based image retrieval system is developed for multi-object images. We model both shape and topology of image objects using a structured representation called curvature tree (CT). The hierarchy of the CT reflects the inclusion relationships between the image objects. To facilitate shape-based matching, triangle-area representation (TAR) of each object is stored at the corresponding node in the CT. The similarity between two multi-object images is measured based on the maximum similarity subtree isomorphism (MSSI) between their CTs. For this purpose, we adopt a recursive algorithm to solve the MSSI problem and a very effective dynamic programming algorithm to measure the similarity between the attributed nodes. Our matching scheme agrees with many recent findings in psychology about the human perception of multi-object images. Experiments on a database of 13500 real and synthesized medical images and the MPEG-7 CE-1 database of 1400 shape images have shown the effectiveness of the proposed method.

Multi-object image retrieval based on shape and topology

We aim at developing a geometry-based retrieval system for multi-object images. We model both shape and topology of image objects including holes using a structured representation called curvature tree (CT); the hierarchy of the CT reflects the inclusion relationships between the objects and holes. To facilitate shape-based matching, triangle-area representation (TAR) of each object and hole is stored at the corresponding node in the CT. The similarity between two multi-object images is measured based on the maximum similarity subtree isomorphism (MSSI) between their CTs. For this purpose, we adapt a continuous optimization approach to solve the MSSI problem and a very effective dynamic programming algorithm to measure the similarity between the attributed nodes. Our matching scheme agrees with many recent findings in psychology about the human perception of multi-object images. Experiments on a database of 1500 logos and the MPEG-7 CE-1 database of 1400 shape images have shown the significance of the proposed method.

A Theoretical Analysis of Tree Edit Distance Measures

The notion of the tree edit distance provides a unifying framework for measuring distance and finding approximate common patterns between two trees. A diversity of tree edit distance measures have been proposed to deal with tree related problems, such as minor containment, maximum common subtree isomorphism, maximum common embedded subtree, and alignment of trees. These classes of problems are characterized by the conditions of the tree mappings, which specify how to associate the nodes in one tree with the nodes in the other. In this paper, we study the declarative semantics of edit distance measures based on the tree mapping. In prior work, the edit distance measures have been not well-formalized. So the relationship among various algorithms based on the tree edit distance has hardly been studied. Our framework enables us to study the relationship. By using our framework, we reveal the declarative semantics of the alignment of trees, which has remained unknown in prior work.

Decision diagrams in machine learning: an empirical study on real-life credit-risk data

Decision trees are a widely used knowledge representation in machine learning. However, one of their main drawbacks is the inherent replication of isomorphic subtrees, as a result of which the produced classifiers might become too large to be comprehensible by the human experts that have to validate them. Alternatively, decision diagrams, a generalization of decision trees taking on the form of a rooted, acyclic digraph instead of a tree, have occasionally been suggested as a potentially more compact representation. Their application in machine learning has nonetheless been criticized, because the theoretical size advantages of subgraph sharing did not always directly materialize in the relatively scarce reported experiments on real-world data. Therefore, in this paper, starting from a series of rule sets extracted from three real-life credit-scoring data sets, we will empirically assess to what extent decision diagrams are able to provide a compact visual description. Furthermore, we will investigate the practical impact of finding a good attribute ordering on the achieved size savings.

Matching of 3D tree structures using association graphs

AbstractMatching of tree structures is equivalent to the problem of searching for maximum cliques in tree association graphs. Pelillo and colleagues have recently proposed a method for searching for maximum weight cliques that gives weight to association graphs on the basis of similarities of the attributes of tree nodes. Here, the obtained maximum weight clique corresponds to the maximum similarity subtree isomorphism. This paper proposes a tree structure‐matching scheme using tree association graphs, as a general scheme for performing nonrigid body registration of tree structures in three‐dimensional medical images. Matching of tree structures of three‐dimensional medical images of bronchi, blood vessels, and the like must be capable of handling changes in body positions accompanying their changes as well as physiological changes due to breathing exercises etc. In addition, it needs to recognize topology errors of pseudo or false branches occurring during image processing. This study, as attributes of tree nodes, constructs similarities using body volume change rates extending over the parents, children and siblings, focusing on the positional relations of the nodes of the parents, children, and siblings. The similarities proposed are those having high general properties that can be applied to changes in positions and shapes that occur inside a body since they focus on their general properties. In addition, as a method for avoiding mismatches due to topology errors, changing the weights of neighboring edges on the basis of similarities is added. The proposed scheme has been confirmed to show a high correct solution rate or true positive fraction regardless of high degree changes in positions and shapes when applied to a bronchial model generated inside a computer. It has also been confirmed that mismatches due to topology errors occur at a very low rate with the proposed scheme. © 2004 Wiley Periodicals, Inc. Electron Comm Jpn Pt 2, 87(2): 59–72, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjb.10216

Faster Subtree Isomorphism

We study the subtree isomorphism problem: Given trees H and G, find a subtree of G which is isomorphic to H or decide that there is no such subtree. We give an O(( k 1.5/log k) n)-time algorithm for this problem, where k and n are the number of vertices in H and G, respectively. This improves over the O( k 1.5 n) algorithms of Chung and Matula. We also give a randomized (Las Vegas) O( k 1.376 n)-time algorithm for the decision problem.

On an algorithm of Zemlyachenko for subtree isomorphism

Zemlyachenko's linear time algorithm for free tree isomorphism is unique in that it also partitions the set of rooted subtrees of a given rooted tree into isomorphism equivalence classes. Unfortunately, his algorithm is very hard to follow. In this note, we use modern data structures to explain and implement Zemlyachenko's scheme. We give a full description of a free rendition of his method using some of his ideas and adding some new ones; in particular, the usage of the data structures is new.

Matching hierarchical structures using association graphs

It is well-known that the problem of matching two relational structures can be posed as an equivalent problem of finding a maximal clique in a (derived) graph. However, it is not clear how to apply this approach to computer vision problems where the graphs are hierarchically organized, i.e., are trees, since maximal cliques are not constrained to preserve the partial order. We provide a solution to the problem of matching two trees by constructing the association graph using the graph-theoretic concept of connectivity. We prove that, in the new formulation, there is a one-to-one correspondence between maximal cliques and maximal subtree isomorphisms. This allows us to cast the tree matching problem as an indefinite quadratic program using the Motzkin-Straus theorem, and we use replicator dynamical systems developed in theoretical biology to solve it. Such continuous solutions to discrete problems are attractive because they can motivate analog and biological implementations. The framework is also extended to the matching of attributed trees by using weighted association graphs. We illustrate the power of the approach by matching articulated and deformed shapes described by shock trees.

General Techniques for Analyzing Recursive Algorithms with Applications

The complexity of divide-and-conquer algorithms is often described by recurrences of various forms. In this paper, we develop general techniques and master theorems for solving several kinds of recurrences, and we give several applications of our results. In particular, almost all of the earlier work on solving the recurrences considered here is subsumed by our work. In the process of solving such recurrences, we establish interesting connections between some elegant mathematics and analysis of recurrences. Using our results and improved bipartite matching algorithms, we also improve existing bounds in the literature for several problems, viz, associative-commutative (AC) matching of linear terms, associative matching of linear terms, rooted subtree isomorphism, and rooted subgraph homeomorphism for trees.

SUBTREE ISOMORPHISM IS IN DLOG FOR NESTED TREES

This research shows subtree isomorphism is in DLOG, and hence [Formula: see text], for nested trees. To our knowledge this result provides the first interesting class of trees for which the problem is in a non-randomized version of [Formula: see text]. We also show that one can determine whether or not an arbitrary tree is a nested tree in DLOG.

Strings, trees, and patterns

In this paper we study subtree isomorphism and its relationships with some important symbolic problems. Recently, a linear time algorithm for ordered subtree isomorphism was given by Mäkinen. We note that a subtree of a rooted tree T, in Mäkinen's paper, refers to the tree rooted at any vertex in T. We consider ordered subtree isomorphism when a broader definition of subtree is used, viz., a subtree of T is any connected subgraph, including v, of the tree rooted at any vertex v in T. We give some evidence to show that a linear time algorithm for this problem is unlikely. We also give an almost linear time reduction from the important problem of pattern matching to ordered subtree isomorphism. Finally, we present simpler linear time algorithms for ordered and unordered subtree isomorphism (with the more restrictive definition of subtree).

Further comments on the subtree isomorphism for ordered trees

Further comments on the subtree isomorphism for ordered trees