Unordered Trees Research Articles

A Novel Coverage Pattern Mining Method for Unordered Tree

Unordered tree is widely used for semi-structured data modeling, coverage pattern mining on it has benefit for finding frequent subtrees without redundant information, so that improve the efficiency of subsequent knowledge discovering. A coverage pattern mining method MCRP is proposed. Firstly, all candidate subtrees are generated on the basis of maximum prefix coding and edge extension.Then coverage patterns are output by introducing δ′- coverage concept. Compared with traditional algorithms such as mining frequent closed tree patterns and maximal frequent tree patterns, the proposed method can output fewer frequent subtrees in the case of preserving all the frequent subtree information, and has a certain superiority in processing efficiency.

Arithmetic for Rooted Trees

We introduce a new arithmetic for non-empty rooted unordered trees. After discussing tree representation and enumeration, we define the operations of tree addition, multiplication, and stretch, and prove their properties. Using these operations all trees can be generated from a starting tree of one vertex. We show how a given tree can be obtained as the sum or as the product of two trees, and define prime trees with respect to addition and multiplication. In both cases we show how primality can be decided in time polynomial in the number of vertices and prove that factorization is unique. We then define negative trees and introduce tree equations whose coefficients are integers and whose unknowns are trees. We show how to solve some tree equations as an introduction to the field, and suggest more advanced examples. Finally we briefly discuss how our arithmetic might be useful in different applications. To the best of our knowledge our proposal is new and may be susceptible of variations and improvements.

Limiting Until in Ordered Tree Query Languages

Marx and de Rijke have shown that the navigational core of the w3c XML query language XPath is not first-order complete; that is, it cannot express every query definable in first-order logic over the navigational predicates. How can one extend XPath to get a first-order complete language? Marx has shown that Conditional XPath—an extension of XPath with an “Until” operator—is first-order complete. The completeness argument makes essential use of the presence of upward axes in Conditional XPath. We examine whether it is possible to get “forward-only” languages that are first-order complete for Boolean queries on ordered trees. It is easy to see that a variant of the temporal logic CTL * is first-order complete; the variant has path quantifiers for downward, leftward, and rightward paths, while along a path one can check arbitrary formulas of Linear Temporal Logic (LTL). This language has two major disadvantages: It requires path quantification in both horizontal directions (in particular, it requires looking backward at the prior siblings of a node), and it requires the consideration of formulas of LTL of arbitrary complexity on vertical paths. This last is in contrast with Marx’s Conditional XPath, which requires only the checking of a single Until operator on a path. We investigate whether either of these restrictions can be eliminated. Our main results are negative ones. We show that if we restrict our CTL * language by having an Until operator in only one horizontal direction, then we lose completeness. We also show that no restriction to a “small” subset of LTL along vertical paths is sufficient for first-order completeness. Smallness here means of bounded “Until Depth,” a measure of complexity of LTL formulas defined by Etessami and Wilke. In particular, it follows from our work that Conditional XPath with only forward axes is not expressively complete; this extends results proved by Rabinovich and Maoz in the context of infinite unordered trees.

Missing Value Imputation Using Stratified Supervised Learning for Cardiovascular Data

Legacy (and current) medical datasets are rich source of information and knowledge. However, the use of most legacy medical datasets is beset with problems. One of the most often faced is the problem of missing data, often due to oversights in data capture or data entry procedures. Algorithms commonly used in the analysis of data often depend on a complete data set. Missing value imputation offers a solution to this problem. This may result in the generation of synthetic data, with artificially induced missing values, but simply removing the incomplete data records often produces the best classifier results. With legacy data, simply removing the records from the original datasets can significantly reduce the data volume and often affect the class balance of the dataset. A suitable method for missing value imputation is very much needed to produce good quality datasets for better analysing data resulting from clinical trials. This paper proposes a framework for missing value imputation using stratified machine learning methods. We explore machine learning technique to predict missing value for incomplete clinical (cardiovascular) data, with experiments comparing this with other standard methods. Two machine learning (classifier) algorithms, fuzzy unordered rule induction algorithm and decision tree, plus other machine learning algorithms (for comparison purposes) are used to train on complete data and subsequently predict missing values for incomplete data. The complete datasets are classified using decision tree, neural network, K-NN and K-Mean clustering. The classification performances are evaluated using sensitivity, specificity, accuracy, positive predictive value and negative predictive value. The results show that final classifier performance can be significantly improved for all class labels when stratification was used with fuzzy unordered rule induction algorithm to predict missing attribute values.

Similar Subtree Search Using Extended Tree Inclusion

This paper considers the problem of identifying all locations of subtrees in a large tree or in a large collection of trees that are similar to a specified pattern tree, where all trees are assumed to be rooted and node-labeled. The tree edit distance is a widely-used measure of tree (dis-)similarity, but is NP-hard to compute for unordered trees. To cope with this issue, we propose a new similarity measure which extends the concept of unordered tree inclusion by taking the costs of insertion and substitution operations on the pattern tree into account, and present an algorithm for computing it. Our algorithm has the same time complexity as the original one for unordered tree inclusion, i.e., it runs in $O(|T_1||T_2|)$ time, where $T_1$ and $T_2$ denote the pattern tree and the text tree, respectively, when the maximum outdegree of $T_1$ is bounded by a constant. Our experimental evaluation using synthetic and real datasets confirms that the proposed algorithm is fast and scalable and very useful for bibliographic matching, which is a typical entity resolution problem for tree-structured data. Furthermore, we extend our algorithm to also allow a constant number of deletion operations on $T_1$ while still running in $O(|T_1||T_2|)$ time.

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.

Grammar-based compression approach to extraction of common rules among multiple trees of glycans and RNAs.

BackgroundMany tree structures are found in nature and organisms. Such trees are believed to be constructed on the basis of certain rules. We have previously developed grammar-based compression methods for ordered and unordered single trees, based on bisection-type tree grammars. Here, these methods find construction rules for one single tree. On the other hand, specified construction rules can be utilized to generate multiple similar trees.ResultsTherefore, in this paper, we develop novel methods to discover common rules for the construction of multiple distinct trees, by improving and extending the previous methods using integer programming. We apply our proposed methods to several sets of glycans and RNA secondary structures, which play important roles in cellular systems, and can be regarded as tree structures. The results suggest that our method can be successfully applied to determining the minimum grammar and several common rules among glycans and RNAs.ConclusionsWe propose integer programming-based methods MinSEOTGMul and MinSEUTGMul for the determination of the minimum grammars constructing multiple ordered and unordered trees, respectively. The proposed methods can provide clues for the determination of hierarchical structures contained in tree-structured biological data, beyond the extraction of frequent patterns.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-015-0558-4) contains supplementary material, which is available to authorized users.

Rumor source detection for rumor spreading on random increasing trees

In a recent paper, Shah and Zaman proposed the rumor center as an effective rumor source estimator for rumor spreading on random graphs. They proved for a very general random tree model that the detection probability remains positive as the number of nodes to which the rumor has spread tends to infinity. Moreover, they derived explicit asymptotic formulas for the detection probability of random $d$-regular trees and random geometric trees. In this paper, we derive asymptotic formulas for the detection probability of grown simple families of random increasing trees. These families of random trees contain important random tree models as special cases, e.g., binary search trees, recursive trees and plane-oriented recursive trees. Our results show that the detection probability varies from $0$ to $1$ across these families. Moreover, a brief discussion of the rumor center for unordered trees is given as well.

Schemas for Unordered XML on a DIME

We investigate schema languages for unordered XML having no relative order among siblings. First, we propose unordered regular expressions (UREs), essentially regular expressions with unordered concatenation instead of standard concatenation, that define languages of unordered words to model the allowed content of a node (i.e., collections of the labels of children). However, unrestricted UREs are computationally too expensive as we show the intractability of two fundamental decision problems for UREs: membership of an unordered word to the language of a URE and containment of two UREs. Consequently, we propose a practical and tractable restriction of UREs, disjunctive interval multiplicity expressions (DIMEs). Next, we employ DIMEs to define languages of unordered trees and propose two schema languages: disjunctive interval multiplicity schema (DIMS), and its restriction, disjunction-free interval multiplicity schema (IMS). We study the complexity of the following static analysis problems: schema satisfiability, membership of a tree to the language of a schema, schema containment, as well as twig query satisfiability, implication, and containment in the presence of schema. Finally, we study the expressive power of the proposed schema languages and compare them with yardstick languages of unordered trees (FO, MSO, and Presburger constraints) and DTDs under commutative closure. Our results show that the proposed schema languages are capable of expressing many practical languages of unordered trees and enjoy desirable computational properties.

On the complexity of finding a largest common subtree of bounded degree

The largest common subtree problem is to find a bijective mapping between subsets of nodes of two input rooted trees of maximum cardinality or weight that preserves labels and ancestry relationship. The problem is known to be NP-hard for unordered trees. In this paper, we consider a restricted unordered case in which the maximum outdegree of a common subtree is bounded by a constant D. We present an O(nD) time algorithm where n is the maximum size of two input trees, which improves a previous O(n2D) time algorithm. We also present an O((H2⋅22H−1⋅D2H)D−1poly(n)) time algorithm, where H is the maximum height of two input trees.

Massive XML Data Mining in Cloud Computing Environment

This paper concentrates on how to mine useful information from massive XML documents in cloud computing environment. The structure of the Cloud computing and the corresponding tree data model of a XML document are analyzed in advance. Afterwards, structure of the proposed XML data mining system is illustrated, which is made up of three layers, such as “Application layer”, “Data processing layer”, and “XML Data converting layer”. In the XML Data converting layer, XML data are collected from databases and documents, and then the source data can be converted to XML file effectively. In the data processing layer, the process of data selection, cleaning and standardization for XML data set is implemented, moreover, a XML data set with higher degree of structure and rich semantics are obtained. In the application layer, “the results report module”, “data query module” and “results analysis module” are included. Next, massive XML data mining algorithm is proposed. The main innovations of this algorithm lie in that 1) the structure of a XML document is represented as an unordered tree, 2) the sub-structures of a XML document are modeled as sub-trees, and XML trees are regarded as a forest which is made up of all the sub-trees. Experimental results show that the proposed method can effectively mine useful information from massive XML documents in cloud computing environment with high efficiency.

New techniques for mining frequent patterns in unordered trees.

We consider a new tree mining problem that aims to discover restrictedly embedded subtree patterns from a set of rooted labeled unordered trees. We study the properties of a canonical form of unordered trees, and develop new Apriori-based techniques to generate all candidate subtrees level by level through two efficient rightmost expansion operations: 1) pairwise joining and 2) leg attachment. Next, we show that restrictedly embedded subtree detection can be achieved by calculating the restricted edit distance between a candidate subtree and a data tree. These techniques are then integrated into an efficient algorithm, named frequent restrictedly embedded subtree miner (FRESTM), to solve the tree mining problem at hand. The correctness of the FRESTM algorithm is proved and the time and space complexities of the algorithm are discussed. Experimental results on synthetic and real-world data demonstrate the effectiveness of the proposed approach.

TRACTABLE AND INTRACTABLE VARIATIONS OF UNORDERED TREE EDIT DISTANCE

In this paper, we investigate the problem of computing structural sensitive variations of an unordered tree edit distance. First, we focus on the variations tractable by the algorithms including the submodule of a network algorithm, either the minimum cost maximum flow algorithm or the maximum weighted bipartite matching algorithm. Then, we show that both network algorithms are replaceable, and hence the time complexity of computing these variations can be reduced to O(nmd) time, where n is the number of nodes in a tree, m is the number of nodes in another tree and d is the minimum degree of given two trees. Next, we show that the problem of computing the bottom-up distance is MAX SNP-hard. Note that the well-known linear-time algorithm for the bottom-up distance designed by Valiente (2001) computes just a bottom-up indel (insertion-deletion) distance allowing no substitutions.

Efficient exponential-time algorithms for edit distance between unordered trees

The edit distance problem for rooted unordered trees is known to be NP-hard. Based on this fact, this paper studies exponential-time algorithms for the problem. For a general case, an O(min(1.26n1+n2,2b1+b2⋅poly(n1,n2))) time algorithm is presented, where n1 and n2 are the numbers of nodes and b1 and b2 are the numbers of branching nodes in two input trees. This algorithm is obtained by a combination of dynamic programming, exhaustive search, and maximum weighted bipartite matching. For bounded degree trees over a fixed alphabet, it is shown that the problem can be solved in O((1+ϵ)n1+n2) time for any fixed ϵ>0. This result is achieved by avoiding duplicate calculations for identical subsets of small subtrees.

A New Structural Approach for Mining Frequent Itemset

problem of mining high quality frequent substructures from a large collection of semi-structured data has recently attracted a lot of research. There are various efficient algorithms available for discovering frequent substructures in a large structured data, where both of the patterns and the data are modeled by labeled unordered trees. In this paper, a new structure based algorithm ENHANCE CLIQUE TREE is proposed which creates the unordered structure of itemsets and gives the perceptive representation of the mining results. It eliminates the limitation of FP-tree which always follows the ordered tree structure and its performance also degrades when the dataset is regularly updated. Here a VERIFY TREE ALGORITHM is added in the proposed algorithm which verifies that whether there is a need to restructure the tree or not when the itemsets are inserted or deleted. It improves the performance so that it will reduce unnecessary space and time used for restructure the tree when there is no need for it.

BOUNDING THE NUMBER OF REDUCED TREES, COGRAPHS, AND SERIES-PARALLEL GRAPHS BY COMPRESSION

We give an efficient encoding and decoding scheme for computing a compact representation of a graph in one of unordered reduced trees, cographs and series-parallel graphs. The unordered reduced trees are rooted trees in which (i) the ordering of children of each vertex does not matter, and (ii) no vertex has exactly one child. This is one of basic models frequently used in many areas. Our algorithm computes a bit string of length 2ℓ – 1 for a given unordered reduced tree with ℓ ≥ 1 leaves in O(ℓ) time, whereas a known folklore algorithm computes a bit string of length 2n – 2 for an ordered tree with n vertices. Note that in an unordered reduced tree, ℓ ≤ n < 2ℓ holds. To the best of our knowledge this is the first of such a compact representation for unordered reduced trees. From the theoretical point of view, the length of the representation gives us an upper bound of the number of unordered reduced trees with ℓ leaves. Precisely, the number of unordered reduced trees with ℓ leaves is at most 22ℓ–2 for ℓ ≥ 1. Moreover, the encoding and decoding can be done in linear time. Therefore, from the practical point of view, our representation is also useful to store a lot of unordered reduced trees efficiently. We also apply the scheme for computing a compact representation to cographs and series-parallel graphs. We show that each of cographs with n vertices has a compact representation in 2n – 1 bits, and the number of cographs with n vertices is at most 22n–1. The resulting number is close to the number of cographs with n vertices obtained by the enumeration for small n that approximates Cdn/n3/2, where C = 0.4126⋯ and d = 3.5608⋯. Series-parallel graphs are well-investigated in the context of the graphs of bounded treewidth. We give a method to represent a series-parallel graph with m edges in ⌈2.5285m – 2⌉ bits. Hence the number of series-parallel graphs with m edges is at most 2⌈2.5285m–2⌉. As far as the authors know, this is the first non-trivial result about the number of series-parallel graphs. The encoding and decoding of the cographs and series-parallel graphs also can be done in linear time.

Coloring based approach for matching unrooted and/or unordered trees

We consider the problem of matching unrooted unordered labeled trees, which refers to the task of evaluating the distance between trees. One of the most famous formalizations of this problem is the computation of the edit distance defined as the minimum-cost sequence of edit operations that transform one tree into another. Unfortunately, this problem has been proved to be NP-complete. In this paper, we propose a new algorithm to measure distance between unrooted unordered labeled trees. This algorithm uses a specific graph coloring to decompose the trees into small components (stars and bistars). Then, it determines a distance between two trees by computing the edit distance between their components. We prove that the proposed distance is a pseudo-metric and we analyze its time complexity. Our experimental evaluations on large synthetic and real world datasets confirm our analytical results and suggest that the distance we propose is accurate and its algorithm is scalable.

Drawing Unordered Trees on k-Grids

We present almost linear area bounds for drawing complete trees on the octagonal grid. For 7-ary trees we establish an upper and lower bound of Θ(n1.129) and for ternary trees the bounds of $\O(n^{1.048})$ and Θ(n), where the latter needs edge bends. We explore the unit edge length and area complexity of drawing unordered trees on k-grids with k∈{4, 6, 8} and generalize the $\mathcal{NP}$ -hardness results of the orthogonal and hexagonal grid to the octagonal grid.

Efficient Streaming Algorithms for Tree Matching Problems

With the growing importance of XML in data exchange, much research has been done in providing flexible query mechanisms to extract data from XML documents. In this paper, we focus on the query evaluation in an XML streaming environment, in which data streams arrive continuously and queries have to be evaluated even before all the data of an XML document are available. Two algorithms will be discussed. One is for the unordered tree matching, by which only ancestor-descendant and parent-child relationships are considered. It requires O(|T'|⋅leafQ) time, where T' is a subtree of document tree T, in which each node matches at least one node in query Q and leafQ is the number of leaf nodes in Q. The other is for the ordered tree matching, by which the left-to-right order of nodes must also be taken into account. It runs in O(|T'|⋅|Q|) time. Furthermore, our algorithms achieve high time performance without trading off space requirements. They have the same caching space and buffering space overhead as state-of-the-art stream-querying algorithm. We show the efficiency and effectiveness of our algorithms by a lot of experiments.

Approximation and parameterized algorithms for common subtrees and edit distance between unordered trees

Given two rooted, labeled, unordered trees, the common subtree problem is to find a bijective matching between subsets of nodes of the trees of maximum cardinality which preserves labels and ancestry relationship. The tree edit distance problem is to determine the least cost sequence of insertions, deletions and substitutions that converts a tree into another given tree. Both problems are known to be hard to approximate within some constant factor in general.We tackle these problems from two perspectives: giving exact algorithms, either for special cases or in terms of some parameters; and approximation algorithms and hardness of approximation. We present a parameterized algorithm in terms of the number of branching nodes that solves both problems and yields polynomial algorithms for several special classes of trees. This is complemented with a tighter APX-hardness proof that holds when the trees are of height one and two, respectively. Furthermore, we present the first approximation algorithms for both problems. In particular, for the common subtree problem for t trees, we present an algorithm achieving a tlog2(bOPT+1) ratio, where bOPT is the number of branching nodes in the optimal solution. We also present constant factor approximation algorithms for both problems in the case of bounded height trees.