Unordered Trees Research Articles

Polyregular Functions on Unordered Trees of Bounded Height

We consider injective first-order interpretations that input and output trees of bounded height. The corresponding functions have polynomial output size, since a first-order interpretation can use a k -tuple of input nodes to represent a single output node. We prove that the equivalence problem for such functions is decidable, i.e. ‍given two such interpretations, one can decide whether, for every input tree, the two output trees are isomorphic. We also give a calculus of typed functions and combinators which derives exactly injective first-order interpretations for unordered trees of bounded height. The calculus is based on a type system, where the type constructors are products, coproducts and a monad of multisets. Thanks to our results about tree-to-tree interpretations, the equivalence problem is decidable for this calculus. As an application, we show that the equivalence problem is decidable for first-order interpretations between classes of graphs that have bounded tree-depth. In all cases studied in this paper, first-order logic and MSO have the same expressive power, and hence all results apply also to MSO interpretations.

Anti Tai mapping for unordered labeled trees

The well-studied Tai mapping between two rooted labeled trees T1=(V1,E1) and T2=(V2,E2) defines a one-to-one mapping between nodes in T1 and T2 that preserves ancestor relationship [1]. For unordered trees the problem of finding a maximum-weight Tai mapping is known to be NP-complete [2]. In this work, we define an anti Tai mapping M⊆V1×V2 as a binary relation between two unordered labeled trees such that any two (x,y),(x′,y′)∈M violate ancestor relationship and thus cannot be part of the same Tai mapping, i.e. (x≤x′⇔y≰y′)∨(x′≤x⇔y′≰y), given an ancestor order x<x′ meaning that x is an ancestor of x′. Finding a maximum-weight anti Tai mapping arises in the cutting plane method for solving the maximum-weight Tai mapping problem via integer programming. We give an efficient polynomial-time algorithm for finding a maximum-weight anti Tai mapping for the case when one of the two trees is a path and further show how to extend this result in order to provide a polynomially computable lower bound on the optimal anti Tai mapping for two unordered labeled trees. The latter result stems from the special class of anti Tai mappings defined by the more restricted condition x∼x′⇔y≁y′, where ∼ denotes that two nodes belong to the same root-to-leaf path. For this class, we give an efficient algorithm that solves the problem exactly on two unordered trees in O(|V1|2|V2|2).

Characterization of random walks on space of unordered trees using efficient metric simulation

The simple random walk on Zp shows two drastically different behaviors depending on the value of p: it is recurrent when p∈{1,2} while it escapes (with a rate increasing with p) as soon as p≥3. This classical example illustrates that the asymptotic properties of a random walk provides some information on the structure of its state space. This paper aims to explore analogous questions on space made up of combinatorial objects with no algebraic structure. We take as a model for this problem the space of unordered unlabeled rooted trees endowed with Zhang edit distance. To this end, it defines the canonical unbiased random walk on the space of trees and provides an efficient algorithm to evaluate its escape rate. Compared to Zhang algorithm, it is incremental and computes the edit distance along the random walk approximately 100 times faster on trees of size 500 on average. The escape rate of the random walk on trees is precisely estimated using intensive numerical simulations, out of reasonable reach without the incremental algorithm.

Two metrics on rooted unordered trees with labels

BackgroundThe early development of a zygote can be mathematically described by a developmental tree. To compare developmental trees of different species, we need to define distances on trees. If children cells after a division are not distinguishable, developmental trees are represented by the space {mathcal {T}} of rooted trees with possibly repeated labels, where all vertices are unordered. If children cells after a division are partially distinguishable, developmental trees are represented by the space {mathcal {P}} of rooted trees with possibly repeated labels, where vertices can be ordered or unordered.ResultsOn {mathcal {T}}, the space of rooted unordered trees with possibly repeated labels, we define two metrics: the best-match metric and the left-regular metric, which show some advantages over existing methods. On {mathcal {P}}, the space of rooted labeled trees with ordered or unordered vertices, there is no metric, and we define a semimetric, which is a variant of the best-match metric. To compute the best-match distance between two trees, the expected time complexity and worst-case time complexity are both {mathcal {O}}(n^2), where n is the tree size. To compute the left-regular distance between two trees, the expected time complexity is {mathcal {O}}(n), and the worst-case time complexity is {mathcal {O}}(nlog n).ConclusionsFor rooted labeled trees with (fully/partially) unordered vertices, we define metrics (semimetric) that have fast algorithms to compute and have advantages over existing methods. Such trees also appear outside of developmental biology, and such metrics can be applied to other types of trees which have more extensive applications, especially in molecular biology.

Enumeration of irredundant forests

Reverse search is a convenient method for enumerating structured objects, that can be used both to address theoretical issues and to solve data mining problems. This method has already been successfully developed to handle unordered trees. If the literature proposes solutions to enumerate singletons of trees, we study in this article a more general problem, the enumeration of sets of trees – forests. Specifically, we mainly study irredundant forests, i.e., where no tree is a subtree of another. By compressing each such forest into a Directed Acyclic Graph (DAG), we develop a reverse search like method to enumerate DAGs compressing irredundant forests. Remarkably, we prove that these DAGs are in bijection with the row-Fishburn matrices, a well-studied class of combinatorial objects. In a second step, we derive our irredundant forest enumeration to provide algorithms for tackling related problems: (i) enumeration of forests in their classical sense (where redundancy is allowed); (ii) the enumeration of “subforests” of a forest, and (iii) the frequent “subforest” mining problem. All the methods presented in this article enumerate each item uniquely, up to isomorphism.

Lossless Compression of Semi-Ordered Trees

In this paper, authors are interested in the problem of lossless compression of unlabeled semi-ordered trees. Semi-ordered trees are a class of trees that present an order between some sibling while some other sibling are unordered. They offer a wide possibility of applications especially for the representation of plants architecture. Authors show that these trees present remarkable compression properties covering those of ordered and unordered trees. To illustrate this approach, authors apply these notions to a particular class of semi-ordered trees which is the most studied branching structure particularly for a botanical motivation, namely axial trees.

A class of higher inductive types in Zermelo‐Fraenkel set theory

AbstractWe define a class of higher inductive types that can be constructed in the category of sets under the assumptions of Zermelo‐Fraenkel set theory without the axiom of choice or the existence of uncountable regular cardinals. This class includes the example of unordered trees of any arity.

New and improved algorithms for unordered tree inclusion

The tree inclusion problem is, given two node-labeled trees P and T (the “pattern tree” and the “target tree”), to locate every minimal subtree in T (if any) that can be obtained by applying a sequence of node insertion operations to P. Although the ordered tree inclusion problem is solvable in polynomial time, the unordered tree inclusion problem is NP-hard. The currently fastest algorithm for the latter is a classic algorithm by Kilpeläinen and Mannila from 1995 that runs in O(d22dmn) time, where m and n are the sizes of the pattern and target trees, respectively, and d is the degree of the pattern tree. Here, we develop a new algorithm that runs in O(d2dmn2) time, improving the exponential factor from 22d to 2d by considering a particular type of ancestor-descendant relationships that is suitable for dynamic programming. We also study restricted variants of the unordered tree inclusion problem.

Cache Oblivious Algorithms for Computing the Triplet Distance between Trees

We consider the problem of computing the triplet distance between two rooted unordered trees with n labeled leaves. Introduced by Dobson in 1975, the triplet distance is the number of leaf triples that induce different topologies in the two trees. The current theoretically fastest algorithm is an O( n log n ) algorithm by Brodal et al. (SODA 2013). Recently, Jansson and Rajaby proposed a new algorithm that, while slower in theory, requiring O( n log 3 n ) time, in practice it outperforms the theoretically faster O( n log n ) algorithm. Both algorithms do not scale to external memory. We present two cache oblivious algorithms that combine the best of both worlds. The first algorithm is for the case when the two input trees are binary trees, and the second is a generalized algorithm for two input trees of arbitrary degree. Analyzed in the RAM model, both algorithms require O( n log n ) time, and in the cache oblivious model O( n / B log 2 n / M ) I/Os. Their relative simplicity and the fact that they scale to external memory makes them achieve the best practical performance. We note that these are the first algorithms that scale to external memory, both in theory and in practice, for this problem.

A Canonical String Encoding for Pure Bigraphs

The bigraph theory, devised by Robin Milner, is a recent mathematical framework for concurrent processes. Its generality is able to subsume many existing process calculi, for example, CCS, CSP, and Petri nets. Further, it provides a uniform proof of bisimilarity, which is a congruence. We present the first canonical string encoding for pure and lean bigraphs by lifting the breadth-first canonical form of rooted unordered trees to a unique representation for bigraphs up to isomorphism (i.e., lean-support equivalence). The encoding’s applicability is limited to atomic alphabets. The time complexity is O(n^{2}k, d log {d}), where n is the number of places, d the degree of the place graph and k the maximum arity of a bigraph’s signature. We provide proof of the correctness of our method and also conduct experimental measurements to assess the complexity.

An Error Analysis Framework for Shallow Surface Realization

Abstract The metrics standardly used to evaluate Natural Language Generation (NLG) models, such as BLEU or METEOR, fail to provide information on which linguistic factors impact performance. Focusing on Surface Realization (SR), the task of converting an unordered dependency tree into a well-formed sentence, we propose a framework for error analysis which permits identifying which features of the input affect the models’ results. This framework consists of two main components: (i) correlation analyses between a wide range of syntactic metrics and standard performance metrics and (ii) a set of techniques to automatically identify syntactic constructs that often co-occur with low performance scores. We demonstrate the advantages of our framework by performing error analysis on the results of 174 system runs submitted to the Multilingual SR shared tasks; we show that dependency edge accuracy correlate with automatic metrics thereby providing a more interpretable basis for evaluation; and we suggest ways in which our framework could be used to improve models and data. The framework is available in the form of a toolkit which can be used both by campaign organizers to provide detailed, linguistically interpretable feedback on the state of the art in multilingual SR, and by individual researchers to improve models and datasets.1

Discovering closed and maximal embedded patterns from large tree data

Many current applications and systems produce large tree datasets and export, exchange, and represent data in tree-structured form. Extracting informative patterns from large data trees is an important research direction with multiple applications in practice. Pattern mining research initially focused on mining induced patterns and gradually evolved into mining embedded patterns. A well-known problem of frequent pattern mining is the huge number of patterns it produces. This affects not only the efficiency but also the effectiveness of mining. A typical solution to this problem is to summarize frequent patterns through closed and maximal patterns. No previous work addresses the problem of mining closed and/or maximal embedded tree patterns, not even in the framework of mining multiple small trees.We address the problem of summarizing embedded tree patterns extracted from large data trees, by defining and mining closed and maximal embedded unordered tree patterns. We design an embedded frequent pattern mining algorithm extended with a local closedness checking technique. This algorithm is called closedEmbTM-eager as it eagerly eliminates non-closed patterns. To mitigate the generation of intermediate patterns, we devise pattern search space pruning rules to proactively detect and prune branches in the pattern search space which do not correspond to closed patterns. The pruning rules are accommodated into the extended embedded pattern miner to produce a new algorithm, called closedEmbTM-prune, for mining all the closed and maximal embedded frequent patterns. Our extensive experiments on synthetic and real large-tree datasets demonstrate that, on dense datasets, closedEmbTM-prune not only generates a complete closed and maximal pattern set which is substantially smaller than that generated by the embedded pattern miner, but also runs much faster with negligible overhead on pattern pruning.

Customizable HMM-based measures to accurately compare tree sets

Trees have been topics of much interest since many decades due to various emerging applications using data represented as trees. Several techniques have been developed to compare two trees. But there is a serious lack of metrics to compare weighted trees. Existing approaches do not also allow to explicitly specify the targeted nodes properties on which the comparison should be performed. Furthermore, the problem of comparing two tree sets is not specifically addressed by existing techniques. This paper attempts to solve these problems by first proposing a distance and a similarity for the comparison of two finite sets of rooted ordered trees which can be labeled or not, as well as weighted or unweighted. To achieve this goal, a hidden Markov model is associated with each tree set for each targeted nodes property. The model associated with a tree set T for the targeted nodes property p learns how much the nodes of the trees in T verify property p. The resulting models are finally compared to derive a distance and similarity between the two sets of trees. The previous measures are then generalized for the comparison of unrooted and unordered trees. Flat classification experiments were carried out on two synthetic databases named FirstLast-L and FirstLast-LW available online. They both contain four classes of 100 rooted ordered trees whose specific and non-trivial nodes properties are clearly defined. When the distance proposed in this paper is selected as metric for the Nearest Neighbor classifier, a perfect accuracy of $$100\%$$ is obtained for these two databases. This performance is $$41\%$$ higher than the accuracy exhibited when the widespread tree Edit distance is selected for FirstLast-L.

A Review on Different Techniques for Mining Frequent Patterns in Unordered Trees

Numerous fields, such as bioinformatics, web mining, and social network analysis, now require frequent pattern mining in unordered trees. Finding repeating patterns and substructures inside a collection of unordered trees is the key to unlocking this technique's potential to provide important information about the underlying data. This paper gives a thorough overview of various methods for mining frequent patterns in unordered trees, highlighting their advantages, disadvantages, and practical uses.The review starts out by defining the basic terms and concepts related to frequent pattern mining in unordered trees. The discussion then moves on to a number of widely used algorithms in this setting, such as graph-based strategies, bottom-up tree traversal techniques, and depth-first search-based techniques. Each approach is thoroughly explained, including its underlying concepts, computational complexity, and applicability to different kinds of tree datasets.The paper also examines recent developments in the subject, including distributed frameworks and scalable parallel algorithms for mining common patterns in big unordered tree collections. In order to improve the mining process and the calibre of patterns found, it also looks at the incorporation of extra constraints and measurements like weighted support and tree edit distance.The paper also talks about recent developments in the subject, namely scalable parallel algorithms and distributed frameworks for finding common patterns in enormous unordered tree collections. In order to strengthen the mining process and raise the calibre of patterns found, it also looks at the incorporation of extra restrictions and metrics like tree edit distance and weighted support.

Exchangeable interval hypergraphs and limits of ordered discrete structures

A hypergraph $(V,E)$ is called an interval hypergraph if there exists a linear order $l$ on $V$ such that every edge $e\in E$ is an interval w.r.t. $l$; we also assume that $\{j\}\in E$ for every $j\in V$. Our main result is a de Finetti-type representation of random exchangeable interval hypergraphs on $\mathbb{N}$ (EIHs): the law of every EIH can be obtained by sampling from some random compact subset $K$ of the triangle $\{(x,y):0\leq x\leq y\leq1\}$ at i.i.d. uniform positions $U_{1},U_{2},\dots$, in the sense that, restricted to the node set $[n]:=\{1,\dots,n\}$ every nonsingleton edge is of the form $e=\{i\in[n]:x<U_{i}<y\}$ for some $(x,y)\in K$. We obtain this result via the study of a related class of stochastic objects: erased-interval processes (EIPs). These are certain transient Markov chains $(I_{n},\eta_{n})_{n\in\mathbb{N}}$ such that $I_{n}$ is an interval hypergraph on $V=[n]$ w.r.t. the usual linear order (called interval system). We present an almost sure representation result for EIPs. Attached to each transient Markov chain is the notion of Martin boundary. The points in the boundary of EIPs can be seen as limits of growing interval systems. We obtain a one-to-one correspondence between these limits and compact subsets $K$ of the triangle with $(x,x)\in K$ for all $x\in[0,1]$. Interval hypergraphs are a generalizations of hierarchies and as a consequence we obtain a representation result for exchangeable hierarchies, which is close to a result of Forman, Haulk and Pitman in (Probab. Theory Related Fields 172 (2018) 1–29). Several ordered discrete structures can be seen as interval systems with additional properties, that is, Schröder trees (rooted, ordered, no node has outdegree one) or even more special: binary trees. We describe limits of Schröder trees as certain tree-like compact sets. These can be seen as an ordered counterpart to real trees, which are widely used to describe limits of discrete unordered trees. Considering binary trees, we thus obtain a homeomorphic description of the Martin boundary of Rémy’s tree growth chain, which has been analyzed by Evans, Grübel and Wakolbinger in (Ann. Probab. 45 (2017) 225–277).

Generating Trees for Comparison

Tree comparisons are used in various areas with various statistical or dissimilarity measures. Given that data in various domains are diverse, and a particular comparison approach could be more appropriate for specific applications, there is a need to evaluate different comparison approaches. As gathering real data is often an extensive task, using generated trees provides a faster evaluation of the proposed solutions. This paper presents three algorithms for generating random trees: parametrized by tree size, shape based on the node distribution and the amount of difference between generated trees. The motivation for the algorithms came from unordered trees that are created from class hierarchies in object-oriented programs. The presented algorithms are evaluated by statistical and dissimilarity measures to observe stability, behavior, and impact on node distribution. The results in the case of dissimilarity measures evaluation show that the algorithms are suitable for tree comparison.

Classification of maps on a finite set under permutation

We characterize when two maps f,g:X→X on a finite set are conjugated, that is, when there is a permutation σ:X→X such that f=σ−1gσ. We build a signature sgn (f) for every map f such that f and g are conjugated if and only if sgn (f)=sgn (g). This signature is an array that includes information about the cycles of the map and the noncyclic elements, called transients, besides data about the insertion of the transients in the cycle. The transient elements form several tree shape graphs. Our characterization is a generalization of the known fact that two permutations are conjugated if and only if they have the same cycle structure. We use elementary facts about finite dynamical systems and about the canonical labeling of unordered trees. The signature can be built by an algorithm of complexity O(n), being n the cardinality of X.

Path histogram distance and complete subtree histogram distance for rooted labelled caterpillars

ABSTRACTA rooted labelled caterpillar (a caterpillar, for short) is a rooted labelled unordered tree transformed to a path after removing all the leaves in it. In this paper, we discuss two histogram distance between caterpillars. One is a path histogram distance as an -distance between the histograms of paths from the root to every leaf and another is a complete subtree histogram distance as an -distance between the histograms of complete subtrees for every node. While the latter is always a metric for general trees, the former is not a metric. In this paper, we show that, for caterpillars, the path histogram distance is always a metric, simply linear-time computable and incomparable with the edit distance. Furthermore, we give experimental results for caterpillars in real data of comparing the path histogram distance and the complete subtree histogram distance with the isolated-subtree distance as the most general tractable variation of the edit distance.

Logics for unordered trees with data constraints

We study monadic second-order logics with counting constraints (CMso) for unordered data trees. Our objective is to enhance this logic with data constraints for comparing string data values. Comparisons between data values at arbitrary positions of a data tree quickly lead to undecidability. Therefore, we restrict ourselves to comparing sibling data values of unordered trees. But even in this case CMso remains undecidable when allowing for data comparisons that can check the equality of string factors. However, for more restricted data constraints that can only check the equality of string prefixes, it becomes decidable. This decidability result is obtained by reduction to WSkS. Furthermore, we exhibit a restricted class of constraints which can be used in transitions of tree automata, resulting in a model with tractable complexity, which can be extended with structural equality tests between siblings. This efficient restriction is relevant to applications such as checking well-formedness properties of file system trees.

NED

Node similarity is fundamental in graph analytics. However, node similarity between nodes in different graphs (inter-graph nodes) has not received enough attention yet. The inter-graph node similarity is important in learning a new graph based on the knowledge extracted from an existing graph (transfer learning on graphs) and has applications in biological, communication, and social networks. In this paper, we propose a novel distance function for measuring inter-graph &lt;u&gt;n&lt;/u&gt;ode similarity with &lt;u&gt;e&lt;/u&gt;dit &lt;u&gt;d&lt;/u&gt;istance, called NED . In NED, two nodes are compared according to their local neighborhood topologies which are represented as unordered k -adjacent trees, without relying on any extra information. Due to the hardness of computing tree edit distance on unordered trees which is NP-Complete, we propose a modified tree edit distance, called TED* , for comparing unordered and unlabeled k -adjacent trees. TED* is a metric distance, as the original tree edit distance, but more importantly, TED* is polynomially computable. As a metric distance, NED admits efficient indexing, provides interpretable results, and shows to perform better than existing approaches on a number of data analysis tasks, including graph deanonymization. Finally, the efficiency and effectiveness of NED are empirically demonstrated using real-world graphs.