Unordered Trees Research Articles

Algorithms for solving the symmetry number problem on trees

The symmetry number of a tree is defined as the number of nodes of the maximum subtree of the tree that exhibits axial symmetry. Chin and Yen have presented an algorithm for solving the problem of finding the symmetry number of unrooted unordered trees in time O( n 4). In this paper we present an improved algorithm for solving the symmetry number problem on trees that runs in time O( n 3). We also show that our algorithm needs O( n 2log n) time for trees with bounded degrees.

Managing and analyzing carbohydrate data

One of the most vital molecules in multicellular organisms is the carbohydrate, as it is structurally important in the construction of such organisms. In fact, all cells in nature carry carbohydrate sugar chains, or glycans, that help modulate various cell-cell events for the development of the organism. Unfortunately, informatics research on glycans has been slow in comparison to DNA and proteins, largely due to difficulties in the biological analysis of glycan structures. Our work consists of data engineering approaches in order to glean some understanding of the current glycan data that is publicly available. In particular, by modeling glycans as labeled unordered trees, we have implemented a tree-matching algorithm for measuring tree similarity. Our algorithm utilizes proven efficient methodologies in computer science that has been extended and developed for glycan data. Moreover, since glycans are recognized by various agents in multicellular organisms, in order to capture the patterns that might be recognized, we needed to somehow capture the dependencies that seem to range beyond the directly connected nodes in a tree. Therefore, by defining glycans as labeled ordered trees, we were able to develop a new probabilistic tree model such that sibling patterns across a tree could be mined. We provide promising results from our methodologies that could prove useful for the future of glycome informatics.

A logic you can count on

We prove the decidability of the quantifier-free, static fragment of ambient logic, with composition adjunct and iteration, which corresponds to a kind of regular expression language for semistructured data. The essence of this result is a surprising connection between formulas of the ambient logic and counting constraints on (nested) vectors of integers.Our proof method is based on a new class of tree automata for unranked, unordered trees, which may result in practical algorithms for deciding the satisfiability of a formula. A benefit of our approach is to naturally lead to an extension of the logic with recursive definitions, which is also decidable. Finally, we identify a simple syntactic restriction on formulas that improves the effectiveness of our algorithms on large examples.

Optimal mappings with minimum number of connected components in tree-to-tree comparison problems

In this paper we consider the problem of finding a minimum cost mapping between two unordered trees which induces a graph with a minimum number of connected components. The proposed algorithm is based on the generalization of an algorithm for computing an edit distance between trees, and it solves the stated problem in sequential time precisely in O(| T 1|×| T 2|×(deg T 1+deg T 2)×log 2(deg T 1+deg T 2)).

SIZES OF ORDERED DECISION TREES

Decision tables provide a natural framework for knowledge acquisition and representation in the area of knowledge based information systems. Decision trees provide a standard method for inductive inference in the area of machine learning. In this paper we show how decision tables can be considered as ordered decision trees: decision trees satisfying an ordering restriction on the nodes. Every decision tree can be represented by an equivalent ordered decision tree, but we show that doing so may exponentially blow up sizes, even if the choice of the order is left free. Our main result states that finding an ordered decision tree of minimal size that represents the same function as a given ordered decision tree is an NP-hard problem; in earlier work we obtained a similar result for unordered decision trees.

Finding approximate patterns in undirected acyclic graphs

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 PN Dd 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.

The symmetry number problem for trees

For trees, we define the notion of the so-called symmetry number to measure the size of the maximum subtree that exhibits an axial symmetry in graph drawing. For unrooted unordered trees, we are able to demonstrate a polynomial time algorithm for computing the symmetry number.

On Exact Learning of Unordered Tree Patterns

Tree patterns are natural candidates for representing rules and hypotheses in many tasks such as information extraction and symbolic mathematics. A tree pattern is a tree with labeled nodes where some of the leaves may be labeled with variables, whereas a tree instance has no variables. A tree pattern matches an instance if there is a consistent substitution for the variables that allows a mapping of subtrees to matching subtrees of the instance. A finite union of tree patterns is called a forest. In this paper, we study the learnability of tree patterns from queries when the subtrees are unordered. The learnability is determined by the semantics of matching as defined by the types of mappings from the pattern subtrees to the instance subtrees. We first show that unordered tree patterns and forests are not exactly learnable from equivalence and subset queries when the mapping between subtrees is one-to-one onto, regardless of the computational power of the learner. Tree and forest patterns are learnable from equivalence and membership queries for the one-to-one into mapping. Finally, we connect the problem of learning tree patterns to inductive logic programming by describing a class of tree patterns called Clausal trees that includes non-recursive single-predicate Horn clauses and show that this class is learnable from equivalence and membership queries.

On Regular Tree Embeddings

Regular trees are a natural extension of finite trees, which have many applications. The path-embedding problem is to determine whether a regular tree S can be obtained from another regular tree T by deleting (probably infinitely many) subtrees of T. This paper explores efficient algorithms for the path-embedding problem in ordered and unordered trees. Given two regular trees S and T represented by rational graphs, our algorithms solve the ordered version of path-embedding problem in O(|E_S||E_T|) time and the unordered version in O(|ES||ET|DSDT) time. Here |ES| denotes the number of edges in the rational graph for S, and DS denotes the maximum outdegree of a vertex inS. We also demonstrate that our approach can be applied to pattern matching problems for regular trees recently studied by Fu J. Algorithms, 22 (1997), pp. 372--391].

Complexity results on the minimum width drawing problem for rooted unordered trees

Complexity results on the minimum width drawing problem for rooted unordered trees

Complexity results on the minimum width drawing problem for rooted unordered trees

The minimum width drawing problem is a problem in which a rooted tree T and an integer B are given, and it must be decided whether or not there exists a drawing of T with width B or less satisfying the specified condition. This problem is widely investigated for the case of an ordered rooted tree. This paper discusses the computational complexity of the minimum width drawing problem for a rooted unordered tree. As the first step, several constraints are introduced for the placement of vertices, and eight sets of constraints are defined. It is shown for each of the eight constraint sets that the minimum width drawing problem is NP complete, whichever integer coordinate system or real coordinate system is used. It is also shown in the case of the integer coordinate system that the problem is NP complete, even if the number of children of each vertex is limited to three or less. © 1998 Scripta Technica. Electron Comm Jpn Pt 3, 81(1): 32–41, 1998

Algorithms for computing the distances between unordered trees

AbstractThis paper proposes algorithms for computing three kinds of distances between two rooted and unordered trees (R‐trees) and those between two unrooted and unordered trees (trees). Those are the distance based on the strongly structure preserving mapping (SSPD), that based on the closest common ancestor mappings (CD) and that based on the maximal closest common ancestor mapping (MCD). the time and space complexities to compute each distance between two R‐trees Ta and Tb are OT(ma Na Nb) and OS(Na Nb), respectively, and those to compute each distance between two trees Ta and Tb are OT(max(ma,mb)2 NaNb) and OS(NaNb), respectively, where ma(mb) and Na(Nb) are the largest degree of a vertex and the number of vertices of Ta(Tb), respectively. the proposed algorithms for SSPD and CD are more efficient than the algorithms previously proposed. MCD is newly proposed in this paper. Those distances can be applied to structureactivity studies in chemistry and various structure comparison problems.

A constrained edit distance between unordered labeled trees

This paper considers the problem of computing a constrained edit distance between unordered labeled trees. The problem of approximate unordered tree matching is also considered. We present dynamic programming algorithms solving these problems in sequential timeO(|T1|×|T2|×(deg(T1)+deg(T2))× log2(deg(T1)+deg(T2))). Our previous result shows that computing the edit distance between unordered labeled trees is NP-complete.

An algebraic technique for generating optimal CMOS circuitry in linear time

We explore a method for quickly generating optimal CMOS functional circuits. The method is based upon an algebra we have derived that describes the composition of parallel-series graphs and their duals simultaneously , and as such, exactly describes the layout of CMOS functional circuits. The method is constructive; it creates the smallest components first, putting them together until the final circuit is realized. The constructed layout is representative of an unordered tree traversal, and is generated in time proportional to the number of input signals. After describing the required concepts from graph theory and CMOS layout practices, we introduce an alternative symbolism for describing parallel-series graphs. We develop, with these symbols, a composition algebra, and demonstrate that the properties in the alternative domain hold in the original. We then use the algebra to implement a linear-time algorithm for generating CMOS functional cells.

Alignment of trees — an alternative to tree edit

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.

Alignment of trees — an alternative to tree edit

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.

Ordered and Unordered Tree Inclusion

The following tree-matching problem is considered: Given labeled trees $P$ and $T$, can $P$ be obtained from $T$ by deleting nodes? Deleting a node $u$ entails removing all edges incident to $u$ and, if $u$ has a parent $v$, replacing the edge from $v$ to $u$ by edges from $v$ to the children of $u$. The problem is motivated by the study of query languages for structured text databases. Simple solutions to this problem require exponential time. For ordered trees an algorithm is presented that requires $O(|P| |T|)$ time and space. The corresponding problem for unordered trees is also considered and a proof of its NP-completeness is given. An algorithm is presented for the unordered problem. This algorithm works in $O(|P| |T|)$ time if the out-degrees of the nodes in $P$ are bounded by a constant, and in polynomial time if they are $O(\log |T|)$.

Exact and approximate algorithms for unordered tree matching

We consider the problem of comparison between unordered trees, i.e., trees for which the order among siblings is unimportant. The criterion for comparison is the distance as measured by a weighted sum of the costs of deletion, insertion and relabel operations on tree nodes. Such comparisons may contribute to pattern recognition efforts in any field (e.g., genetics) where data can naturally be characterized by unordered trees. In companion work, we have shown this problem to be NP-complete. This paper presents an efficient enumerative algorithm and several heuristics leading to approximate solutions. The algorithms are based on probabilistic hill climbing and bipartite matching techniques. The paper evaluates the accuracy and time efficiency of the heuristics by applying them to a set of trees transformed from industrial parts based on a previously proposed morphological model. >

Some MAX SNP-hard results concerning unordered labeled trees

Given two rooted, labeled, and unordered trees, we consider the problem of finding the largest common sub-tree and the problem of finding the edit distance between them. We show that both problems are MAX SNP-hard. This means that neither problem has a polynomial time approximation scheme (PTAS) unless P = NP.

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/).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>