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 $O(\min ({1.26}^{n_1+n_2},2^{b_1+b_2}\cdot \text{<mi mathvariant="italic" is="true">poly</mi>}(n_1,n_2)))$ time algorithm is presented, where $n_1$ and $n_2$ are the numbers of nodes and $b_1$ and $b_2$ 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 $O({(1+ϵ)}^{n_1+n_2})$ time for any fixed $ϵ>0$. This result is achieved by avoiding duplicate calculations for identical subsets of small subtrees.     

A bijection between ordered trees and bicoloured ordered trees

A bijection is introduced between ordered trees and bicoloured ordered trees, which maps leaves in an ordered tree to odd height vertices in the related tree. Consequently, a bijection between two sets of bridges specified with four parameters is derived.     

A dynamic edit distance table

In this paper we consider the incremental/decremental version of the edit distance problem: given a solution to the edit distance between two strings A and B, find a solution to the edit distance between A and B′ where B′=aB (incremental) or bB′=B (decremental). As a solution for the edit distance between A and B, we define the difference representation of the D-table, which leads to a simple and intuitive algorithm for the incremental/decremental edit distance problem.     

A distance approximating trees

A 1-approximation of connected graph G=(V,E) is a tree T=(V,E^′) with the same vertex set such that for every two vertices |d_G(u,v)−d_T(u,v)|1. A polynomial time algorithm is designed for finding such a tree.     

A quadratic distance bound on sliding between crossing-free spanning trees

Let S be a set of n points in the plane and let be the set of all crossing-free spanning trees of S. We show that it is possible to transform any two trees in into each other by O(n²) local and constant-size edge slide operations. Previously no polynomial upper bound for this task was known, but in [O. Aichholzer, F. Aurenhammer, F. Hurtado, Sequences of spanning trees and a fixed tree theorem, Comput. Geom.: Theory Appl. 21 (1–2) (2002) 3–20] a bound of O(n²logn) operations was conjectured.     

Lexicographically ordered trees

We characterize trees whose lexicographic ordering produces an order isomorphic copy of some sets of real numbers, or an order isomorphic copy of some set of ordinal numbers. We characterize trees whose lexicographic ordering is order complete, and we investigate lexicographically ordered ω-splitting trees that, under the open-interval topology of their lexicographic orders, are of the first Baire category. Finally we collect together some folklore results about the relation between Aronszajn trees and Aronszajn lines, and use earlier results of the paper to deduce some topological properties of Aronszajn lines.     

On the rotation distance between binary trees

We develop combinatorial methods for establishing lower bounds on the rotation distance between binary trees, i.e., equivalently, on the flip distance between triangulations of a polygon. These methods lead to sharp estimates for certain particular pairs of trees. As an application, we prove that, for each n, there exist size n trees at distance , i.e., the diameter of the nth associahedron has at least this value.     

Balanced ordered trees

A 0-balanced ordered three is an ordered tree all of whose root-to-leaf paths have the same length. In this article, we persent exact and asymptotic enumeration and distribution ersults for classes of 0-balanced ordered trees with n nodes. These results enable us to compute various expected values of interesting parameters defined on these trees. © 1994 John Wiley & Sons, Inc.     

Counterexamples in measuring the distance between binary trees

Examples are given to show that the closest partition distance measure need not agree with the nearest neighbor interchange distance for unordered labeled binary trees. Proposed algorithms for computing the closest partition distance are shown to be of exponential complexity and hence may not be useful in approximating the nearest neighbor interchange distance.     

A bijection between ordered trees and 2-Motzkin paths and its many consequences

A new bijection between ordered trees and 2-Motzkin paths is presented, together with its numerous consequences regarding ordered trees as well as other combinatorial structures such as Dyck paths, bushes, {0,1,2}-trees, Schröder paths, RNA secondary structures, noncrossing partitions, Fine paths, and Davenport-Schinzel sequences.RésuméUne nouvelle bijection entre arbres ordonnés et chemins de Motzkin bicolorés est présentée, avec ses nombreuses conséquences en ce qui concerne les arbres ordonnés ainsi que d'autres structures combinatoires telles que chemins de Dyck, buissons, arbres de type {0,1,2}, chemins de Schröder, structures secondaires de type RNA, partitions non croisées, chemins de Fine, et enfin suites de Davenport-Schinzel.     

Restructuring ordered binary trees

We consider the problem of restructuring an ordered binary tree T, preserving the in-order sequence of its nodes, so as to reduce its height to some target value h. Such a restructuring necessarily involves the downward displacement of some of the nodes of T. Our results, focusing both on the maximum displacement over all nodes and on the maximum displacement over leaves only, provide (i) an explicit tradeoff between the worst-case displacement and the height restriction (including a family of trees that exhibit the worst-case displacements) and (ii) efficient algorithms to achieve height-restricted restructuring while minimizing the maximum node displacement.     

Convex graph invariant relaxations for graph edit distance

Mathematical Programming - The edit distance between two graphs is a widely used measure of similarity that evaluates the smallest number of vertex and edge deletions/insertions required to...     

The maximum edit distance from hereditary graph properties

For a graph property , the edit distance of a graph G from , denoted , is the minimum number of edge modifications (additions or deletions) one needs to apply to G in order to turn it into a graph satisfying . What is the largest possible edit distance of a graph on n vertices from ? Denote this distance by .A graph property is hereditary if it is closed under removal of vertices. In a previous work, the authors show that for any hereditary property, a random graph essentially achieves the maximal distance from , proving: with high probability. The proof implicitly asserts the existence of such , but it does not supply a general tool for determining its value or the edit distance.In this paper, we determine the values of and for some subfamilies of hereditary properties including sparse hereditary properties, complement invariant properties, (r,s)-colorability and more. We provide methods for analyzing the maximum edit distance from the graph properties of being induced H-free for some graphs H, and use it to show that in some natural cases G(n,1/2) is not the furthest graph. Throughout the paper, the various tools let us deduce the asymptotic maximum edit distance from some well studied hereditary graph properties, such as being Perfect, Chordal, Interval, Permutation, Claw-Free, Cograph and more. We also determine the edit distance of G(n,1/2) from any hereditary property, and investigate the behavior of as a function of p.The proofs combine several tools in Extremal Graph Theory, including strengthened versions of the Szemerédi Regularity Lemma, Ramsey Theory and properties of random graphs.     

Decomposition algorithms for the tree edit distance problem

We study the behavior of dynamic programming methods for the tree edit distance problem, such as [P. Klein, Computing the edit-distance between unrooted ordered trees, in: Proceedings of 6th European Symposium on Algorithms, 1998, p. 91–102; K. Zhang, D. Shasha, SIAM J. Comput. 18 (6) (1989) 1245–1262]. We show that those two algorithms may be described as decomposition strategies. We introduce the general framework of cover strategies, and we provide an exact characterization of the complexity of cover strategies. This analysis allows us to define a new tree edit distance algorithm, that is optimal for cover strategies.     

Fuzzy ordered distance measures

Xu and Chen (J Syst Sci Syst Eng 17:432–445, 2008) introduced a type of ordered weighted distance measures called ordered weighted distance (OWD) measures whose fundamental aspect is the reordering step, which can be used in many actual fields, including group decision making, medical diagnosis, data mining, and pattern recognition, etc. The OWD measures are very suitable to deal with situations where the input data are expressed in exact numerical values. In this paper, we consider situations with linguistic, interval or fuzzy preference information, and develop some fuzzy ordered distance measures, such as linguistic ordered weighted distance measure, uncertain ordered weighted distance measure, linguistic hybrid weighted distance measure, and uncertain hybrid weighted distance measure, etc. After that, based on hybrid weighted distance measures, we establish a consensus reaching process of group decision making with linguistic, interval, triangular or trapezoidal fuzzy preference information.