A rotation in a binary tree is a local restructuring of the tree that changes it into another tree. One can execute a rotation by collapsing an internal edge of the tree to a point, thereby obtaining a node with three children, and then re-expanding the node of order three in the alternative way into two nodes of

Different phylogenetic trees for the same group of species are often produced either by procedures that use diverse optimality criteria [18] or from different genes [12] in the study of molecular evolution. Comparing these trees to find their similarities (e.g. agreement or consensus) and dissimilarities, i.e. distance, is thus an important issue in computational molecular biology. The nearest neighbor interchange (nni) distance [26, 24, 32, 4, 5, 3, 16, 17, 19, 30, 20, 21, 23] and the subtree-transfer distance [12, 13, 15] are two major distance metrics that have been proposed and extensively studied for different reasons. Despite their many appealing aspects such as simplicity and sensitivity to tree topologies, computing these distances has remained very challenging. This article studies the complexity and efficient approximation algorithms for computing the nni distance and a natural extension of the subtree-transfer distance, called the linear-cost subtree-transfer distance. The ...

Documents can be represented as ordered labelled trees. Finding the editing distance between documents is a particular case of the general problem for trees. We give a detailed survey of previous results, presenting them in a single notation to elucidate their commonalities. We then discuss two ways of extending these results---first, by changing the set of primitive editing operations used by existing algorithms and, second, by post-processing the output of the algorithms to recognize patterns of change significant to documents. Finally, we provide extensions of the first type. Our algorithm allows subtree operations but is otherwise similar to that of Zhang and Shasha. This is a corrected and expanded version of Technical Report 91-315. y This report was completed during a sabbatical at INRIA (Institute National de Recherche en Informatique et en Automatique) in Rocquencourt, France. Contents 1 Introduction 3 2 Background 5 2.1 String-to-String Correction: Wagner and Fischer ...

. We survey results on self-organizing data structures for the search problem and concentrate on two very popular structures: the unsorted linear list, and the binary search tree. For the problem of maintaining unsorted lists, also known as the list update problem, we present results on the competitiveness achieved by deterministic and randomized on-line algorithms. For binary search trees, we present results for both on-line and off-line algorithms. Self-organizing data structures can be used to build very effective data compression schemes. We summarize theoretical and experimental results. 1 Introduction This paper surveys results in the design and analysis of self-organizing data structures for the search problem. The general search problem in pointer data structures can be phrased as follows. The elements of a set are stored in a collection of nodes. Each node also contains O(1) pointers to other nodes and additional state data which can be used for navigation and self-organizati...

In the practice of molecular evolution, different phylogenetic trees for the same group of species are often produced either by procedures that use diverse optimality criteria [24] or from different genes [15, 16, 17, 18, 14]. Comparing these trees to find their similarities (e.g. agreement or consensus) and dissimilarities, i.e. distance, is thus an important issue in computational molecular biology. The nearest neighbor interchange (nni) distance [29, 28, 34, 3, 6, 2, 19, 20, 23, 33, 22, 21, 26] is a natural distance metric that has been extensively studied. Despite its many appealing aspects such as simplicity and sensitivity to tree topologies, computing this distance has remained very challenging, and many algorithmic and complexity issues about computing this distance have remained unresolved. This paper studies the complexity and e#cient approximation algorithms for computing the nni distance and a natural extension of this distance on weighted phylogenies. The foll...

We present some new results on a well-known distance measure between evolutionary trees. The trees we consider are free 3-trees having n leaves labeled 0,...,n − 1 (representing species), and n −2 internal nodes of degree 3. The distance between two trees is the minimum number of nearest neighbour interchange (NNI) operations required to transform one into the other. First, we improve an upper bound on the nni-distance between two arbitrary n-node trees from 4n log n (Culik & Wood, 1982, Inf. Pro. Letts. 15, 39–42) to n log n. Second, we present a counterexample disproving several theorems in (Waterman & Smith, 1978, J. theor. Biol. 73, 789–800). Roughly speaking, finding an equal partition between two trees does not imply decomposability of the distance finding problem. Third, we present a polynomial-time approximation algorithm that, given two trees, finds a transformation between them of length O(log n) times their distance. We also present some results of computations we performed on small size trees. � 1996 Academic Press Limited

We investigate the problem of transforming one binary tree into another by rotations, subject to certain weight constraints on the nodes of the trees. These constraints arise in the problem of "morphing" one simple polygon to another simple polygon by continuous deformations (translations and scalings) that preserve the turn angles and the simplicity of the polygon; the two polygons must have the same sequence of turn angles. Our main theorem is that two arbitrary n-leaf binary trees satisfying our weight constraint can be morphed into each other with O(n log n) rotations. Furthermore, we also present an O(n log n) time algorithm to determine these rotations. The previous best algorithm for this problem used O(n 4=3+ffl ) rotations. 1 Introduction "Morphing," the continuous deformation of one shape to another, is a popular theme in computer graphics [1, 4, 5, 6]. A recent paper by Guibas and Hershberger [3] considers the problem of morphing a simple polygon P to another simple poly...

this paper are degree-3 trees with unique labels on leaves. An edge of a tree is external if it is incident on a leaf, otherwise it is internal. 2 The Nni and Subtree-transfer Distances

. We present some new results on a well known distance measure between evolutionary trees. The trees we consider are free 3-trees having n leaves labeled 0; : : : ; n \Gamma 1 (representing species), and n \Gamma 2 internal nodes of degree 3. The distance between two trees is the minimum number of nearest neighbour interchange (NNI) operations required to transform one into the other. First, we improve an upper bound on the nni-distance between two arbitrary n-node trees from 4n log n [2] to n log n. Second, we present a counterexample disproving several theorems in [13]. Roughly speaking, finding an equal partition between two trees doesn't imply decomposability of the distance finding problem. Third, we present a polynomial-time approximation algorithm that, given two trees, finds a transformation between them of length O(log n) times their distance. We also present some results of computations we performed on small size trees. 1 Introduction In a free 3-tree, n leaf nodes, labeled...

Introduction The modern era of molecular biology began with the discovery of the double helical structure of DNA. Today, sequencing nucleic acids, the determination of genetic information at the most fundamental level, is a major tool of biological research [79]. This revolution in biology has created a huge amount of data at great speed by directly reading DNA sequences. The growth rate of data volume is exponential. For instance, the volume of DNA and protein sequence data is currently doubling every 22 months [55]. One important reason for this exceptional growth rate of biological data is the medical use of such information in the design of diagnostics and therapeutics [24, 50]. For example, identification of genetic markers in DNA sequences would provide Supported in part by Hong Kong Research Council. important informations regarding which portions of the DNA are significant, and would allow the researchers to find many disease genes of