Approximating Tree Edit Distance through String Edit Distance

We present an algorithm to approximate edit distance between two ordered and rooted trees of bounded degree. In this algorithm, each input tree is transformed into a string by computing the Euler string, where labels of some edges in the input trees are modified so that structures of small subtrees are reflected to the labels. We show that the edit distance between trees is at least 1/6 and at most O(n ^3/4) of the edit distance between the transformed strings, where n is the maximum size of two input trees and we assume unit cost edit operations for both trees and strings. The algorithm works in O(n ²) time since computation of edit distance and reconstruction of tree mapping from string alignment takes O(n ²) time though transformation itself can be done in O(n) time.     

Online shape learning using binary search trees

In this paper we propose an online shape learning algorithm based on the self-balancing binary search tree data structure for the storage and retrieval of shape templates. This structure can also be used for classification purposes. We introduce a similarity measure with which we can make decisions on how to traverse the tree and even backtrack through the search path to find more candidate matches. Then we describe every basic operation a binary search tree can perform adapted to such a tree of shapes. Note that as a property of binary search trees, all operations can be performed in O(logn)$O(\log n)$ time and are very efficient. Finally, we present experimental data evaluating the performance of the proposed algorithm and demonstrating the suitability of this data structure for the purpose it was designed to serve.     

A New Approach to Graph Recognition and Applications to Distance-Hereditary Graphs

Algorithms used in data mining and bioinformatics have to deal with huge amount of data efficiently.In many applications,the data are supposed to have explicit or implicit structures.To develop efficient algorithms for such data,we have to propose possible structure models and test if the models are feasible.Hence,it is important to make a compact model for structured data,and enumerate all instances efficiently.There are few graph classes besides trees that can be used for a model.In this paper,we inves...     

Perfect load balancing on the star interconnection network

In this paper, we use the regular distribution method to design a perfect load balancing algorithm for an n-star with a maximum error of 1 and a time complexity of 3n(n+1). This algorithm is based on the novel notion of leader trees. A second algorithm proposed in this paper as an enhancement to our first algorithm and uses an arbitrary spanning tree as the leader tree and has a worst time complexity of 2.25n ²−3n+0.75. We also discuss the issue of dynamically selecting the leader tree and hybrid load balancing algorithms in general. Furthermore, we present a hybrid algorithm for load balancing on the star interconnection network which benefits from a diffusion load balancing preprocessing phase and shows a smaller mean time complexity than our two first algorithms.     

Trade-offs Between the Size of Advice and Broadcasting Time in Trees

We study the problem of the amount of information required to perform fast broadcasting in tree networks. The source located at the root of a tree has to disseminate a message to all nodes. In each round each informed node can transmit to one child. Nodes do not know the topology of the tree but an oracle knowing it can give a string of bits of advice to the source which can then pass it down the tree with the source message. The quality of a broadcasting algorithm with advice is measured by its competitive ratio: the worst case ratio, taken over n-node trees, between the time of this algorithm and the optimal broadcasting time in the given tree. Our goal is to find a trade-off between the size of advice and the best competitive ratio of a broadcasting algorithm for n-node trees. We establish such a trade-off with an approximation factor of O(n ^ε ), for an arbitrarily small positive constant ε. This is the first communication problem for which a trade-off between the size of advice and the efficiency of the solution is shown for arbitrary size of advice.     

A Flexible Algorithm for Generating All the Spanning Trees in Undirected Graphs

In this paper we propose an algorithm for generating all the spanning trees in undirected graphs. The algorithm requires O (n+m+ τ n) time where the given graph has n vertices, m edges, and τ spanning trees. For outputting all the spanning trees explicitly, this time complexity is optimal. Our algorithm follows a special rooted tree structure on the skeleton graph of the spanning tree polytope. The rule by which the rooted tree structure is traversed is irrelevant to the time complexity. In this sense, our algorithm is flexible. If we employ the depth-first search rule, we can save the memory requirement to O (n+m). A breadth-first implementation requires as much as O (m+ τ n) space, but when a parallel computer is available, this might have an advantage. When a given graph is weighted, the best-first search rule provides a ranking algorithm for the minimum spanning tree problem. The ranking algorithm requires O (n+ m + τ n) time and O (m+ τ n) space when we have a minimum spanning tree. Received January 21, 1995; revised February 19, 1996.     

Optimal search trees using two-way key comparisons

A generalization of binary search trees and binary split trees is developed that takes advantage of two-way key comparisons: the two-way comparison tree. The two-way comparison tree has little use for dynamic situations but is an improvement over the optimal binary search tree and the optimal binary split tree for static data sets. AnO(n) time and space algorithm is presented for constructing an optimal two-way comparison tree when access probabilities are equal, and an exact formula for the optimal cost is developed. The construction of the optimal two-way comparison tree for unequal access frequencies, both successful and unsuccessful, is computable inO(n ⁵) time andO(n ³) space using algorithms similar to those for the optimal binary split tree. The optimal two-way comparison tree can improve search cost by up to 50% over the optimal binary search tree.     

A VLSI Algorithm for Calculating the Tree to Tree Distance

Given two ordered, labeled trees β and α, to find the distance from tree β to tree α is an important problem in many fields, for example, the pattern recognition field. In this paper, a VLSI algorithm for calculating the tree to tree distance is presented. The computation structure of the algorithm is a 2-D Mesh with the sizem*n and the time isO(m+n), wherem,n are the numbers of nodes of the tree β and tree α, respectively.     

On a Nearest-Neighbor Problem Under Minkowski and Power Metrics for Large Data Sets

The paper presents the sequential and the parallel algorithm for solving the nearest-neighbor problem in the plane, based on the generalized Voronoi diagram construction. The applications of the problem are found in the areas of networking, communications, distributed systems, computer modeling and information retrieval. The input for the problem is the set of circular sites S with varying radii, the query point p and the metric (Minkowski or power) according to which the site, neighboring the query point, is to be reported. The sequential algorithm takes O(n) time to build the data structure and O(log n) time for each query. The parallel algorithm requires O(log n log) preprocessing time and O(log) query time on EREW PRAM architecture with n/log n processors. The IDG/NNM software was developed for an experimental study of the problem. The experimental results demonstrate that the Voronoi diagram method outperforms the k – d tree method for all tested input configurations. The tests were conducted on large data sets, comprising thousands of generators.     

Dynamic multi-dimensional data structures based on quad- and k—d trees

Summary Quad-trees and k—d trees have been noted for their lack of dynamic properties as data structures for multi-dimensional point sets. We describe a method to insert points in a quad-tree while keeping the tree balanced that achieves an average time complexity of O(log² N) per insertion, where N is the number of updates performed on the quad-tree. We define a structure similar to a quad-tree, called a pseudo quad-tree, and show how it can be used to handle both insertions and deletions in O(log² N) average time. We also discuss how quad-trees and pseudo quadtrees can be extended for use in configurations of points in which more than one point may have a same value in some equal coordinate, without altering the earlier time bounds for insertions, deletions and queries. Similar algorithms are given for k—d trees and the same average time bounds for insertion and deletion are achieved.The work of this author was partially supported by the Netherlands Organization for the Advancement of Pure Research (ZWO).