Ross M. McConnell

Modular decomposition and transitive orientation

A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linear-space representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a direction to each edge of a graph so that the resulting digraph is transitive. A graph is a comparability graph if such an assignment is possible. We give O(n+m) algorithms for modular decomposition and transitive orientation, where n and m are the number of vertices and edges of the graph. This gives linear time bounds for recognizing permutation graphs, maximum clique and minimum vertex coloring on comparability graphs, and other combinatorial problems on comparability graphs and their complements. c 1999 Published by Elsevier Science B.V. All rights reserved

Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing

By making use of lexicographic breadth rst search (Lex-BFS) and partition renement with pivots, we obtain very simple algorithms for some well-known problems in graph theory. We give a O(n+mlogn) algorithm for transitive orientation of a comparability graph, and simple linear algorithms to recognize interval graphs, convex graphs, Y-semichordal graphs and matrices that have the consecutive ones property. Previous approaches to these problems used dicult preprocessing steps, such as computing PQ-trees or modular decomposition. The algorithms we give are easy to understand and straightforward to prove. They do not make use of sophisticated data structures, and the complexity analysis is straightforward. c 2000 Elsevier Science B.V. All rights reserved.

Complete inverted files for efficient text retrieval and analysis

Given a finite set of texts <italic>S</italic> = {<italic>w</italic>1, … , <italic>w<subscrpt>k</subscrpt></italic>} over some fixed finite alphabet &Sgr;, a complete inverted file for <italic>S</italic> is an abstract data type that provides the functions <italic>find</italic>(<italic>w</italic>), which returns the longest prefix of <italic>w</italic> that occurs (as a subword of a word) in <italic>S</italic>; <italic>freq</italic>(<italic>w</italic>), which returns the number of times <italic>w</italic> occurs in <italic>S</italic>; and <italic>locations</italic>(<italic>w</italic>), which returns the set of positions where <italic>w</italic> occurs in <italic>S</italic>. A data structure that implements a complete inverted file for <italic>S</italic> that occupies linear space and can be built in linear time, using the uniform-cost RAM model, is given. Using this data structure, the time for each of the above query functions is optimal. To accomplish this, techniques from the theory of finite automata and the work on suffix trees are used to build a deterministic finite automaton that recognizes the set of all subwords of the set <italic>S</italic>. This automaton is then annotated with additional information and compacted to facilitate the desired query functions. The result is a data structure that is smaller and more flexible than the suffix tree.

An ice-motion tracking system at the Alaska SAR facility

An operational system for extracting ice-motion information from synthetic aperture radar (SAR) imagery is being developed as part of the Alaska SAR Facility. This geophysical processing system (GPS) will derive ice-motion information by automated analysis of image sequences acquired by radars on the European ERS-1, Japanese ERS-1, and Canadian RADARSAT remote sensing satellites. The algorithm consists of a novel combination of feature-based and area-based techniques for the tracking of ice floes that undergo translation and rotation between imaging passes. The system performs automatic selection of the image pairs for input to the matching routines using an ice-motion estimator. It is designed to have a daily throughput of ten image pairs. A description is given of the GPS system, including an overview of the ice-motion-tracking algorithm, the system architecture, and the ice-motion products that will be available for distribution to geophysical data users. >

Linear-time recognition of circular-arc graphs

Abstract A graph G is a circular-arc graph if it is the intersection graph of a set of arcs on a circle. That is, there is one arc for each vertex of G, and two vertices are adjacent in G if and only if the corresponding arcs intersect. We give a linear-time algorithm for recognizing this class of graphs. When G is a member of the class, the algorithm gives a certificate in the form of a set of arcs that realize it.

PC trees and circular-ones arrangements

A 0-1 matrix has the consecutive-ones property if its columns can be ordered so that the ones in every row are consecutive. It has the circular-ones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all consecutive-ones orderings of the columns of a matrix that have the consecutive-ones property. We give an analogous structure, called a PC tree, for representing all circular-ones orderings of the columns of a matrix that has the circular-ones property. No such representation has been given previously. In contrast to PQ trees, PC trees are unrooted. We obtain a much simpler algorithm for computing PQ trees that those that were previously available, by adding a zero column, x, to a matrix, computing the PC tree, and then picking the PC tree up by x to root it.

Efficient and Practical Algorithms for Sequential Modular Decomposition

A module of an undirected graph G=(V,E) is a set X of vertices that have the same set of neighbors in V\X. The modular decomposition is a unique decomposition of the vertices into nested modules. We give a practical algorithm with an O(n+m?(m,n)) time bound and a variant with a linear time bound.

Certifying Algorithms for Recognizing Interval Graphs and Permutation Graphs

A certifying algorithm for a decision problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give linear-time certifying algorithms for recognition of interval graphs and permutation graphs. Previous algorithms fail to provide supporting evidence when they claim that the input graph is not a member of the class. We show that our certificates of non-membership can be authenticated in O(|V|) time.

psi -s correlation and dynamic time warping: two methods for tracking ice floes in SAR images

The authors present two algorithms for performing shape matching on ice floe boundaries in SAR (synthetic aperture radar) images. These algorithms quickly produce a set of ice motion and rotation vectors that can be used to guide a pixel value correlator. The algorithms match a shape descriptor known as the psi -s curve. The first algorithm uses normalized correlation to match the psi -s curves, while the second uses dynamic programming to compute an elastic match that better accommodates ice floe deformation. Some empirical data on the performance of the algorithms on Seasat SAR images are presented. >

Linear-time modular decomposition of directed graphs

Modular decomposition of graphs is a powerful tool with many applications in graph theory and optimization. There are efficient linear-time algorithms that compute the decomposition for undirected graphs. The best previously published time bound for directed graphs is O(n + m log n), where n is the number of vertices and m is the number of edges. We give an O(n + m)-time algorithm.