Avraham A. Melkman

On-line construction of the convex hull of a simple polyline

After McCallum and Avis [4] showed that the convex hull of a simple polygon P with n vertices can be constructed in O(n) time, several authors [1,2,3] devised simplified algorithms for this prob- lem. Graham and Yao [2] presented a particularly simple and elegant one. After finding two points of the convex hull, their algorithm generated all other hull vertices using only one stack for inter- mediate storage. It is the purpose of this short article to show that a slightly modified version of their algorithm constructs, on-line, the convex hull of any simple polyline in O(n) time. In contrast, the on-line construction of a nonsimple polyline requires O(n log n) time, as shown by Preparata [5]. In the special case of a simple polygon our algorithm produces the convex hull without first identifying two of the hull vertices, as was re- quired in [2]. The price that we pay is the use of a deque instead of a queue. After this article was submitted, we learned of a similar approach taken by Tor and Middleditch [6], who embed it in an algorithm for the convex decomposition of .a sim- ple polygon. To keep this article short, we use where possi- ble the definitions and notations of Graham and Yao. The polyline P = (vl,... , Vm) is assumed to

On Polygonal Chain Approximation

Imai and Iri recently described a clever algorithm for approximating a polygonal chain within a given tolerance. Their algorithm requires O ( n 3 ) time in the worst case. In this note it is shown that their algorithm can be improved to O ( n 2 log n ) by exploiting the geometrical constraints of the problem.

Seeing the forest for the trees

Motivation: There is a growing interest in improving the cluster analysis of expression data by incorporating into it prior knowledge, such as the Gene Ontology (GO) annotations of genes, in order to improve the biological relevance of the clusters that are subjected to subsequent scrutiny. The structure of the GO is another source of background knowledge that can be exploited through the use of semantic similarity. Results: We propose here a novel algorithm that integrates semantic similarities (derived from the ontology structure) into the procedure of deriving clusters from the dendrogram constructed during expression-based hierarchical clustering. Our approach can handle the multiple annotations, from different levels of the GO hierarchy, which most genes have. Moreover, it treats annotated and unannotated genes in a uniform manner. Consequently, the clusters obtained by our algorithm are characterized by significantly enriched annotations. In both cross-validation tests and when using an external index such as protein–protein interactions, our algorithm performs better than previous approaches. When applied to human cancer expression data, our algorithm identifies, among others, clusters of genes related to immune response and glucose metabolism. These clusters are also supported by protein–protein interaction data. Contact: dotna@cs.bgu.ac.il Supplementary information: Supplementary data are available at Bioinformatics online.

Biological Process Linkage Networks

Background The traditional approach to studying complex biological networks is based on the identification of interactions between internal components of signaling or metabolic pathways. By comparison, little is known about interactions between higher order biological systems, such as biological pathways and processes. We propose a methodology for gleaning patterns of interactions between biological processes by analyzing protein-protein interactions, transcriptional co-expression and genetic interactions. At the heart of the methodology are the concept of Linked Processes and the resultant network of biological processes, the Process Linkage Network (PLN). Results We construct, catalogue, and analyze different types of PLNs derived from different data sources and different species. When applied to the Gene Ontology, many of the resulting links connect processes that are distant from each other in the hierarchy, even though the connection makes eminent sense biologically. Some others, however, carry an element of surprise and may reflect mechanisms that are unique to the organism under investigation. In this aspect our method complements the link structure between processes inherent in the Gene Ontology, which by its very nature is species-independent. As a practical application of the linkage of processes we demonstrate that it can be effectively used in protein function prediction, having the power to increase both the coverage and the accuracy of predictions, when carefully integrated into prediction methods. Conclusions Our approach constitutes a promising new direction towards understanding the higher levels of organization of the cell as a system which should help current efforts to re-engineer ontologies and improve our ability to predict which proteins are involved in specific biological processes.

Hierarchical tree snipping

MOTIVATION Hierarchical clustering is widely used to cluster genes into groups based on their expression similarity. This method first constructs a tree. Next this tree is partitioned into subtrees by cutting all edges at some level, thereby inducing a clustering. Unfortunately, the resulting clusters often do not exhibit significant functional coherence. RESULTS To improve the biological significance of the clustering, we develop a new framework of partitioning by snipping--cutting selected edges at variable levels. The snipped edges are selected to induce clusters that are maximally consistent with partially available background knowledge such as functional classifications. Algorithms for two key applications are presented: functional prediction of genes, and discovery of functionally enriched clusters of co-expressed genes. Simulation results and cross-validation tests indicate that the algorithms perform well even when the actual number of clusters differs considerably from the requested number. Performance is improved compared with a previously proposed algorithm. AVAILABILITY A java package is available at http://www.cs.bgu.ac.il/~dotna/ TreeSnipping

Finding a Periodic Attractor of a Boolean Network

In this paper, we study the problem of finding a periodic attractor of a Boolean network (BN), which arises in computational systems biology and is known to be NP-hard. Since a general case is quite hard to solve, we consider special but biologically important subclasses of BNs. For finding an attractor of period 2 of a BN consisting of n OR functions of positive literals, we present a polynomial time algorithm. For finding an attractor of period 2 of a BN consisting of n AND/OR functions of literals, we present an O(1.985n) time algorithm. For finding an attractor of a fixed period of a BN consisting of n nested canalyzing functions and having constant treewidth w, we present an O(n2p(w+1)poly(n)) time algorithm.

Determining a Singleton Attractor of a Boolean Network with Nested Canalyzing Functions

In this article, we study the problem of finding a singleton attractor for several biologically important subclasses of Boolean networks (BNs). The problem of finding a singleton attractor in a BN is known to be NP-hard in general. For BNs consisting of n nested canalyzing functions, we present an O(1.799(n)) time algorithm. The core part of this development is an O(min(2(k/2) · 2(m/2), 2(k)) · poly(k, m)) time algorithm for the satisfiability problem for m nested canalyzing functions over k variables. For BNs consisting of chain functions, a subclass of nested canalyzing functions, we present an O(1.619(n)) time algorithm and show that the problem remains NP-hard, even though the satisfiability problem for m chain functions over k variables is solvable in polynomial time. Finally, we present an o(2(n)) time algorithm for bounded degree BNs consisting of canalyzing functions.

Determining a singleton attractor of an AND/OR Boolean network in O (1.587n) time

The Boolean network (BN) is a discrete model of gene regulatory networks [5]. Each node in this network corresponds to a gene, and takes on a value of 1 or 0, meaning that the gene is or is not expressed. The value of a node at a given time instant is determined according to a regulation rule that is a Boolean function of the values of the predecessors of the node at the previous time, or their negations. The values of nodes change synchronously. We focus here on AND/OR Boolean networks, in which the regulation rule assigned to each node is restricted to be either a conjunction or a disjunction of literals. An important characteristic of any BN is the existence of an attractor, whether it is a singleton attractor, i.e. a stable state, or a cyclic attractor, i.e. a state that repeats periodically. Here the state of a network at a given time instant is the set of its node values. Unfortunately, the problem of detection of a singleton attractor (or an attractor of the shortest

Sleeved coclustering

A coCluster of a m x n matrix X is a submatrix determined by a subset of the rows and a subset of the columns. The problem of finding coClusters with specific properties is of interest, in particular, in the analysis of microarray experiments. In that case the entries of the matrix X are the expression levels of

Singleton and 2-periodic attractors of sign-definite Boolean networks

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