MOTIVATION: Large-scale evolutionary events such as genomic rearrange.ments and segmental duplications form an important part of the evolution of genomes and are widely studied from both biological and computational perspectives. A basic computational problem is to infer these events in the evolutionary history for given modern genomes, a task for which many algorithms have been proposed under various constraints. Algorithms that can handle both rearrangements and content-modifying events such as duplications and losses remain few and limited in their applicability. RESULTS: We study the comparison of two genomes under a model including general rearrangements (through double-cut-and-join) and segmental duplications. We formulate the comparison as an optimization problem and describe an exact algorithm to solve it by using an integer linear program. We also devise a sufficient condition and an efficient algorithm to identify optimal substructures, which can simplify the problem while preserving optimality. Using the optimal substructures with the integer linear program (ILP) formulation yields a practical and exact algorithm to solve the problem. We then apply our algorithm to assign in-paralogs and orthologs (a necessary step in handling duplications) and compare its performance with that of the state-of-the-art method MSOAR, using both simulations and real data. On simulated datasets, our method outperforms MSOAR by a significant margin, and on five well-annotated species, MSOAR achieves high accuracy, yet our method performs slightly better on each of the 10 pairwise comparisons. AVAILABILITY AND IMPLEMENTATION: http://lcbb.epfl.ch/softwares/coser.

MOTIVATION: Comparative genomics aims to understand the structure and function of genomes by translating knowledge gained about some genomes to the object of study. Early approaches used pairwise comparisons, but today researchers are attempting to leverage the larger potential of multi-way comparisons. Comparative genomics relies on the structuring of genomes into syntenic blocks: blocks of sequence that exhibit conserved features across the genomes. Syntenic blocs are required for complex computations to scale to the billions of nucleotides present in many genomes; they enable comparisons across broad ranges of genomes because they filter out much of the individual variability; they highlight candidate regions for in-depth studies; and they facilitate whole-genome comparisons through visualization tools. However, the concept of syntenic block remains loosely defined. Tools for the identification of syntenic blocks yield quite different results, thereby preventing a systematic assessment of the next steps in an analysis. Current tools do not include measurable quality objectives and thus cannot be benchmarked against themselves. Comparisons among tools have also been neglected-what few results are given use superficial measures unrelated to quality or consistency. RESULTS: We present a theoretical model as well as an experimental basis for comparing syntenic blocks and thus also for improving or designing tools for the identification of syntenic blocks. We illustrate the application of the model and the measures by applying them to syntenic blocks produced by three different contemporary tools (DRIMM-Synteny, i-ADHoRe and Cyntenator) on a dataset of eight yeast genomes. Our findings highlight the need for a well founded, systematic approach to the decomposition of genomes into syntenic blocks. Our experiments demonstrate widely divergent results among these tools, throwing into question the robustness of the basic approach in comparative genomics. We have taken the first step towards a formal approach to the construction of syntenic blocks by developing a simple quality criterion based on sound evolutionary principles.

MOTIVATION: We have witnessed an enormous increase in ChIP-Seq data for histone modifications in the past few years. Discovering significant patterns in these data is an important problem for understanding biological mechanisms. RESULTS: We propose probabilistic partitioning methods to discover significant patterns in ChIP-Seq data. Our methods take into account signal magnitude, shape, strand orientation and shifts. We compare our methods with some current methods and demonstrate significant improvements, especially with sparse data. Besides pattern discovery and classification, probabilistic partitioning can serve other purposes in ChIP-Seq data analysis. Specifically, we exemplify its merits in the context of peak finding and partitioning of nucleosome positioning patterns in human promoters. AVAILABILITY AND IMPLEMENTATION: The software and code are available in the supplementary material. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

BACKGROUND: In cell differentiation, a cell of a less specialized type becomes one of a more specialized type, even though all cells have the same genome. Transcription factors and epigenetic marks like histone modifications can play a significant role in the differentiation process. RESULTS: In this paper, we present a simple analysis of cell types and differentiation paths using phylogenetic inference based on ChIP-Seq histone modification data. We precisely defined the notion of cell-type trees and provided a procedure of building such trees. We propose new data representation techniques and distance measures for ChIP-Seq data and use these together with standard phylogenetic inference methods to build biologically meaningful cell-type trees that indicate how diverse types of cells are related. We demonstrate our approach on various kinds of histone modifications for various cell types, also using the datasets to explore various issues surrounding replicate data, variability between cells of the same type, and robustness. We use the results to get some interesting biological findings like important patterns of histone modification changes during cell differentiation process. CONCLUSIONS: We introduced and studied the novel problem of inferring cell type trees from histone modification data. The promising results we obtain point the way to a new approach to the study of cell differentiation. We also discuss how cell-type trees can be used to study the evolution of cell types.

We study the problem of sorting genomes under an evolutionary model that includes genomic rearrangements and segmental duplications. We propose an iterative algorithm to improve any initial evolutionary trajectory between two genomes in terms of parsimony. Our algorithm is based on a new graphical model, the trajectory graph, which models not only the final states of two genomes but also an existing evolutionary trajectory between them. We show that redundant rearrangements in the trajectory correspond to certain cycles in the trajectory graph, and prove that our algorithm converges to an optimal trajectory for any initial trajectory involving only rearrangements.

Alternative splicing is now recognized as a major mechanism for transcriptome and proteome diversity in higher eukaryotes, yet its evolution is poorly understood. Most studies focus on the evolution of exons and introns at the gene level, while only few consider the evolution of transcripts. In this paper, we present a framework for transcript phylogenies where ancestral transcripts evolve along the gene tree by gains, losses, and mutation. We demonstrate the usefulness of our method on a set of 805 genes and two different topics. First, we improve a method for transcriptome reconstruction from ESTs (ASPic), then we study the evolution of function in transcripts. The use of transcript phylogenies allows us to double the precision of ASPic, whereas results on the functional study reveal that conserved transcripts are more likely to share protein domains than functional sites. These studies validate our framework for the study of evolution in large collections of organisms from the perspective of transcripts; for this purpose, we developed and provide a new tool, TrEvoR.

TIBA is a tool to reconstruct phylogenetic trees from rearrangement data that consist of ordered lists of synteny blocks (or genes), where each synteny block is shared with all of its homologues in the input genomes. The evolution of these synteny blocks, through rearrangement operations, is modelled by the uniform Double-Cut-and-Join model. Using a true distance estimate under this model and simple distance-based methods, TIBA reconstructs a phylogeny of the input genomes. Unlike any previous tool for inferring phylogenies from rearrangement data, TIBA uses novel methods of robustness estimation to provide support values for the edges in the inferred tree.

Alternative splicing, an unknown mechanism 20 years ago, is now recognized as a major mechanism for proteome and transcriptome diversity, particularly in mammals–some researchers conjecture that up to 90% of human genes are alternatively spliced. Despite much research on exon and intron evolution, little is known about the evolution of transcripts. In this paper, we present a model of transcript evolution and an associated algorithm to reconstruct transcript phylogenies. The evolution of the gene structure–exons and introns–is used as basis for the reconstruction of transcript phylogenies. We apply our model and reconstruction algorithm on two well-studied genes, MAG and PAX6, obtaining results consistent with current knowledge and thereby providing evidence that a phylogenetic analysis of transcripts is feasible and likely to be informative.

Comparing two or more phylogenetic trees is a fundamental task in computational biology. The simplest outcome of such a comparison is a pairwise measure of similarity, dissimilarity, or distance. A large number of such measures have been proposed, but so far all suffer from problems varying from computational cost to lack of robustness; many can be shown to behave unexpectedly under certain plausible inputs. For instance, the widely used Robinson-Foulds distance is poorly distributed and thus affords little discrimination, while also lacking robustness in the face of very small changes--reattaching a single leaf elsewhere in a tree of any size can instantly maximize the distance. In this paper, we introduce a new pairwise distance measure, based on matching, for phylogenetic trees. We prove that our measure induces a metric on the space of trees, show how to compute it in low polynomial time, verify through statistical testing that it is robust, and finally note that it does not exhibit unexpected behavior under the same inputs that cause problems with other measures. We also illustrate its usefulness in clustering trees, demonstrating significant improvements in the quality of hierarchical clustering as compared to the same collections of trees clustered using the Robinson-Foulds distance.

The experimental determination of transcriptional regulatory networks in the laboratory remains difficult and timeconsuming, while computational methods to infer these networks provide only modest accuracy. The latter can be attributed partly to the limitations of a single-organism approach. Computational biology has long used comparative and evolutionary approaches to extend the reach and accuracy of its analyses. In this paper, we describe ProPhyC, a probabilistic phylogenetic model and associated inference algorithms, designed to improve the inference of regulatory networks for a family of organisms by using known evolutionary relationships among these organisms. ProPhyC can be used with various network evolutionary models and any existing inference method. Extensive experimental results on both biological and synthetic data confirm that our model (through its associated refinement algorithms) yields substantial improvement in the quality of inferred networks over all current methods. We also compare ProPhyC with a transfer learning approach we design. This approach also uses phylogenetic relationships while inferring regulatory networks for a family of organisms. Using similar input information but designed in a very different framework, this transfer learning approach does not perform better than ProPhyC, which indicates that ProPhyC makes good use of the evolutionary information.

Alternative splicing, an unknown mechanism 20 years ago, is now recognized as a major mechanism for proteome and transcriptome diversity, particularly in mammals-some researchers conjecture that up to 90% of human genes are alternatively spliced. Despite much research on exon and intron evolution, little is known about the evolution of transcripts. In this paper, we present a model of transcript evolution and an associated algorithm to reconstruct transcript phylogenies. The evolution of the gene structure-exons and introns-is used as basis for the reconstruction of transcript phylogenies. We apply our model and reconstruction algorithm on two well-studied genes, MAG and PAX6, obtaining results consistent with current knowledge and thereby providing evidence that a phylogenetic analysis of transcripts is feasible and likely to be informative.

Many of the steps in phylogenetic reconstruction can be confounded by “rogue” taxa—taxa that cannot be placed with assurance anywhere within the tree, indeed, whose location within the tree varies with almost any choice of algorithm or parameters. Phylogenetic consensus methods, in particular, are known to suffer from this problem. In this paper, we provide a novel framework to define and identify rogue taxa. In this framework, we formulate a bicriterion optimization problem, the relative information criterion, that models the net increase in useful information present in the consensus tree when certain taxa are removed from the input data. We also provide an effective greedy heuristic to identify a subset of rogue taxa and use this heuristic in a series of experiments, with both pathological examples from the literature and a collection of large biological data sets. As the presence of rogue taxa in a set of bootstrap replicates can lead to deceivingly poor support values, we propose a procedure to recompute support values in light of the rogue taxa identified by our algorithm; applying this procedure to our biological data sets caused a large number of edges to move from “unsupported” to “supported” status, indicating that many existing phylogenies should be recomputed and reevaluated to reduce any inaccuracies introduced by rogue taxa. We also discuss the implementation issues encountered while integrating our algorithm into RAxML v7.2.7, particularly those dealing with scaling up the analyses. This integration enables practitioners to benefit from our algorithm in the analysis of very large data sets (up to 2,500 taxa and 10,000 trees, although we present the results of even larger analyses).

Reassortments are events in the evolution of the genome of influenza (flu). As reassortments have been implicated in major human pandemics of the last century, their identification has become a health priority. While such identification can be done on a small dataset by a biologist (using phylogenetic trees), databases of flu sequences are growing exponentially, so that it is imperative to develop automated identification methods. However, current methods are limited to pairwise segment comparisons. We present FluRF, a fully automated flu virus reassortment finder. FluRF is inspired by the visual approach to reassortment identification and thus uses the reconstructed phylogenetic trees of the individual segments and the full genome. We also present a simple flu evolution simulator, used to produce synthetic datasets to tune the FluRF parameters. On synthetic datasets produced by our flu evolution simulator, FluRF, tuned for a 0% false positive rate, yielded false negative rates of 10% or less. FluRF corroborated two new reassortments identified by visual analysis of 75 Human H3N2 New York flu strains from 2005-2008 and gave partial verification of reassortments found using another bioinformatics method. FluRF finds reassortments by a bottom-up search of the full-genome and segment-based phylogenetic trees for candidate clades-groups of one or more sampled viruses that are separated from the other variants from the same season. Candidate clades in each tree are tested to guarantee confidence values, using the lengths of key edges as well as other tree parameters; clades with reassortments must have validated incongruencies among segment trees. FluRF demonstrates robustness of prediction for geographically and temporally expanded datasets, and is not limited to finding reassortments with previously collected sequences. The complete source code is available from lcbb.epfl.ch/software.html.

BACKGROUND: The rapidly increasing availability of whole-genome sequences has enabled the study of whole-genome evolution. Evolutionary mechanisms based on genome rearrangements have attracted much attention and given rise to many models; somewhat independently, the mechanisms of gene duplication and loss have seen much work. However, the two are not independent and thus require a unified treatment, which remains missing to date. Moreover, existing rearrangement models do not fit the dichotomy between most prokaryotic genomes (one circular chromosome) and most eukaryotic genomes (multiple linear chromosomes). RESULTS: To handle rearrangements, gene duplications and losses, we propose a new evolutionary model and the corresponding method for estimating true evolutionary distance. Our model, inspired from the DCJ model, is simple and the first to respect the prokaryotic/eukaryotic structural dichotomy. Experimental results on a wide variety of genome structures demonstrate the very high accuracy and robustness of our distance estimator. CONCLUSION: We give the first robust, statistically based, estimate of genomic pairwise distances based on rearrangements, duplications and losses, under a model that respects the structural dichotomy between prokaryotic and eukaryotic genomes. Accurate and robust estimates in true evolutionary distances should translate into much better phylogenetic reconstructions as well as more accurate genomic alignments, while our new model of genome rearrangements provides another refinement in simplicity and verisimilitude.

BACKGROUND: The study of genome rearrangements has become a mainstay of phylogenetics and comparative genomics. Fundamental in such a study is the median problem: given three genomes find a fourth that minimizes the sum of the evolutionary distances between itself and the given three. Many exact algorithms and heuristics have been developed for the inversion median problem, of which the best known is MGR. RESULTS: We present a unifying framework for median heuristics, which enables us to clarify existing strategies and to place them in a partial ordering. Analysis of this framework leads to a new insight: the best strategies continue to refer to the input data rather than reducing the problem to smaller instances. Using this insight, we develop a new heuristic for inversion medians that uses input data to the end of its computation and leverages our previous work with DCJ medians. Finally, we present the results of extensive experimentation showing that our new heuristic outperforms all others in accuracy and, especially, in running time: the heuristic typically returns solutions within 1% of optimal and runs in seconds to minutes even on genomes with 25'000 genes--in contrast, MGR can take days on instances of 200 genes and cannot be used beyond 1'000 genes. CONCLUSION: Finding good rearrangement medians, in particular inversion medians, had long been regarded as the computational bottleneck in whole-genome studies. Our new heuristic for inversion medians, ASM, which dominates all others in our framework, puts that issue to rest by providing near-optimal solutions within seconds to minutes on even the largest genomes.

BACKGROUND: Reconstructing complete ancestral genomes (at least in terms of their gene inventory and arrangement) is attracting much interest due to the rapidly increasing availability of whole genome sequences. While modest successes have been reported for mammalian and even vertebrate genomes, more divergent groups continue to pose a stiff challenge, mostly because current models of genomic evolution support too many choices. RESULTS: We describe a novel type of genomic signature based on rearrangements that characterizes evolutionary changes that must be common to all minimal rearrangement scenarios; by focusing on global patterns of rearrangements, such signatures bypass individual variations and sharply restrict the search space. We present the results of extensive simulation studies demonstrating that these signatures can be used to reconstruct accurate ancestral genomes and phylogenies even for widely divergent collections. CONCLUSION: Focusing on genome triples rather than genomes pairs unleashes the full power of evolutionary analysis. Our genomic signature captures shared evolutionary events and thus can form the basis of a robust analysis and reconstruction of evolutionary history.

BACKGROUND: Given three signed permutations, an inversion median is a fourth permutation that minimizes the sum of the pairwise inversion distances between it and the three others. This problem is NP-hard as well as hard to approximate. Yet median-based approaches to phylogenetic reconstruction have been shown to be among the most accurate, especially in the presence of long branches. Most existing approaches have used heuristics that attempt to find a longest sequence of inversions from one of the three permutations that, at each step in the sequence, moves closer to the other two permutations; yet very little is known about the quality of solutions returned by such approaches. RESULTS: Recently, Arndt and Tang took a step towards finding longer such sequences by using sets of commuting inversions. In this paper, we formalize the problem of finding such sequences of inversions with what we call signatures and provide algorithms to find maximum cardinality sets of commuting and noninterfering inversions. CONCLUSION: Our results offer a framework in which to study the inversion median problem, faster algorithms to obtain good medians, and an approach to study characteristic events along an evolutionary path.

MOTIVATION: Modern techniques can yield the ordering and strandedness of genes on each chromosome of a genome; such data already exists for hundreds of organisms. The evolutionary mechanisms through which the set of the genes of an organism is altered and reordered are of great interest to systematists, evolutionary biologists, comparative genomicists and biomedical researchers. Perhaps the most basic concept in this area is that of evolutionary distance between two genomes: under a given model of genomic evolution, how many events most likely took place to account for the difference between the two genomes? RESULTS: We present a method to estimate the true evolutionary distance between two genomes under the 'double-cut-and-join' (DCJ) model of genome rearrangement, a model under which a single multichromosomal operation accounts for all genomic rearrangement events: inversion, transposition, translocation, block interchange and chromosomal fusion and fission. Our method relies on a simple structural characterization of a genome pair and is both analytically and computationally tractable. We provide analytical results to describe the asymptotic behavior of genomes under the DCJ model, as well as experimental results on a wide variety of genome structures to exemplify the very high accuracy (and low variance) of our estimator. Our results provide a tool for accurate phylogenetic reconstruction from multichromosomal gene rearrangement data as well as a theoretical basis for refinements of the DCJ model to account for biological constraints. AVAILABILITY: All of our software is available in source form under GPL at http://lcbb.epfl.ch.

Rearrangements of genes and other syntenic blocks have become a topic of intensive study by phylogenists, comparative genomicists, and computational biologists: they are a feature of many cancers, must be taken into account to align highly divergent sequences, and constitute a phylogenetic marker of great interest. The mathematics of rearrangements is far more complex than for indels and mutations in sequences. Inversions have been well characterized through 20 years of work, but transpositions still await comparable results. We can compute inversion and DCJ (a combination of inversions and block exchanges) distances, and bounds on the transposition distance. The first has been extensively used in comparative genomics and phylogenetics, the second is quite new, and the third has not seen significant use to date. We present here a detailed experimental study of these three distance measures within the context of genome comparison (pairwise distances) and phylogenetic reconstruction. We used data generated through simulated evolution along various trees, using various evolutionary rates and various mixes of inversions and transpositions. Our main finding is that inversion and DCJ measures return very similar results even on data generated using only transpositions, while the measure based on Hartman's bound is often too loose to provide comparable accuracy in genomic comparisons or phylogenetic reconstruction

SUMMARY: NetGen is an event-driven simulator that creates phylogenetic networks by extending the birth-death model to include diploid hybridizations. DNA sequences are evolved in conjunction with the topology, enabling hybridization decisions to be based on contemporary evolutionary distances. NetGen supports variable rate lineages, root sequence specification, outgroup generation and many other options. This note describes the NetGen application and proposes an extension of the Newick format to accommodate phylogenetic networks. AVAILABILITY: NetGen is written in C and is available in source form at http://www.phylo.unm.edu/~morin/.

Phylogenetic reconstruction from gene-order data has attracted attention from both biologists and computer scientists over the last few years. So far, our software suite GRAPPA is the most accurate approach, but it requires that all genomes have identical gene content, with each gene appearing exactly once in each genome. Some progress has been made in handling genomes with unequal gene content, both in terms of computing pair-wise genomic distances and in terms of reconstruction. In this paper, we present a new approach for computing the median of three arbitrary genomes and apply it to the reconstruction of phylogenies from arbitrary gene-order data. We implemented these methods within GRAPPA and tested them on simulated datasets under various conditions as well as on a real dataset of chloroplast genomes; we report the results of our simulations and our analysis of the real dataset and compare them to reconstructions made by using neighbor-joining and using the original GRAPPA on the same genomes with equalized gene contents. Our new approach is remarkably accurate both in simulations and on the real dataset, in contrast to the distance-based approaches and to reconstructions using the original GRAPPA applied to equalized gene contents.

Phylogenetic networks model the evolutionary history of sets of organisms when events such as hybrid speciation and horizontal gene transfer occur. In spite of their widely acknowledged importance in evolutionary biology, phylogenetic networks have so far been studied mostly for specific data sets. We present a general definition of phylogenetic networks in terms of directed acyclic graphs (DAGs) and a set of conditions. Further, we distinguish between model networks and reconstructible ones and characterize the effect of extinction and taxon sampling on the reconstructibility of the network. Simulation studies are a standard technique for assessing the performance of phylogenetic methods. A main step in such studies entails quantifying the topological error between the model and inferred phylogenies. While many measures of tree topological accuracy have been proposed, none exist for phylogenetic networks. Previously, we proposed the first such measure, which applied only to a restricted class of networks. In this paper, we extend that measure to apply to all networks, and prove that it is a metric on the space of phylogenetic networks. Our results allow for the systematic study of existing network methods, and for the design of new accurate ones.

We analyze the performance of likelihood-based approaches used to reconstruct phylogenetic trees. Unlike other techniques such as Neighbor-joining (NJ) and Maximum Parsimony (MP), relatively little is known regarding the behavior of algorithms founded on the principle of likelihood. We study the accuracy, speed, and likelihood scores of four representative likelihood-based methods (fastDNAml, Mr Bayes, PAUP*-ML, and TREE-PUZZLE) that use either Maximum Likelihood (ML) or Bayesian inference to find the optimal tree. NJ is also studied to provide a baseline comparison. Our simulation study is based on random birth-death trees, which are deviated from ultrametricity, and uses the Kimura 2-parameter +Gamma model of sequence evolution. We find that Mr Bayes (a Bayesian inference approach) consistently outperforms the other methods in terms of accuracy and running time.

Motivated by applications in cartography and computer graphics, we study a version of the map-labeling problem that we call the k-Position Map-Labeling Problem: given a set of points in the plane and, for each point, a set of up to k allowable positions, place uniform and nonintersecting labels of maximum size at each point in one of the allowable positions.This version combines an aesthetic criterion and a legibility criterion and comes close to actual practice while generalizing the fixed-point and slider models found in the literature.We then extend our approach to arbitrary positions, obtaining an algorithm that is easy to implement and also dramatically improves the best approximation bounds.We present a general heuristic which runs in time O(n log n+ nlogR*) , where R* is the size of the optimal label, and which guarantees a fixed-ratio approximation for any regular labels.For circular labels, our technique yields a 3.6approximation, a dramatic improvement in the case of arbitrary placement over the previous bound of 19.35 given by Strijk and Wolff [11].Our technique combines several geometric and combinatorial properties, which might be of independent interest.

The authors present exact characterizations of structures on which the greedy algorithm produces optimal solutions. Our characterization, which are called matroid embeddings, complete the partial characterizations of Rado [A note on independent functions, Proc. London Math. Soc., 7 (1957), pp. 300–320], Gale [Optimal assignments in an ordered set, J. Combin. Theory, 4 (1968), pp. 176–180], and Edmonds [Matroids and the greedy algorithm, Math. Programming, 1 (1971), pp. 127–136], (matroids), and of Korte and Lovasz [Greedoids and linear object functions, SIAM J. Alg. Discrete Meth., 5 (1984), pp. 239–248] and [Mathematical structures underlying greedy algorithms, in Fundamentals of Computational Theory, LNCS 177, Springer-Verlag, 1981, pp. 205–209] (greedoids). It is shown that the greedy algorithm optimizes all linear objective functions if and only if the problem structure (phrased in terms of either accessible set systems or hereditary languages) is a matroid embedding. An exact characterization of the objective functions optimized by the greedy algorithm on matroid embeddings is also presented. Finally, the authors present an exact characterization of the structures on which the greedy algorithm optimizes all bottleneck functions, structures that are less constrained than matroid embeddings.

The authors propose a method based on kernel density estimates to partition sequentially the feature space along the best feature axis (either one of the original axes or one obtained by a carefully developed one-dimensional linear feature transformation). This method alleviates the storage and classification speed problems of traditional kernel-based classifiers without losing their flexibility and their relative insensitivity to dimensionality. The authors present a simple procedure and a distribution-free criterion for finding a good smoothing parameter for the kernel density estimate and develop a one-dimensional feature linear transformation based on correlation between density functions, which can be applied regardless of the geometrical structure of the data. The authors' proposals are validated by theoretical results and by simulations. An application to the severely under-sampled problem of texture classification (only 32 design samples per class in 22-dimensional space) is presented.>

The worst-case number of variable evaluations (testing cost) of Boolean functions is examined. Following up on a result by Rivest and Vuillemin, we show that all symmetric as well as all linearly separable Boolean functions are exhaustive, that is, have a pessimal worst-case testing cost.

The construction of sequential testing procedures from functions of discrete arguments is a common problem in switching theory, software engineering, pattern recognition, and management. The concept of the activity of an argument is introduced, and a theorem is proved which relates it to the expected testing cost of the most general type of decision trees. This result is then extended to trees constructed from relations on finite sets and to decision procedures with cycles. These results are used, in turn, as the basis for a fast heuristic selection rule for constructing testing procedures. Finally, some bounds on the performance of the selection rule are developed.