Hannenhalli and Pevzner gave the first polynomial-time algorithm for computing the inversion distance between two signed permutations, as part of the larger task of determining the shortest sequence of inversions needed to transform one permutation into the other. Their algorithm (restricted to distance calculation) proceeds in two stages: in the first stage, the overlap graph induced by the permutation is decomposed into connected components; then, in the second stage, certain graph structures (hurdles and others) are identified. Berman and Hannenhalli avoided the explicit computation of the overlap graph and gave an O(n alpha(n)) algorithm, based on a Union-Find structure, to find its connected components, where a is the inverse Ackerman function. Since for all practical purposes alpha(n) is a constant no larger than four, this algorithm has been the fastest practical algorithm to date. In this paper, we present a new linear-time algorithm for computing the connected components, which is more efficient than that of Berman and Hannenhalli in both theory and practice. Our algorithm uses only a stack and is very easy to implement. We give the results of computational experiments over a large range of permutation pairs produced through simulated evolution; our experiments show a speed-up by a factor of 2 to 5 in the computation of the connected components and by a factor of 1.3 to 2 in the overall distance computation.

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Motivation: Phylogenetic reconstruction from gene-order data has attracted increasing attention from both biologists and computer scientists over the last few years. Methods used in reconstruction include distance-based methods (such as neighbor-joining), parsimony methods using sequence-based encodings, Bayesian approaches, and direct optimization. The latter, pioneered by Sankoff and extended by us with the software suite GRAPPA, is the most accurate approach, but cannot handle more than about 15 genomes of limited size (e.g., organelles). Results: We report here on our successful efforts to scale up direct optimization through a two-step approach: the first step decomposes the dataset into smaller pieces and runs the direct optimization (GRAPPA) on the smaller pieces, while the second step builds a tree from the results obtained on the smaller pieces. We used the sophisticated disk-covering method (DCM) pioneered by Warnow and her group, suitably modified to take into account the computational limitations of GRAPPA. We find that DCM-GRAPPA scales gracefully to at least 1,000 genomes of a few hundred genes each and retains surprisingly high accuracy throughout the range: in our experiments, the topological error rate rarely exceeded a few percent. Thus, reconstruction based on gene-order data can now be accomplished with high accuracy on datasets of significant size. Availability: All of our software is available in source form under GPL at www.compbio.unm.edu Contact:

Phylogenies (that is, tree-of-life relationships) derived from gene order data may prove crucial in answering some fundamental open questions in biomolecular evolution. Real-world interest is strong in determining these relationships. For example, pharmaceutical companies may use phylogeny reconstruction in drug discovery for finding plants with similar gene production. Health organizations study the evolution and spread of viruses such as HIV to gain understanding of future outbreaks. And governments are interested in aiding the production of foodstuffs like rice, wheat, and corn, by understanding the genetic code. Yet very few techniques are available for such phylogenetic reconstructions. Appropriate tools for analyzing such data may help resolve some difficult phylogenetic reconstruction problems; indeed, this new source of data has been embraced by many biologists in their phylogenetic work. With the rapid accumulation of whole genome sequences for a wide diversity of taxa, phylogenetic reconstruction based on changes in gene order and gene content is showing promise, particularly for resolving deep (i.e., old) branches. However, reconstruction from gene-order data is even more computationally intensive than reconstruction from sequence data, particularly in groups with large numbers of genes and highly rearranged genomes. We have developed a software suite, GRAPPA, that extends the breakpoint analysis (BPAnalysis) method of Sankoff and Blanchette while running much faster: in a recent analysis of a collection of chloroplast data for species of Campanulaceae on a 512-processor Linux supercluster with Myrinet, we achieved a one-million-fold speedup over BPAnalysis. GRAPPA currently can use either breakpoint or inversion distance (computed exactly) for its computati...

As more and more genomes are sequenced, evolutionary biologists are becoming increasingly interested in evolution at the level of whole genomes, in scenarios in which the genome evolves through insertions, deletions, and movements of genes along its chromosomes. In the mathematical model pioneered by Sankoff and others, a unichromosomal genome is represented by a signed permutation of a multi-set of genes; Hannenhalli and Pevzner showed that the edit distance between two signed permutations of the same set can be computed in polynomial time when all operations are inversions. El-Mabrouk extended that result to allow deletions and a limited form of insertions (which forbids duplications). In this paper we extend El-Mabrouk's work to handle duplications as well as insertions and present an alternate framework for computing (near) minimal edit sequences involving insertions, deletions, and inversions. We derive an error bound for our polynomial-time distance computation under various assumptions and present preliminary experimental results that suggest that performance in practice may be excellent, within a few percent of the actual distance.

As more and more genomes are sequenced, evolutionary biologists are becoming increasingly interested in evolution at the level of whole genomes, in scenarios in which the genome evolves through insertions, duplications, deletions, and movements of genes along its chromosomes. In the mathematical model pioneered by Sankoff and others, a unichromosomal genome is represented by a signed permutation of a multiset of genes; Hannenhalli and Pevzner showed that the edit distance between two signed permutations of the same set can be computed in polynomial time when all operations are inversions. El-Mabrouk extended that result to allow deletions and a limited form of insertions (which forbids duplications); in turn we extended it to compute a nearly optimal edit sequence between an arbitrary genome and the identity permutation. In this paper we generalize our approach to compute distances between two arbitrary genomes, but focus on approximating the true evolutionary distance rather than the edit distance. We present experimental results showing that our algorithm produces excellent estimates of the true evolutionary distance up to a (high) threshold of saturation; indeed, the distances thus produced are good enough to enable the simple

A phylogeny is the evolutionary history of a group of organisms; systematists (and other biologists) attempt to reconstruct this history from various forms of data about contemporary organisms. Phylogeny reconstruction is a crucial step in the understanding of evolution as well as an important tool in biological, pharmaceutical, and medical research. Phylogeny reconstruction from molecular data is very difficult: almost all optimization models give rise to NP-hard (and thus computationally intractable) problems. Yet approximations must be of very high quality in order to avoid outright biological nonsense. Thus many biologists have been willing to run farms of processors for many months in order to analyze just one dataset. High-performance algorithm engineering offers a battery of tools that can reduce, sometimes spectacularly, the running time of existing phylogenetic algorithms, as well as help designers produce better algorithms. We present an overview of algorithm engineering techniques, illustrating them with an application to the "breakpoint analysis" method of Sankoff et al., which resulted in the GRAPPA software suite. GRAPPA demonstrated a speedup in running time by over eight orders of magnitude over the original implementation on a variety of real and simulated datasets. We show how these algorithmic engineering techniques are directly applicable to a large variety of challenging combinatorial problems in computational biology.

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