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.

We give a quadratic-time algorithm for finding the minimum number of reversals needed to sort a signed permutation. Our algorithm is faster than the previous algorithm of Hannenhalli and Pevzner and its faster implementation of Berman and Hannenhalli. The algorithm is conceptually simple and does not require special data structures. Our study also considerably simplifies the combinatorial structures used by the analysis.

We prove a number of improved inaproximability results, including the best up to date explicit approximation thresholds for MIS problem of bounded degree, bounded occurrences MAX-2SAT, and bounded degree Node Cover. We prove also for the first time inapproximability of the problem of Sorting by Reversals and display an explicit approximation threshold. This last problem was proved only recently to be NP-hard, in contrast to its signed version which is computable in polynomial time.

The breakpoint distance between two n-permutations is the number of pairs that appear consecutively in one but not in the other. In the median problem for breakpoints one is given a set of permutations and has to construct a permutation that minimizes the sum of breakpoint distances to all the original ones. Recently, the problem was suggested as a model for determining the evolutionary history of several species based on their gene orders. We show that the problem is already NP-hard for three permutations, and that this result holds both for signed and for unsigned permutations.

In 1995, Hannenhalli and Pevzner gave a first polynomial solution to the problem of finding the minimum number of reversals needed to sort a signed permutation. Their solution, as well as subsequent ones, relies on many intermediary constructions, such as simulations with permutations on 2n elements, and manipulation of various graphs.

The first heuristic for reconstructing phylogenetic trees from gene order data was introduced by Blanchette et al.. It sought to reconstruct the breakpoint phylogeny and was applied to a variety of datasets. We present a new heuristic for estimating the breakpoint phylogeny which, although not polynomial-time, is much faster in practice than BPAnalysis. We use this heuristic to conduct a phylogenetic analysis of chloroplast genomes in the flowering plant family Campanulaceae. We also present and discuss the results of experimentation on this real dataset with three methods: our new method, BPAnalysis, and the neighbor-joining method, using breakpoint distances, inversion distances, and inversion plus transposition distances. 1

Analysis of genomes evolving by inversions leads to a general combinatorial problem of Sorting by Reversals, MIN-SBR, the problem of sorting a permutation by a minimum number of reversals. This combinatorial problem has a long history, and a number of other motivations. It was studied in a great detail recently in computational molecular biology. Following a series of preliminary results, Hannenhalli and Pevzner developed the first exact polynomial time algorithm for the problem of sorting signed permutations by reversals, and a polynomial time algorithm for a special case of unsigned permutations. The best known approximation algorithm for MIN-SBR, due to Christie, gives a performance ratio of 1.5. In this paper, by exploiting the polynomial time algorithm for sorting signed permutations and by developing a new approximation algorithm for maximum cycle decomposition of breakpoint graphs, we improve the performance ratio for MIN-SBR to 1.375.

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

Improved inaproximability results are given, including the best up to date explicit approximation thresholds for bounded occurence satisfiability problems, like MAX-2SAT and E2-LIN-2, and problems in bounded degree graphs, like MIS, Node Cover and MAX CUT. We prove also for the first time inapproximability of the problem of Sorting by Reversals and display an explicit approximation threshold for this problem.