Finding Nested Common Intervals Efficiently

International audienceIn this paper, we study the problem of effi ciently fi nding gene clusters formalized by nested common intervals between two genomes represented either as permutations or as sequences. Considering permutations, we give several algorithms whose running time depends on the size of the actual output rather than the output in the worst case. Indeed, we first provide a straightforward O(n^3) time algorithm for finding all nested common intervals. We reduce this complexity by providing an O(n^2) time algorithm computing an irredundant output. Finally, we show, by providing a third algorithm, that fi nding only the maximal nested common intervals can be done in linear time. Considering sequences, we provide solutions (modi cations of previously de ned algorithms and a new algorithm) for di fferent variants of the problem, depending on the treatment one wants to apply to duplicated genes

Easy identification of generalized common and conserved nested intervals

In this paper we explain how to easily compute gene clusters, formalized by classical or generalized nested common or conserved intervals, between a set of K genomes represented as K permutations. A b-nested common (resp. conserved) interval I of size |I| is either an interval of size 1 or a common (resp. conserved) interval that contains another b-nested common (resp. conserved) interval of size at least |I|-b. When b=1, this corresponds to the classical notion of nested interval. We exhibit two simple algorithms to output all b-nested common or conserved intervals between K permutations in O(Kn+nocc) time, where nocc is the total number of such intervals. We also explain how to count all b-nested intervals in O(Kn) time. New properties of the family of conserved intervals are proposed to do so

On the distribution of the number of cycles in the breakpoint graph of a random signed permutation

International audienceWe use the finite Markov chain embedding technique to obtain the distribution of the number of cycles in the breakpoint graph of a random uniform signed permutation. This further gives a very good approximation of the distribution of the reversal distance between two random genomes

MinMax-Profiles: A Unifying View of Common Intervals, Nested Common Intervals and Conserved Intervals of K Permutations

Common intervals of K permutations over the same set of n elements were firstly investigated by T. Uno and M.Yagiura (Algorithmica, 26:290:309, 2000), who proposed an efficient algorithm to find common intervals when K=2. Several particular classes of intervals have been defined since then, e.g. conserved intervals and nested common intervals, with applications mainly in genome comparison. Each such class, including common intervals, led to the development of a specific algorithmic approach for K=2, and - except for nested common intervals - for its extension to an arbitrary K. In this paper, we propose a common and efficient algorithmic framework for finding different types of common intervals in a set P of K permutations, with arbitrary K. Our generic algorithm is based on a global representation of the information stored in P, called the MinMax-profile of P, and an efficient data structure, called an LR-stack, that we introduce here. We show that common intervals (and their subclasses of irreducible common intervals and same-sign common intervals), nested common intervals (and their subclass of maximal nested common intervals) as well as conserved intervals (and their subclass of irreducible conserved intervals) may be obtained by appropriately setting the parameters of our algorithm in each case. All the resulting algorithms run in O(Kn+N)-time and need O(n) additional space, where N is the number of solutions. The algorithms for nested common intervals and maximal nested common intervals are new for K>2, in the sense that no other algorithm has been given so far to solve the problem with the same complexity, or better. The other algorithms are as efficient as the best known algorithms.Comment: 25 pages, 2 figure

Finding Nested Common Intervals Efficiently

International audienceIn this paper, we study the problem of effi ciently fi nding gene clusters formalized by nested common intervals between two genomes represented either as permutations or as sequences. Considering permutations, we give several algorithms whose running time depends on the size of the actual output rather than the output in the worst case. Indeed, we first provide a straightforward O(n^3) time algorithm for finding all nested common intervals. We reduce this complexity by providing an O(n^2) time algorithm computing an irredundant output. Finally, we show, by providing a third algorithm, that fi nding only the maximal nested common intervals can be done in linear time. Considering sequences, we provide solutions (modi cations of previously de ned algorithms and a new algorithm) for di fferent variants of the problem, depending on the treatment one wants to apply to duplicated genes

Finding Nested Common Intervals Efficiently

International audienceIn this paper, we study the problem of efficiently finding gene clusters formalized by nested common intervals between two genomes represented either as permutations or as sequences. Considering permutations, we give several algorithms whose running time depends on the size of the actual output rather than the output in the worst case. Indeed, we first provide a straightforward cubic time algorithm for finding all nested common intervals. We reduce this complexity by providing a quadratic time algorithm computing an irredundant output. We then show, by providing a third algorithm, that finding only the maximal nested common intervals can be done in linear time. Finally, we prove that finding approximate nested common intervals is fixed parameter tractable. Considering sequences, we provide solutions (modifications of previously defined algorithms and a new algorithm) for different variants of the problem, depending on the treatment one wants to apply to duplicated genes. This includes a polynomial-time algorithm for a variant implying a matching of the genes in the cluster, a setting that for other problems often leads to hardness

Finding Nested Common Intervals Efficiently

Blin G, Faye D, Stoye J. Finding Nested Common Intervals Efficiently. Journal of Computational Biology. 2010;17(9):1183-1194