Conference Paper

# Common Intervals of Two Sequences.

DOI: 10.1007/978-3-540-39763-2_2 Conference: Algorithms in Bioinformatics, Third International Workshop, WABI 2003, Budapest, Hungary, September 15-20, 2003, Proceedings
Source: DBLP

ABSTRACT Looking for the subsets of genes appearing consecutively in two or more genomes is an useful approach to identify clusters
of genes functionally associated. A possible formalization of this problem is to modelize the order in which the genes appear
in all the considered genomes as permutations of their order in the first genome and find k-tuples of contiguous subsets of these permutations consisting of the same elements: the common intervals. A drawback of this
approach is that it doesn’t allow to take into account paralog genes and genomic internal duplications (each element occurs
only once in a permutation). To do it we need to modelize the order of genes by sequences which are not necessary permutations.

In this work, we study some properties of common intervals between two general sequences. We bound the maximum number of common
intervals between two sequences of length n by n
2 and present an O(n
2log(n)) time complexity algorithm to enumerate their whole set of common intervals. This complexity does not depend on the size
of the alphabets of the sequences.

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ABSTRACT: An interval of a permutation is a consecutive substring consisting of consecutive symbols. For example, 4536 is an interval in the permutation 71453682. These arise in genetic applications. For the applications, it makes sense to generalize so as to allow gaps of bounded size δ-1, both in the locations and the symbols. For example, 4527 has gaps bounded by 1 (since 3 and 6 are missing) and is therefore a δ-interval of 389415627 for δ=2. After analyzing the distribution of the number of intervals of a uniform random permutation, we study the number of 2-intervals. This is exponentially large, but tightly clustered around its mean. Perhaps surprisingly, the quenched and annealed means are the same. Our analysis is via a multivariate generating function enumerating pairs of potential 2-intervals by size and intersection size.
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##### Article: Extending Common Intervals Searching from Permutations to Sequences
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ABSTRACT: Common intervals have been defined as a modelisation of gene clusters in genomes represented either as permutations or as sequences. Whereas optimal algorithms for finding common intervals in permutations exist even for an arbitrary number of permutations, in sequences no optimal algorithm has been proposed yet even for only two sequences. Surprisingly enough, when sequences are reduced to permutations, the existing algorithms perform far from the optimum, showing that their performances are not dependent, as they should be, on the structural complexity of the input sequences. In this paper, we propose to characterize the structure of a sequence by the number $q$ of different dominating orders composing it (called the domination number), and to use a recent algorithm for permutations in order to devise a new algorithm for two sequences. Its running time is in $O(q_1q_2p+q_1n_1+q_2n_2+N)$, where $n_1, n_2$ are the sizes of the two sequences, $q_1,q_2$ are their respective domination numbers, $p$ is the alphabet size and $N$ is the number of solutions to output. This algorithm performs better as $q_1$ and/or $q_2$ reduce, and when the two sequences are reduced to permutations (i.e. when $q_1=q_2=1$) it has the same running time as the best algorithms for permutations. It is also the first algorithm for sequences whose running time involves the parameter size of the solution. As a counterpart, when $q_1$ and $q_2$ are of $O(n_1)$ and $O(n_2)$ respectively, the algorithm is less efficient than other approaches.
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