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Additional info for A branch-and-cut algorithm for multiple sequence alignment
Example text
S j | + 1, each having l weight w(v j ,v j m p−1 )i m l p j j and representing the realization of gap arc (vm , vp−1 )i in the align- ment. It is easy to check that the optimal alignment between strings s i and s j corresponds to the longest (arc-weighed) path from n{v i ,v j } to n{v i ,v j in Qij with the property } 1 1 |s i |+1 |s j |+1 that no consecutive arcs in the path are red. Such a path can be found in linear time in the size of Qij , which is O(|s i ||s j | max{|s i |, |s j |}), by using two labels for each node, one for the longest path and the other for the longest path whose last arc is green.
S|s j | of s j . The sum of the values of all the pairwise optimal alignments is clearly an upper bound on the value of the optimal alignment, called the pairwise upper bound. For convenience, let σ ij denote the sum of the values of the optimal alignments between all string pairs different from {s i , s j }. After having computed the above values, for every alignment variable xe correspondj ing to edge e = {vli , vm }, we let Uxe = σ ij + π ij (l − 1, m − 1) + we + ωij (l + 1, m + 1), and perform the associated reduction of the alignment variables in O(n2 ) time, whereas i )j , we let for every gap variable ya corresponding to gap arc a = (vli , vm Uya = σ ij + max (π ij (l − 1, p) + ωij (m + 1, p + 1)).
S|s j | of s j . The sum of the values of all the pairwise optimal alignments is clearly an upper bound on the value of the optimal alignment, called the pairwise upper bound. For convenience, let σ ij denote the sum of the values of the optimal alignments between all string pairs different from {s i , s j }. After having computed the above values, for every alignment variable xe correspondj ing to edge e = {vli , vm }, we let Uxe = σ ij + π ij (l − 1, m − 1) + we + ωij (l + 1, m + 1), and perform the associated reduction of the alignment variables in O(n2 ) time, whereas i )j , we let for every gap variable ya corresponding to gap arc a = (vli , vm Uya = σ ij + max (π ij (l − 1, p) + ωij (m + 1, p + 1)).