Fluctuations of the Longest Common Subsequence for Sequences of Independent Blocks

The problem of the fluctuation of the Longest Common Subsequence (LCS) of two i.i.d. sequences of length $n>0$ has been open for decades. There exist contradicting conjectures on the topic. Chvatal and Sankoff conjectured in 1975 that asymptotically the order should be $n^{2/3}$, while Waterman conjectured in 1994 that asymptotically the order should be $n$. A contiguous substring consisting only of one type of symbol is called a block. In the present work, we determine the order of the fluctuation of the LCS for a special model of sequences consisting of i.i.d. blocks whose lengths are uniformly distributed on the set $\{l-1,l,l+1\}$, with $l$ a given positive integer. We showed that the fluctuation in this model is asymptotically of order $n$, which confirm Waterman's conjecture. For achieving this goal, we developed a new method which allows us to reformulate the problem of the order of the variance as a (relatively) low dimensional optimization problem.Comment: PDFLatex, 40 page

Letter Change Bias and Local Uniqueness in Optimal Sequence Alignments

Considering two optimally aligned random sequences, we investigate the effect on the alignment score caused by changing a random letter in one of the two sequences. Using this idea in conjunction with large deviations theory, we show that in alignments with a low proportion of gaps the optimal alignment is locally unique in most places with high probability. This has implications in the design of recently pioneered alignment methods that use the local uniqueness as a homology indicator