Fluctuations of the longest common subsequence for sequences of independent blocks

The problem of the order of the fluctuation of the Longest Common Subsequence (LCS) of two independent sequences has been open for decades. There exist contradicting conjectures on the topic, [1] and [2]. Lember and Matzinger [3] showed that with i.i.d. binary strings, the standard deviation of the length of the LCS is asymptotically linear in the length of the strings, provided that 0 and 1 have very different probabilities. Nonetheless, with two i.i.d. sequences and a finite number of equiprobable symbols, the typical size of the fluctuation of the LCS remains unknown. In the present article, we determine the order of the fluctuation of the LCS for a special model of i.i.d. sequences made out of blocks. A block is a contiguous substring consisting only of one type of symbol. Our model allows only three possible block lengths, each been equiprobable picked up. For i.i.d. sequences with equiprobable symbols, the blocks are independent of each other. In order to study the fluctuation of the LCS in this model, we developed a method which reformulates the fluctuation problem as a (relatively) low dimensional optimization problem. We finally proved that for our model, the fluctuation of the ength of the LCS coincides with the Waterman’s conjecture [2]. We belive that our method can be applied to any other case dealing with i.i.d. sequences, only that the optimization problem might be more complicated to formulate and to solve.