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The nature of the alignment with gaps corresponding to a Longest Common Subsequence (LCS) of two independent iid random sequences drawn from a finite alphabet is investigated. It is shown that such an optimal alignment typically matches pieces of similar length. This is of importance to understand the structure of optimal alignments of two sequences. Moreover, it is also shown that a property, common to two subsequences, typically holds in most parts of the optimal alignment whenever this same property holds, with high probability, for strings of similar short length. Our results should, in particular, prove useful for simulations since they imply that the re-scaled two dimensional representation of the LCS gets uniformly close to the diagonal as the length of the sequences grows without bound.

Closeness to the diagonal for longest common subsequences in random words