O(mn log σ) Time Transposition Invariant LCS Computation

Given strings A and B of lengths m and n over a finite alphabet Σ ⊂ Z of size O(σ), the length of the longest common transposition invariant subsequence is LCTS(A,B) = maxt∈Z{LCS(A+ t, B)}, where A + t = (a1 + t)(a2 + t) . . . (am + t) and LCS(A + t, B) is the length of the longest common subsequence between A+ t and B. LCTS(A,B) can be computed naively in O(mnσ) time. We present a simple and easy to implement algorithm obtaining O(mn log σ) time. We also show that transposition invariant Levenshtein distance can be computed in O(mn √ σ) time.