Adaptive Computation of the Swap-Insert Correction Distance

The Swap-Insert Correction distance from a string S of length n to another string L of length m ≥ n on the alphabet [1‥σ] is the minimum number of insertions, and swaps of pairs of adjacent symbols, converting S into L. Contrarily to other correction distances, computing it is NP-Hard in the size σ of the alphabet. We describe an algorithm computing this distance in time within O(σ ²nmg^σ−1), where for each α ∈ [1‥σ] there are n_α occurrences of α in S, m_α occurrences of α in L, and where g = max _{α∈[1‥σ]} min{n_α, m_α − n_α} is a new parameter of the analysis, measuring one aspect of the difficulty of the instance. The difficulty g is bounded by above by various terms, such as the length n of the shortest string S, and by the maximum number of occurrences of a single character in S (max_{α ∈[1‥σ]} n_α). This result illustrates how, in many cases, the correction distance between two strings can be easier to compute than in the worst case scenario.