Adaptive Computation of the Swap-Insert Correction Distance

Abstract

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 2 nmg 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.