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 $m\ge n$ on the alphabet [1..d] 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 d of the alphabet. We describe an algorithm computing this distance in time within $O(d^2 nm g^{d-1})$, where there are $n_\alpha$ occurrences of $\alpha$ in S, $m_\alpha$ occurrences of $\alpha$ in L, and where $g=\max _{\alpha \in [1..d]} \min \{n_\alpha ,m_\alpha -n_\alpha \}$ measures the difficulty of the instance. The difficulty g is bounded by above by various terms, such as the length of the shortest string S, and by the maximum number of occurrences of a single character in S. The latter bound yields a running time within $O(d(n+m)+(d/(d-1)^{d-2})\cdot n^{d}(m-n))$ in the worst case over instances of fixed lengths n and m for S and L, which further simplifies to within $O(n^d(m-n)+m)$ when d is fixed, the state of the art for this problem. This illustrates how, in many cases, the correction distance between two strings can be easier to compute than in the worst case scenario.

J. Barbay and P. Pérez-Lantero—Partially supported by Millennium Nucleus Information and Coordination in Networks ICM/FIC RC130003.