Edit Distance with Duplications and Contractions Revisited

Abstract

In this paper, we propose three algorithms for the problem of string edit distance with duplication and contraction operations, which improve the time complexity of previous algorithms for this problem. These include a faster algorithm for the general case of the problem, and two improvements which apply under certain assumptions on the cost function. The general algorithm is based on fast min-plus multiplication of square matrices, and obtains the running time of $O\left(\frac{|\Sigma|n^3 \log^3\log n}{\log^2n}\right)$, where n is the length of the input strings and |Σ| is the alphabet size. This algorithm is further accelerated, under some assumption on the cost function, to $O\left(|\Sigma|\left(n^2+\frac{nn'^2\log^3\log n'}{\log^2 n'}\right)\right)$ time, where n′ is the length of the run-length encoding of the input. Another improvement is based on a new fast matrix-vector min-plus multiplication under a certain discreteness assumption, and yields an $O\left( |\Sigma| \frac{n^3}{\log^2n}\right)$ time algorithm. Furthermore, this algorithm is online, in the sense that one of the strings may be given letter by letter. As part of this algorithm we present the currently fastest online algorithm for weighted CFG parsing for discrete weighted grammars. This result is useful on its own.