Metric Weighted Edit Distance: $(3+\varepsilon)$-Approximation in $\widetilde O_\varepsilon(N^{1.6})$ Time

2026-09-17

AI Summary

本文提出了一种在度量加权编辑距离问题上的(3+ε)近似算法，通过字符串分解和采样框架等方法，在~O(N^1.6)时间内实现高效求解。

Abstract

For every $0 < \varepsilon \le 1$, we give a randomized $(3+\varepsilon)$-approximation to weighted edit distance when the costs form a metric on the alphabet augmented with a gap symbol. For strings of total length $N$, the running time is $\widetilde{O}(N^{8/5}/\varepsilon^{16/5})$, where $\widetilde{O}$ suppresses factors polynomial in $\log(N/\varepsilon)$. The dependence on $N$ matches that of the fastest known $(3+\varepsilon)$-approximation for unit-cost edit distance. The algorithm never underestimates the edit distance and achieves the approximation guarantee with inverse-polynomial failure probability in $N$. The running time bound assumes constant-time exact arithmetic operations and metric queries, and it is independent of the numerical range of the edit costs. We build on three tools: the sampling framework of Chakraborty, Das, Goldenberg, Koucký, and Saks (J. ACM, 2020), with subsequent refinements by Andoni (2020); Kuszmaul's removal of inexpensive characters (ICALP 2019); and Klein's data structure for distances in planar graphs (SODA 2005). Our new ingredients include, among others, a decomposition of one string into pieces of bounded length with highly structured total deletion costs. This decomposition lets us compare all pieces against a small family of substrings of the other string.

Problem

• weighted edit distance
• metric costs
• approximation algorithm
• time complexity

Innovation

• Metric Weighted Edit Distance
• Approximation Algorithm
• Randomized Algorithm
• String Decomposition