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.
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