text processing engineers

Abstract: For fixed nonempty words $u,v$, the minimum insertion degree of a shuffle is the least number of inserted-source runs among its representations. We prove that, after a block exchange from a minimum mask of degree $r$, the minimum over masks with commuting source projections is exactly $r-1$; a larger decrease occurs precisely when a noncommuting mask has fewer runs. If either source is unary, every such exchange is exact. We prove an explicit repair for an infinite periodic degree-three family. Computer-assisted results establish first-mask descent through combined length seventeen over every alphabet, degree-three first-mask descent through length thirty, and a uniform repair of all degree-three failures through length thirty, including all further letter identifications. Counterexamples to arbitrary, first, and last minimum-mask choices explain the need for alternative masks. Finally, we prove that minimum iteration depths have no gaps for arbitrary languages. General block-exchange descent and Singleton No-Gap remain open.

Abstract: We give randomized algorithms for $3$-approximate edit distance in $\widetilde{\mathcal{O}}(N^{11/6})$ time for unweighted edit distance and in $\widetilde{\mathcal{O}}(N^{40/21})$ time for arbitrary metric edit weights, where $N$ is the total input length. For non-metric costs, we prove an unconditional $Ω(N^2)$ oracle-query lower bound for every approximation factor depending only on $N$, even for symmetric weights or weights satisfying the triangle inequality (but not both). Under the Orthogonal Vectors Hypothesis, we show a similar result for constant-size alphabets. This holds even for symmetric weights over a size-$3$ alphabet or triangle-inequality weights over a size-$2$ alphabet. In contrast, for symmetric weights over a binary alphabet we show an $\widetilde{\mathcal{O}}(N^{40/21})$-time $3$-approximation algorithm.

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.

Abstract: We study weighted edit distance between two strings of total length $n$, where edit costs are induced by an arbitrary metric. For equal-length inputs, Kuszmaul (2019) gave an $O(n^δ)$-approximation with $\widetilde{O}(n^{2-δ})$ running time for every fixed $0 < δ< 1$. We give the first constant-factor approximation for weighted edit distance over arbitrary metrics in strongly subquadratic running time. For every $0 < \varepsilon \le 1$, our randomized algorithm runs in $\widetilde{O}(n^{7/4}/\varepsilon^8)$ time and returns a $(3+\varepsilon)$-approximation with probability at least $1-n^{-10}$. The algorithm allows unequal input lengths and places no bound on the ratio between edit costs.