Approximating Edit Distance in Truly Subquadratic Time: Quantum and MapReduce

The edit distance between two strings is defined as the smallest number of insertions , deletions , and substitutions that need to be made to transform one of the strings to another one. Approximating edit distance in subquadratic time is “one of the biggest unsolved problems in the field of combinatorial pattern matching” [37]. Our main result is a quantum constant approximation algorithm for computing the edit distance in truly subquadratic time. More precisely, we give an quantum algorithm that approximates the edit distance within a factor of 3. We further extend this result to an quantum algorithm that approximates the edit distance within a larger constant factor. Our solutions are based on a framework for approximating edit distance in parallel settings. This framework requires as black box an algorithm that computes the distances of several smaller strings all at once. For a quantum algorithm, we reduce the black box to metric estimation and provide efficient algorithms for approximating it. We further show that this framework enables us to approximate edit distance in distributed settings. To this end, we provide a MapReduce algorithm to approximate edit distance within a factor of , with sublinearly many machines and sublinear memory. Also, our algorithm runs in a logarithmic number of rounds.

Combinatorics of Nahm sums, quiver resultants and the K-theoretic condition

Algebraic Nahm equations, considered in the paper, are polynomial equations, governing the q → 1 limit of the q-hypergeometric Nahm sums. They make an appearance in various fields: hyperbolic geometry, knot theory, quiver representation theory, topological strings and conformal field theory. In this paper we focus primarily on Nahm sums and Nahm equations that arise in relation with symmetric quivers. For a large class of them, we prove that quiver A-polynomials — specialized resultants of the Nahm equations, are tempered (the so-called K-theoretic condition). This implies that they are quantizable. Moreover, we find that their face polynomials obey a remarkable combinatorial pattern. We use the machinery of initial forms and mixed polyhedral decompositions to investigate the edges of the Newton polytope. We show that this condition holds for the diagonal quivers with adjacency matrix C = diag(α, α, . . . , α), α ≥ 2, and provide several checks for non-diagonal quivers. Our conjecture is that the K-theoretic condition holds for all symmetric quivers.