A Brief Announcement: Regular and Dyck Languages in MPC

Regular languages are some of the most widely studied languages in computer science history. Given a regular language L subset of {0, 1}star. and string w is an element of{0, 1}star, two of the most fundamental regular language problems are recognition, the problem of determining if w is in L, and testing, the problem of determining if w is at most epsilon-far from L. In this paper we modernize regular language recognition and testing algorithms for the Massively Parallel Computations (MPC) model used everyday in big data engineering. First we give a regular language testing algorithm, which succeeds with high probability using (O) over tilde (1/epsilon) queries to the input string. Following the testing algorithm, we give a simple dynamic programming solution for regular language recognition. Both algorithms run in constant communication rounds and O(n) total memory in MPC where n is the size of the input. Beyond regular language recognition, we also provide a Dyck language recognition algorithm in MPC. Dyck languages are the sets containing strings composed of only balanced parentheses, which are important to problems such as RNA folding and error detection in common document types such as XML, JSON, and LaTeX. Our Dyck language recognition algorithm is a constant round, O(n) total memory MPC algorithm. We also show that the same algorithm can be used to find the edit distance of an unbalanced parenthesis string over a single type of parentheses.