Streaming algorithms for recognizing nearly well-parenthesized expressions

Abstract

We study the streaming complexity of the membership problem of 1-turn-Dyck2 and Dyck2 when there are a few errors in the input string. 1-turn-Dyck2 with errors: We prove that there exists a randomized one-pass algorithm that given x checks whether there exists a string x′ ε 1-turn-Dyck2 such that x is obtained by flipping at most k locations of x′ using: - O(k log n) space, O(k log n) randomness, and poly(k log n) time per item and with error at most 1/nΩ(1). - O(k 1+ε + log n) space for every 0 ≤ ε ≤ 1, O(log n) randomness, O((log O(1) n + kO(1))) time per item, with error at most 1/8. Here, we also prove that any randomized one-pass algorithm that makes error at most k/n requires at least Ω(k log(n/k)) space to accept strings which are exactly k-away from strings in 1-turn-Dyck2 and to reject strings which are exactly k+2-away from strings in 1-turn-Dyck2. Since 1-turn-Dyck2 and the Hamming Distance problem are closely related we also obtain new upper and lower bounds for this problem. Dyck2 with errors: We prove that there exists a randomized one-pass algorithm that given x checks whether there exists a string x′ε Dyck2 such that x is obtained from x′ by changing (in some restricted manner) at most k positions using: - O(k log n + √n log n) space, O(k log n) randomness, poly(k log n)time per element and with error at most 1/nΩ(1). - O(k1+ε + √n log n) space for every 0