Streaming Algorithms for Recognizing Nearly Well-Parenthesized Expressions

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′ ∈ 1-turn-Dyck2 such that x is obtained by flipping at most k locations of x′ using: – O(k logn) space, O(k logn) randomness, and poly(k logn) time per item and with error at most 1/n. – O(k + logn) space for every 0 ≤ ≤ 1, O(logn) randomness, O((log n + k)) 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′ ∈ Dyck2 such that x is obtained from x′ by changing (in some restricted manner) at most k positions using: – O(k logn+ √ n logn) space, O(k logn) randomness, poly(k logn) time per element and with error at most 1/n. – O(k + √ n logn) space for every 0 ≤ ≤ 1, O(logn) randomness, O((log n + k)) time per element, with error at most 1/8.