2012/06/01 by Andreas Krebs, Krebs, Andreas, Nutan Limaye +3 · 1 citation
Computer Science · #Algorithms and Data Compression #Complexity and Algorithms in Graphs #Computational Complexity (cs.CC) #Cryptography and Data Security #FOS: Computer and information sciences #cs.CC #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1206.0206
arxiv created 2012/06/01 · openalex publication_date 2012/06/01 · arxiv updated 2012/06/04 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
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' in 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/poly(n). - O(k1+epsilon + log n) space for every 0 <= epsilon <= 1, O(log n) randomness, O(polylog(n) + poly(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 Omega(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' in Dyck2 such that x is obtained from x' by changing (in some restricted manner) at most k positions using: - O(k log n + sqrt(n log n)) space, O(k log n) randomness, poly(k log n) time per element and with error at most 1/poly(n). - O(k^(1+epsilon)+ sqrt(n log n)) space for every 0 <= epsilon <= 1, O(log n) randomness, O(polylog(n) + poly(k)) time per element, with error at most 1/8.