Dynamic and approximate pattern matching in 2D

Abstract

We consider dynamic and online variants of 2D pattern matching between an m×m pattern and an n×n text. All the algorithms we give are randomised and give correct outputs with at least constant probability. -For dynamic 2D exact matching where updates change individual symbols in the text, we show updates can be performed in O(log2 n) time and queries in O(log2 m) time. -We then consider a model where an update is a new 2D pattern and a query is a location in the text. For this setting we show that Hamming distance queries can be answered in O(logm + H) time, where H is the relevant Hamming distance. -Extending this work to allow approximation, we give an efficient algorithm which returns a (1+ε) approximation of the Hamming distance at a given location in O(ε−2 log2 mlog log n) time. Finally, we consider a different setting inspired by previous work on locality sensitive hashing (LSH). Given a threshold k and after building the 2D text index and receiving a 2D query pattern, we must output a location where the Hamming distance is at most (1 + ε)k as long as there exists a location where the Hamming distance is at most k. -For our LSH inspired 2D indexing problem, the text can be preprocessed in O(n2(4/3+1/(1+ε)) log3 n) time into a data structure of size O(n2(1+1/(1+ε))) with query time O(n2(1/(1+ε))m2).