Dynamic and Approximate Pattern Matching in 2D

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(log n) time and queries in O(log 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 log m log 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(n log n) time into a data structure of size O(n) with query time O(nm).