Computing Covers of 2D-Strings

We consider two notions of covers of a two-dimensional string T . A (rectangular) subarray P of T is a 2D-cover of T if each position of T is in an occurrence of P in T . A one-dimensional string S is a 1D-cover of T if its vertical and horizontal occurrences in T cover all positions of T . We show how to compute the smallest-area 2D-cover of an m × n array T in the optimal O(N) time, where N = mn, all aperiodic 2D-covers of T in O(N log N) time, and all 2D-covers of T in N4/3 · logO(1) N time. Further, we show how to compute all 1D-covers in the optimal O(N) time. Along the way, we show that the Klee’s measure of a set of rectangles, each of width and height at least √ n, on an n × n grid can be maintained in √ n · logO(1) n time per insertion or deletion of a rectangle, a result which could be of independent interest. 2012 ACM Subject Classification Theory of computation → Pattern matching