WithinSubmatrix

WithinSubmatrix, in a mathematical or computer science context, refers to the property that a given matrix A appears as a contiguous block within a larger matrix B. Formally, if A is an m by n matrix and B is a p by q matrix with p ≥ m and q ≥ n, A is said to be within a submatrix of B if there exists a row offset r (0 ≤ r ≤ p − m) and a column offset c (0 ≤ c ≤ q − n) such that for all i in 0..m−1 and j in 0..n−1, A[i][j] = B[r+i][c+j]. In other words, A matches exactly a submatrix of B when positioned at (r, c).

Search methods for detecting within-submatrix relationships range from straightforward to advanced. The naive approach checks all possible placements of A inside B, with a worst-case time complexity of O(pq mn). More sophisticated techniques include two-dimensional pattern matching and 2D rolling hashes (an extension of the Rabin–Karp method) to achieve improved expected performance. Some variants allow additional flexibility, such as matching under rotations or reflections, or permitting approximate matches with tolerance for noise, which alters the algorithmic approach and complexity.

Examples help illustrate the concept: if A is a 2×2 matrix and B is a larger matrix, A is within a submatrix of B when there exists a 2×2 block in B that exactly equals A. If B contains multiple occurrences of A, the property holds for several positions, and overlapping occurrences are possible.

Applications of within-submatrix detection appear in pattern recognition, image processing, data mining, and verification tasks where a template or pattern must be located within a larger data structure. Related topics include submatrix, submatrix search, and two-dimensional pattern matching.