### Algorithms for Pattern Containment in 0-1 Matrices [article]

P.A. CrowdMath
2017 arXiv   pre-print
We say a zero-one matrix A avoids another zero-one matrix P if no submatrix of A can be transformed to P by changing some ones to zeros. A fundamental problem is to study the extremal function ex(n,P), the maximum number of nonzero entries in an n × n zero-one matrix A which avoids P. To calculate exact values of ex(n,P) for specific values of n, we need containment algorithms which tell us whether a given n × n matrix A contains a given pattern matrix P. In this paper, we present optimal
more » ... thms to determine when an n × n matrix A contains a given pattern P when P is a column of all ones, an identity matrix, a tuple identity matrix, an L-shaped pattern, or a cross pattern. These algorithms run in Θ(n^2) time, which is the lowest possible order a containment algorithm can achieve. When P is a rectangular all-ones matrix, we also obtain an improved running time algorithm, albeit with a higher order.