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We say a zero-one matrix $A$ avoids another zero-one matrix $P$ if nosubmatrix of $A$ can be transformed to $P$ by changing some ones to zeros. Afundamental problem is to study the extremal function $ex(n,P)$, the maximumnumber of nonzero entries in an $n \times n$ zero-one matrix $A$ which avoids$P$. To calculate exact values of $ex(n,P)$ for specific values of $n$, we needcontainment algorithms which tell us whether a given $n \times n$ matrix $A$contains a given pattern matrix $P$. In this paper, we present optimalalgorithms to determine when an $n \times n$ matrix $A$ contains a givenpattern $P$ when $P$ is a column of all ones, an identity matrix, a tupleidentity matrix, an $L$-shaped pattern, or a cross pattern. These algorithmsrun in $\Theta(n^2)$ time, which is the lowest possible order a containmentalgorithm can achieve. When $P$ is a rectangular all-ones matrix, we alsoobtain an improved running time algorithm, albeit with a higher order.

Algorithms for Pattern Containment in 0-1 Matrices