A linear time algorithm to verify strong structural controllability

We prove that strong structural controllability of a pair of structural matrices (A, B) can be verified in time linear in n + r + v, where A is square, n and r denote the number of columns of A and B, respectively, and v is the number of non-zero entries in (A, B). We also present an algorithm realizing this bound, which depends on a recent, high-level method to verify strong structural controllability and uses sparse matrix data structures. Linear time complexity is actually achieved by separately storing both the structural matrix (A, B) and its transpose, linking the two data structures through a third one, and a novel, efficient scheme to update all the data during the computations. We illustrate the performance of our algorithm using systems of various sizes and sparsity.