Powers of matrices over an extremal algebra with applications to periodic graphs

Consider the extremal algebra ℝ = (ℝ∪{∞}, min, +), using + and min instead of addition and multiplication. This extremal algebra has been successfully applied to a lot of scheduling problems. In this paper the behavior of the powers of a matrix over &Rmacr; is studied. The main result is a representation of the complete sequence (A^m)m ∈ N which can be computed within polynomial time complexity. In the second part we apply this result to compute a minimum cost path in a 1-dimensional periodic graph.