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Max-plus algebra is a semiring with addition $a\oplus b = \max(a,b)$ andmultiplication $a\otimes b = a+b$. It is applied in cases, such ascombinatorial optimization and discrete event systems. We consider the power ofmax-plus square matrices, which is equivalent to obtaining the all-pair maximumweight paths with a fixed length in the corresponding weighted digraph. Each$n$-by-$n$ matrix admits the CSR expansion that decomposes the matrix into asum of at most $n$ periodic terms after $O(n^{2})$ times of powers. In thisstudy, we propose an $O(n(m+n \log n))$ time algorithm for the CSR expansion,where $m$ is the number of nonzero entries in the matrix, which improves the$O(n^{4} \log n)$ algorithm known for this problem. Our algorithm is based onfinding the roots of the characteristic polynomial of the max-plus matrix.These roots play a similar role to the eigenvalues of the matrix, and becomethe growth rates of the terms in the CSR expansion.

Algorithm for the CSR expansion of max-plus matrices using the characteristic polynomial