Efficient Computation of the State Equation for Sparse Adjacency Matrices in Max-Plus Algebraic Systems

This research proposes a useful framework for efficiently computing the state equations in scheduling problems for a class of discrete event systems. We focus on systems in which the precedence relationships are represented by a sparse adjacency matrix. The state equations are linear ones in max-plus algebra, and give the earliest event occurrence times. Even if we use the most efficient algorithm proposed thus far, the time complexity for computing the state equations is O(n^2), where n represents the number of nodes. Moreover, the values of the adjacency matrix have to be given by a full matrix, which is inefficient in terms of both time and space. Our proposed framework adopts a compressed matrix format, the use of which reduces the time complexity for the computation to O(n+m), where m represents the number of arcs. Thus, the framework is especially beneficial if the adjacency matrix is sparse.