High-Speed Computation of the Kleene Star in Max-Plus Algebraic System Using a Cell Broadband Engine

Abstract

This research addresses a high-speed computation method for the Kleene star of the weighted adjacency matrix in a max-plus algebraic system. We focus on systems whose precedence constraints are represented by a directed acyclic graph and implement it on a Cell Broadband Engine™ (CBE) processor. Since the resulting matrix gives the longest travel times between two adjacent nodes, it is often utilized in scheduling problem solvers for a class of discrete event systems. This research, in particular, attempts to achieve a speedup by using two approaches: parallelization and SIMDization (Single Instruction, Multiple Data), both of which can be accomplished by a CBE processor. The former refers to a parallel computation using multiple cores, while the latter is a method whereby multiple elements are computed by a single instruction. Using the implementation on a Sony PlayStation 3™ equipped with a CBE processor, we found that the SIMDization is effective regardless of the system's size and the number of processor cores used. We also found that the scalability of using multiple cores is remarkable especially for systems with a large number of nodes. In a numerical experiment where the number of nodes is 2000, we achieved a speedup of 20 times compared with the method without the above techniques.