A Method for Order/Degree Problem Based on Graph Symmetry and Simulated Annealing with MPI/OpenMP Parallelization

Abstract

The network topology in various systems, such as large-scale data centers, high-performance computing systems, and Network on Chip, is strongly related to network latency. Designing a network topology with low latency can be defined as an order/degree problem (ODP) in graph theory by modeling the network topology as an undirected graph. This study proposes a method for efficiently solving ODPs based on graph symmetry and simulated annealing (SA). This method makes the network topology symmetrical, thereby improving the solution search performance of SA and drastically reducing the calculation time. The proposed method is applied to several problems from an international competition for ODPs called Graph Golf to find network topologies with sufficiently low latency. The symmetry-based calculation achieves a speed up of 31.76 times for one of the problems. Furthermore, to reduce calculation time, the proposed method is extended to use hybrid parallelization with MPI and OpenMP. As a result, a maximum speed up of 209.80 times was achieved on 20 compute nodes consisting of 400 CPU cores. Even faster performance was achieved by combining the symmetry-based calculation and hybrid parallelization.