Unification of Spectral and Inertial Bisection

We discuss algorithms for graph bisection which are relevant to the distribution of tasks, such as the elements and nodes of an unstructured mesh to the processors of a parallel computer. Starting with a cost function consisting of a part to ensure equal numbers of tasks for each processor, and a part to minimize communication time between processors, we derive the spectral bisection method, which consists of finding an eigenvector of a large sparse matrix. Inertial bisection is derived as a statistical approximation from the spectral method, via the moments of the node density and moments of the pair distribution function. Introduction When a parallel computer has many tasks to perform simultaneously, each task is given to one of the processors. If we assume that each task communicates continually with a subset of the others, then the tasks take the structure of a graph. If the task graph remains constant or changes slowly, it may be advantageous to decide carefully to which processor each task should be assigned. The resulting optimization problem is a graph-partitioning problem. The task graph is to be partitioned among the processors such that the number of cut edges (edges with one end in one processor and the other end in another processor) of the graph is minimized. The number of tasks per processor should be approximately equal to maximize parallelism of the computation, and the number of cut edges minimized to reduce communication costs. In this paper, we shall assume that the graph-partitioning problem is to be solved by recursive bisection [8] where the graph is partitioned into two equal halves, then into halves again, and so on. We shall thus consider only the problem of partitioning the graph into two equal pieces. There are several ways to solve approximately this graph bisection problem, and two popular algorithms are: spectral bisection, and geometric methods such as inertial bisection. The spectral method [1,2,7,9] uses properties of the task graph itself, such as the Laplacian matrix of the graph. In contrast, geometric methods [9] make a statistical hypothesis, assuming that each task is associated with a definite position in some geometric space, and ignore the detailed structure of