A Task Mapping Method for a Hypercube by Combining Subcubes

Abstract

This paper presents a new algorithm for mapping of tasks onto a hypercube. Given a weighted task graph, the algorithm finds good mapping in a reasonable computation time. When the target computer is ndimensional cube (n-cube), the proposed algorithm is composed of n stages. The algorithm starts with an initial state in which the tasks are mapped onto 2n 0cubes. At each stage k, the task graph is mapped onto 2n-k k-cubes. At the beginning of stage k, the tasks have already been mapped onto 2n-(k-1) (k-1)-cubes. The tasks are mapped onto k-cubes by combining a pair of (k-1)-cubes. 2n-k pairs of (k-1)-cubes are determined, and they are combined so that the mapping onto the k-cubes makes the communication cost as low as possible. When the target computer is n-dimensional cube (ncube), the proposed algorithm is composed of n stages. The algorithm starts with an initial state in which the tasks are mapped onto 2" 0-cubes. At each stage k (k=1,2,..,n), the task graph is mapped onto 2n-k k-cubes. At the beginning of stage k, the tasks are already mapped onto 2n-(k-1) (k-1)-cubes. The mapping onto k-cubes can be done by combining a pair of (k-1)-cubes. 2n-k pairs are determined among 2n-(k-1) (k-1)-cubes, and they are combined so that mapping onto the k-cubes makes the communication cost as low as possible.