On incomplete hypercubes

Abstract

New approach to the fragmentation problem of hypercube multiprocessors with dynamic allocation of subcubes is proposed. It is based on constructing Hamiltonian circuits of incomplete hypercubes. The main result is a constructive proof that an n-cube from which up to n-2 vertex-disjoint subcubes are removed so that it remains connected is a Hamiltonian graph and its Hamiltonian circuit can be constructed. If the communication subsystem can use a circuit as an efficient broadcasting and communication graph, then the operating system can use this algorithm for allocating nodes of a fragmented hypercube to a user task when there is no subcube large enough to accommodate the task and in the same time the total number of free nodes is sufficient - instead of forcing the user to wait for releasing allocated subcubes or forcing the system to move the running tasks to other parts of the hypercube.