Computing Large Subcubes in Residual Hypercubes

A residual hypercube is an arbitrary subgraph induced by a subset of the nodes of the hypercube. Residual hypercubes can be used to model the availability of only some of the nodes of a hypercube, e.g., in fault tolerance and processor allocation problems. A residual hypercube may be specified either by listing all the nodes included in the subgraph, or else by specifying a collection of subcube descriptors, which are n-element strings over {0, 1, ?} describing all of the available nodes; these variations in representation are useful in designing processor allocation algorithms 1, 3]. We present an efficient algorithm for determining a subcube of maximum dimension in the residual hypercube when all the available nodes are listed explicitly. Given a residual hypercube of an n-dimensional cube with m available nodes, our algorithm takes time O(nm2) time; this compares favorably with the O(2m22m) algorithm of Ozguner and Aykanat (Inform. Process. Lett. 29 (1988), 247-254) in the case where large numbers of nodes are unavailable. (The algorithm of Ozguner and Aykanat is superior for small numbers of faults.) We show how to parallelize our algorithm to run on the residual hypercube in O(n) parallel steps regardless of the size of the residual hypercube. We show that when the available nodes are described by subcube descriptors rather than by an explicit list of all available nodes, the problem of finding a maximal-dimension subcube becomes co-NP-hard. Finally, we show that the related problem of finding long cycles in residual hypercubes is NP-complete.