Reconfigurability and Reliability of Systolic/Wavefront Arrays

In this paper, we study fault-tolerant redundant structures for maintaining reliable arrays. In particular, we assume the desired array (application graph) is embedded in a certain class of regular, bounded-degree graphs called dynamic graphs. We define the degree of reconfigurability DR, and DR with distance DRd, of a redundant graph. When DR (respectively, DRd) is independent of the size of the application graph, we say the graph is finitely reconfigurable, FR (respectively, locally reconfigurable, LR). We show that DR provides a natural lower bound on the time complexity of any distributed reconfiguration algorithm and that there is no difference between being FR and LR on dynamic graphs. We then show that if we wish to maintain both local reconfigurability and a fixed level of reliability, a dynamic graph must be of dimension at least one greater than the application graph. Thus, for example, a one-dimensional systolic array cannot be embedded in a one-dimensional dynamic graph without sacrificing either reliability or locality of reconfiguration.