Rate-Optimality of Cyclo-Static Schedules

1 Deenitions and Preliminaries A data-ow graph (DFG) is represented by a directed weighted graph G = (V , E, d, t) where V is the set of computation nodes, E is the edge set which deenes the directed edges (or precedence relation) from nodes in V to nodes in V , and d(e) is the delay count for edge e 2 E. Each node v in V is associated with a positive integer t(v) which may represent the computation time for the node v. The graph in Figure ?? is an example of DFG, where the number attaches to a node is its computation time. The delay count, say i, on an edge (u; v) represents the sequenced relation between computation nodes u and v. For a meaningful data-ow graph, the total delay count of any loop is nonzero. The execution of all computation nodes once in V is called one iteration. An edge e from u to v with delay count d(e) means that the computation of node v at iteration j depends on the computation of node u at iteration j ? d(e). Time and processor schedules of the DFG in a parallel system are formally modeled as follows. A time schedule is represented by a function S : V N ?! R. The starting time of node v in the i-th iteration is S(v; i). A time schedule is legal if for every edge u e ?!v and iteration i, we have S(u; i) + t(u) S(v; i + d(e)). A time schedule is said to have an unfolding factor f and a cycle period c if S(v; i+f) = S(v; i)+c for every v in V and iteration i. Thus, such a time schedule can be represented by the partial schedule of the rst f iterations. A new instance of this partial schedule of f iterations can be initiated for every interval of length c to form a legal complete schedule.