| Abstract: | Data Flow Graphs (DFGs) [1, 2] are often used to model iterative concurrent activities. The acyclic precedence graph (APG) constructed from a single-rate Data-Flow Graph (SRDFG) can be used to construct an admissible schedule. The iteration period (IP) from this schedule is, in general, not the shortest. Therefore, it is necessary to find the minimum IP of a schedule [2, 20, 35, 40]. Transformation techniques such as retiming [24, 28, 42] and program unfolding [23, 26-27, 35, 41] can uncover hidden concurrency by altering the structure of the SRDFG, resulting in a new APG and, thus, more efficient scheduling due to the shortening of the IP. These techniques cannot, however, shorten the IP indefinitely. The shortest IP (hence, the maximum throughput), called the iteration bound (IB), is a function of the parameters (the number of delay elements and node computation times) of the original SRDFG. The IB for a SRDFG equals the maximum loop bound (LB) of all the loops as shown in [35, 38]. This approach, however, cannot be extended to the multi-rate DFG (MRDFG). Parhi [35] suggested converting the MRDFG into an equivalent SRDFG to find the IB. No explicit formula for IB was presented. How to perform such conversion is not obvious in [35]. Further, the IB of an MRDFG may be a linear additive combination of the loop bounds (LB`s) of more than one loop (called loop - combination [12]). It is unclear what the equivalent SRDFG and be since the corresponding IB depends solely on the critical loop (CL). The contributions of this work include (1) developing a procedure to derive Petri Net (PN) models from the MRDFG, (2) finding a procedure with polynomial time complexity to obtain the components of the reproduction vector for an MRDFG, (3) giving a sufficient condition for an MRDFG to be loop-combination free, (4) showing that retiming in an MRDFG is equivalent to getting another reachable marking of its equivalent PN from its initial marking, (5) extending the technique to determine the IB and GL for SRDFG to that for an MRDFG meeting the sufficient condition, and (6) achieving rate-optimal scheduling, i.e., where the IP is identical to the IB, for MRDFG. |
