# Two Cycle Related Problems for Regular Dataflow Graphs: Complexity and Heuristics

Praveen K. Murthy, University of California at Berkeley

Edward A. Lee, University of California at Berkeley

Technical Memorandum UCB/ERL M97/76, Electronics Research Laboratory, University of California, Berkeley, Ca 94720, October 1997.

## ABSTRACT

A regular dataflow graph (RDFG) is a graph with a highly regular structure that enables its description to be exponentially smaller than the description size for an ordinary graph. Such graphs arise when certain regular iterative algorithms (like matrix multiplication or convolution) are modeled using dependence graphs. These graphs can be implemented either on systolic arrays, or wavefront arrays (WA). Systolic arrays have a global time clock; operations are scheduled statically and executed according to this schedule. The global clocking however, presents problems due to clock skewing in large circuits; hence, wavefront arrays are an attractive alternative. Wavefront arrays use a dataflow method of execution, and hence, do not require global synchronization. Array elements start computing whenever they have all of their inputs.In a systolic implementation, the dependence graph cannot have any cycles since the existence of a schedule depends on the existence of a schedule vector that has non-negative dot product with each dependency edge. However, a graph implemented on a WA may have cycles provided that the cycles do not deadlock. There are a couple of computational problems that arise in this context: the first is the detection of deadlock; that is, to determine whether the graph to be implemented has a delay-free cycle. The second is to determine the maximum cycle mean; this represents the iteration rate with which the graph can be executed. While both of these problems are well known and well studied for ordinary static, homogeneous dataflow graphs, and can be solved with polynomial time algorithms, they have not been studied in the context of RDFGs. Since RDFGs have an exponentially more compact representation, we determine the complexity of these two problems in terms of this lower representation size. We show that the problems are NP-complete, and hence, no advantage can be theoretically gained from the smaller input size. We develop some heuristics that should work well even if not technically in polynomial time with respect to the specification size, especially for large RDFGs.

Send comments to Praveen K. Murthy at praveen at cadence com .