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Figure 317 The unrolled ow graph of Figure 316(a) for three iterations
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Figure 317 shows the unrolling example of Section 34, where the ow graph of Figure 316(a) is unrolled for three iterations (ie, N = 3) in the underlying code of Example 10 One clearly sees the three distinct kernel task graphs for the three iterations and the three interiteration communications There is no leaving edge from node U in the second iteration, since the potential destination node (in iteration 4) is not part of the computation The same holds for the interiteration communications of the nodes of the third iteration A graph constructed from a ow graph without the attribution of weights to the nodes and edges can be interpreted as the dependence graph re ecting only ow data dependences of the iterative computation This close relationship between task graph and DG was examined earlier More precisely, the DG obtained by unrolling a ow graph is the iteration DG of the computation, comprising only uniform dependence relations Sometimes the unrolling is done only for a fraction of the total number of iterations This partial unrolling, which is suf cient for some purposes (Sandnes and Megson [164], Yang and Fu [208]), has two advantages: (1) the total number of iterations does not need to be known and (2) the size of the unrolled graph, in terms of the number of nodes, is not proportional to the number of iterations However, the resulting graph remains a ow graph; it is not a task graph For this reason, partial unrolling is at times employed as a prestage to the extraction of the iterative kernel as described earlier (eg, Yang and Fu [208]) Projection Iteration DG to Flow Graph The countertechnique to unrolling is the projection of an iteration DG to a ow graph (Kung [109]) Multiple equal nodes of the iteration DG (ie, nodes that represent the same type of task) are projected into one node of the ow graph An illustrative example is the projection of a two-dimensional iteration DG into a ow graph, reducing the iteration DG by one dimension Figure 318 shows such a projection for the two-dimensional iteration DG of Figure 310 along the i-axis, resulting in the depicted ow graph Essentially, the projection is performed by merging all nodes along the projection direction into one node and by transforming the communication edges into new edges with delays A delay substitutes the spatial component of the distance vector of the
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35 TASK GRAPH (DAG)
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Figure 318 The iteration DG of Figure 310 is projected along the i-axis into a ow graph
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edge that is parallel to the direction of the projection In other words, a spatial dimension of the distance vectors is transformed into a temporal one In the example, the i-dimension is transformed into the temporal dimension of the delays The projection in the above example is linear along one of the axes of the iteration DG In general, the projection is not required to be along an axis, in fact, it is not even required to be linear (Kung [109]) However, an inherent iterative structure must be present in the DG; otherwise the computation cannot be described as a ow graph, so normally a general DG cannot be transformed into a ow graph A common application of projection is in VLSI array processor design (Kung [109]), where the iteration DG often serves as an initial model to obtain the ow graph, which is a description of the application closer to the hardware level The conversions and transformations demonstrate the close relationships of the various graph models To conclude and summarize the discussion of these techniques, the relationships are illustrated in Figure 319 Shown are the three major graph models DG, ow graph, and task graph linked by the conversions and transformations discussed in this section
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