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Algorithm A linear clustering can be achieved with a simple and intuitive algorithm proposed by Kim and Browne [104] Consider as a basis the generic clustering Algorithm 15: in each re nement step a new longest path p, consisting of previously unexamined edges, is selected in task graph G The nodes of p are merged into one cluster and the edges incident on these nodes are marked examined This is repeated until all edges of G have been examined Clearly, such an algorithm belongs to the path type in terms of the multiplicity of edge zeroing Algorithm 16 outlines the above described procedure Eunex E is the set of unexamined edges and Gunex = (Vunex , Eunex ) is the task graph spawned by these edges and the nodes Vunex V, which are incident on these edges Hence, each Gunex is a subgraph of G In each step the critical path cp of the current unexamined graph Gunex is determined and its edges are merged into one cluster Observe that all edges incident on the nodes of cp, {eij Eunex : ni Vcp nj Vcp }, are marked as examined at the end of each step, and not only those edges eij Ecp that are part of cp This is important, because otherwise, in a later step, one of the edges eij Ecp / not being part of cp could be zeroed This would mean that the edge s two incident nodes ni and nj , of which at least one is part of a cluster with more than one node, are merged into the same cluster, potentially resulting in a nonlinear cluster Algorithm 16 Linear Clustering Algorithm (G = (V, E, w, c)) Create initial clustering C0 : allocate each node n V to a distinct cluster C C, |C| = |V| Eunex E Eunex : set of unexamined edges, Eunex E while Eunex = do there is at least one unexamined edge Vunex : set of nodes Find a critical path cp of graph Gunex = (Vunex , Eunex ) n V on which edges Eunex are incident Merge all nodes Vcp of cp into one cluster Mark edges that are Eunex Eunex {eij Eunex : ni Vcp nj Vcp } incident on nodes of Vcp as examined end while In comparison to the general structure of a clustering algorithm, as outlined in Algorithm 15, there is no check of whether the schedule length of the new clustering is not worse than the current one The check is unnecessary, as Lemma 52 guarantees that this is the case Finding the critical path of the task graph Gunex in each step is essentially the determination of the nodes top and bottom levels This can be performed using the algorithms presented in Section 442, with a complexity of O(V + E) Merging the nodes and marking the corresponding nodes and edges as examined has lower complexity than O(V + E) As there are at most O(V) steps, Algorithm 16 s total complexity is thus O(V(V + E)) As a last comment on linear clustering, note the similarity between linear projections of dependence graphs as analyzed in Section 351 and linear clustering The linear projection of DGs for example, is used in VLSI array processor design (Kung [109])
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FUNDAMENTAL HEURISTICS
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Figure 55 Linear clusterings of a simple task graph (Figure 53(a)): (a) initial clustering; (b) clustering after rst step; (c) nal clustering; (d) schedule of nal clustering (Examined elements are dotted)
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Example As an example, the linear clustering algorithm is applied to the task graph of Figure 53(a) The usual initial clustering is depicted in Figure 55(a)
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All edges are unexamined at this stage and the critical path is cp = a, b, g with a length of len(cp) = 28 These nodes are merged into one cluster and they are marked examined as are their incident edges, eab , ebg , eac , efg The resulting clustering is illustrated in Figure 55(b), where the examined nodes and edges are drawn with dotted lines The current unexamined graph Gunex consists of the nodes Vunex = {c, d, e, f } and the edges Eunex = {ecd , ece , edf , eef } Its critical path is cp = c, d, f with a length of len(cp) = 19 These nodes are merged into one cluster and they are marked examined as are their incident edges, ecd , ece , edf , eef The resulting clustering is illustrated in Figure 55(c) There are no unexamined edges left, Eunex = , and the algorithm terminates Therefore, the nal clustering C nal is the one given in Figure 55(c), with a schedule length of sl(C nal ) = len(cp(C nal ), C nal ) = len( a, c, e, f , g , C nal ) = 25 The resulting implicit schedule is displayed in Figure 55(d)
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533 Single Edge Clustering The second approach to clustering discussed here considers one single edge at a time for zeroing A simple algorithm based on this approach was proposed by Sarkar
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