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expected that the foregoing conjecture is always true, yet it is an intuitive reasoning for a heuristic In the following, the rst step of Algorithm 14, the actual clustering, is studied Steps 2 and 3 are treated in Section 54
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531 Clustering Algorithms Strictly speaking, the rst step of Algorithm 14 can be performed by any scheduling algorithm suitable for an unlimited number of processors However, in the literature the term clustering designates a certain kind of algorithm, with several characteristic aspects The following discussion is based on Darte et al [52], El-Rewini et al [65], and Gerasoulis and Yang [76] Principle of Clustering Algorithms Clustering algorithms start with an initial clustering C0 of the task graph G Usually each node n V is allocated to a distinct cluster C C The clustering algorithm then performs incremental steps of re nement going from clustering Ci 1 to clustering Ci , in which clusters are merged This means that the nodes of these clusters are merged into one single cluster If communicating nodes are executed in the same cluster, their communication becomes local and hence its cost is zero according to the target system model (De nition 43) This can be bene cial for the total execution time of the graph, as it eliminates communication costs Normally, a merging of clusters is performed only if the schedule length of the new clustering Ci decreases or at least remains the same compared to the schedule length of the current clustering Ci 1 The steps of re nement are performed until all candidates for merging have been considered Algorithm 15 summarizes this principle of clustering algorithms Algorithm 15 Principle of Clustering Algorithms (G = (V, E, w, c)) Create initial clustering C0 : allocate each node n V to a distinct cluster C C, |C| = |V| i 0 repeat i i+1 Select candidate clusters for merging Create new clustering Ci : merge candidate clusters into one cluster if sl(Ci ) > sl(Ci 1 ) then merging increases current schedule length reject new clustering Ci Ci Ci 1 end if until all candidates for merging have been considered Figure 53 shows some example clusterings of a simple task graph (Figure 53(a)), which has often been used to illustrate clustering (Gerasoulis andYang [76]) An initial clustering is depicted in Figure 53(b) Each node is allocated to a distinct cluster,
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FUNDAMENTAL HEURISTICS
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Figure 53 Clusterings of a simple task graph: (a) simple task graph for clustering algorithms (Gerasoulis and Yang [76]); (b) initial clustering; (c) clustering after clusters of nodes a and b have been merged; (d) clustering with only two clusters
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symbolized by the gray dashed line enclosing each node Figure 53(c) depicts the clustering, where the clusters of nodes a and b of the initial clustering have been merged into one cluster This might be the clustering produced by an algorithm after the rst re nement step Lastly, Figure 53(d) illustrates a clustering with only two clusters Implicit Schedule An essential characteristic of clustering algorithms is that the clustering obtained at each step is a feasible schedule Otherwise it would not be possible to determine the current schedule length, on which clustering decisions are based In clustering algorithms, the scheduling of the nodes is often given implicitly by the allocation of the nodes to the clusters The rst example for this is the initial clustering, where each node starts as-soon-as-possible, that is, at its top level (Section 442; see also Section 432): ts (n) = tl(n) n V (56)
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The resulting schedule is shown in Figure 54(a), which is the implicit schedule of the clustering in Figure 53(b) In general, the starting times of the nodes (ie, their scheduling) are implicitly given whenever the node order within each cluster is well de ned This is the case when there are no independent nodes in the same cluster If this criterion is met, the start time of each node in clustering Ci is its allocated top level (Section 44): ts (n) = tl(n, Ci ) n V (57)
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Figure 54(b) displays the implicit schedule for the clustering of Figure 53(c) It is not clear at this point what the implicit schedule of the clustering shown in
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