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hence using the end technique Thus, Eq (55) is an implementation of lines 5 and 6 of the simple list scheduling (Algorithm 9) or the dynamic list scheduling (Algorithm 11) As the selection of the processor is not performed anymore in these algorithms, the complexity of the second part of list scheduling reduces by the factor |P| The second part of simple list scheduling is then O(V + E) From Theorem 51 it is known that the end technique is optimal for a given node order and processor allocation As a result, scheduling with a given processor allocation reduces to nding the best node order Both static and dynamic priorities can be employed to order the nodes for their scheduling However, since the processor allocations are already determined, task graph characteristics, like node levels and the critical path, can be computed using the allocated path length (Section 44), that is, the path length based on the known processor allocations These characteristics do not change during the entire scheduling; hence, dynamic priorities are only sensible when considering the state of the partial schedules, for example, choosing the node among the free nodes that can start earliest, that is, the ready node (Section 513) One might wonder whether this scheduling problem with a given preallocation is still NP-hard After all, it is only about nding the best node order Unfortunately it is still NP-hard, even for task graphs without communication costs, unit execution time, and very simple graph structures, such as forest (Goyal [79]) or chains (Rayward-Smith et al [159]); see also Hoogeveen et al [91] 53 CLUSTERING As mentioned before in Section 422, task scheduling under the classic model is a trade-off between minimizing interprocessor communication costs and maximizing the concurrency of the task execution A natural idea is therefore to determine rst before the actual scheduling which nodes should always be executed on the same processor Obvious candidates for grouping are nodes that depend on each other, especially nodes of the critical path Clustering is a technique that follows this idea It is therefore only suitable for scheduling with communication costs In its core it is a scheduling technique.



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Operating system · Linux, Windows, OS/2 · Type · Optical character recognition · License · GNU General Public License. Website, www-e.uni-magdeburg.de/​jschulen/ocr/ jocr.sourceforge.net(may out of date). GOCR (or JOCR) is a free optical character recognition program, initially written by Jörg ...

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for an unlimited number of processors Nevertheless, it is often proposed as an initial step in scheduling for a limited number of processors To distinguish between the limited number of (physical) processors and the unlimited number of (virtual) processors assumed in the clustering step, the latter are called clusters, hence the term clustering In clustering, the nodes of the task graph are mapped and scheduled into these clusters De nition 53 (Clustering) Let G = (V, E, w, c) be a task graph A clustering C is a schedule of G on an implicit parallel system C (De nition 43) with an unlimited number of processors; that is, |C| = |V| The processors C C are called clusters Clustering based scheduling algorithms for a limited number of processors consist of several steps, similar to the two-phase scheduling outlined by Algorithm 13 in Section 52 Algorithm 13 comprises two phases, where the rst phase is the mapping of the nodes and the second is their scheduling In clustering based algorithms, three steps are necessary: (1) clustering, (2) mapping of the clusters to the (physical) processors, and (3) scheduling of the nodes Algorithm 14 outlines a generic three-step clustering based scheduling heuristic for a limited number of processors Algorithm 14 Generic Three-Step Clustering Based Scheduling Algorithm (G = (V, E, w, c), P) Clustering nodes (1) Find a clustering C of G Mapping clusters to processors (2) Assign clusters of C to (physical) processors of P Scheduling/ordering nodes (3) Attribute start time ts (n) to each node n V, adhering to Condition 41 (processor constraint) and Condition 42 (precedence constraint) In contrast to the two-phase Algorithm 13, where the rst step is the pure processor allocation of the nodes, clustering also includes the scheduling of the nodes in the clusters As will be seen later, this is done for an accurate estimation of the execution time of the task graph It also makes clustering a complete scheduling algorithm for an unlimited number of processors While the third step is theoretically identical to the second phase of Algorithm 13, its actual implementation may differ, since the partial node orders as established by the clustering C might be considered in determining the nal node order In terms of parallel programming terminology, clustering correlates to the step of parallelization designated by orchestration (Culler and Singh [48]) or agglomeration (Foster [69]) as described in Section 23 The often cited motivation for clustering was given by Sarkar [167]: if tasks are best executed in the same processor (cluster) of an ideal system, that is, a system that possesses more processors (clusters) than tasks, they should also be executed on the same processor in any real system Due to the NP-hardness of scheduling, it cannot be.

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(Editor), Nirmal Chidambaram, Shashikiran Guruprasad, Alex Homer, Ric Howell, Stephan Kasippillai, Rob Machin, Tom Myers, Alexander Nakhimovsky, Luca Passani, Chris Pedley, Richard Taylor, Marco Toschi MAILING LISTS

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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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