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Dynamic priorities are often a mixture of task graph based metrics and the earliest start time, whereby the task graph characteristics are only quanti ed at the beginning of the algorithm and remain unmodi ed For instance, the DLS algorithm (Sih and Lee [169]) de nes its dynamic level using blw and the node s earliest start time on all processors The ETF algorithm by Hwang et al [94] also selects the node processor pair with the earliest start time, breaking ties with blw 52 SCHEDULING WITH GIVEN PROCESSOR ALLOCATION At the de nition of a schedule (De nition 42) in Section 42, it was stated that scheduling is the spatial and temporal assignment of the tasks to the processors List scheduling does this assignment in an integrated process, that is, at the same time a node is assigned to a processor and attributed a start time In fact, when referring to scheduling, normally both the spatial and temporal assignments are meant Yet, it is a natural idea to split this process into its two fundamental aspects: processor allocation and attribution of start times When doing so, there are two alternatives: (1) rst processor allocation, then attribution of start times; and (2) rst attribution of start times, then processor allocation For a limited number of processors, the second alternative is extremely dif cult, because nodes with overlapping execution times have to be executed on different processors Furthermore, without knowledge of the processor allocation, it is not known which communications will be local and which remote Hence, adhering to the processor and precedence constraints (Conditions 41 and 42) would be very dif cult As a consequence, the two-phase scheduling starts with the allocation (or mapping) of the nodes to the processors Based on these considerations, the generic structure of a two-phase heuristic is shown in Algorithm 13 Algorithm 13 Generic Two-Phase Scheduling Algorithm (G = (V, E, w, c), P) Processor allocation/mapping (1) Assign each node n V to a processor P P Scheduling/ordering nodes (2) Attribute a start time ts (n) to each node n V, adhering to Condition 41 (processor constraint) and Condition 42 (precedence constraint) Next it is analyzed how list scheduling can e utilized to handle the second phase of this algorithm The processor allocation (ie, the rst phase) can be determined in many different ways One possibility is to extract it from another schedule, as is done, for example, in the proof of Theorem 51 This is especially interesting when a more sophisticated scheduling model is used in the second phase, like those discussed in s 7 and 8 With such an approach, the scheduling in the rst phase can be considered an estimate, which is re ned in the second phase A genetic algorithm based heuristic for nding a processor allocation is referenced in Section 842 More on processor allocation heuristics can be found in El-Rewini et al [65] and Rayward-Smith et al [159].

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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 hen 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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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 ctual 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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