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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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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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Processor Allocation Being a value encoded chromosome, the mutation operator can simply change the values of randomly picked genes This can be done with any procedure, like increasing or decreasing the current values, or by assigning a randomly determined value The only condition is that the resulting values are in the range 1 |P| Figure 622 illustrates such a simple mutation operator, where two genes are mutated Another alternative is to swap the values of two randomly picked genes as visualized in Figure 623 In terms of processor allocation, this has the effect that the number of nodes on ach processor remains constant While this is good for. Figure 622 Illustration of simple mutation operator for rocessor allocation chromosome . Generation In Visual C# Using Barcode encoder for .NET .Related: 

This being so, at the hit point the arm has a simple choice: to alk along the virtual boundary in one or the opposite direction along the curve. Since no information is available beforehand as to which of the two directions is better, one direction, called the local direction, will be chosen once and for all. While following the obstacle virtual boundary, the arm may meet the M-line again. If it does, and if this occurs at a distance (measured appropriately along the M-line) from point T shorter than the distance from the latest hit point Hj to T , the arm will de ne a leave point, Lj . Hit and leave points will play an important role in the path planning procedure. We will see below that these points come in pairs, (Hj , Lj ), j = 1, 2, . . . . For convenience, denote Lo = S, Start, with no corresponding Ho . The motion planning algorithm proper, the proof of its convergence, and the test for target reachability will emerge from our analysis of the described scheme and of C-space properties. Similar to the mobile robot case ( 3), under our scheme the arm will need no beforehand information about the obstacles in order to move properly. The C-space presentation is used primarily for the analysis, the algorithm development, and the proof of convergence. No explicit mapping of any kind from W -space to C-space and no explicit calculation of C-space will ever take place before or during the actual arm motion. Generating International Standard Serial Number In .NET Using . In Visual Basic .NET Using Barcode encoder for .NET .Related: 

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