Figure 611 Example of series parallel graph in .NET

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Figure 612 Example of bipartie graph
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2 Computation Costs This sub eld relates to the computation costs of the tasks
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Each node ni V is associated with a computation cost w(ni ) pi If symbol pi is present, the computation costs of the nodes are restricted in some form pi stands for processing requirement (time) of task i; hence, it corresponds directly to w(ni ) Typical restrictions are:
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pi = p Each node has the same computation cost, w(ni ) = w, ni V pi = 1 Each node has unit computation cost, that is, w(ni ) = 1, ni V This is also known as UET (unit execution time) If pi = {1, 2} the computation cost of each node is either 1 or 2
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3 Communication Costs This sub eld relates to the communication costs associated with the edges of the task graph No communication delays occur There are no weights associated with the edges of G; that is, G = (V, E, w) or, for G = (V, E, w, c), c(eij ) = 0 eij E c Each edge e E is associated with a communication cost c(e ) ij ij ij Restrictions might be speci ed for the values of c(eij ), for instance, as in the next point c p ij min Each edge has a communication cost that is smaller than or equal to the minimum computation cost of the nodes, c(eij ) minn V w(n), eij E In Chr tienne et al [34] this is referred to as SCT (small communication time) It implies minn V w(n)/maxeij E c(eij ) 1, which is of course the de nition of coarse granularity, minn V w(n)/ maxeij E c(eij ) = g(G) 1 (Section 443, De nition 420) Coarser granularity is speci ed with cij r pmin , where r is a constant and 0 < r 1 (designated r-SCT in Chr tienne et al [34]) c Each edge has the same communication cost, c(e ) = c, e E If the ij ij eld is c = 1, each edge has unit communication time (UCT ), that is, c(eij ) = 1, eij E 4 Task Duplication Task duplication is not allowed dup Task duplication is allowed
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Optimality Criterion The eld speci es the optimality criterion for the scheduling problem and there are no sub elds All scheduling problems discussed in this book are about nding the shortest schedule length sl, which is represented by the symbol Cmax in the eld Here Ci stands for the completion time of task ni ; that is, Ci = tf (ni ), with Cmax = maxni V Ci being the maximum completion time of all tasks, which is the de nition of the schedule length (De nition 410) Examples The following examples illustrate the utilization of the | | notation A short | | notation without any optional eld, P||Cmax , refers to the problem of scheduling independent tasks with arbitrary computation costs on |P| identical processors such that the schedule length is minimized P2|prec, pi = 1|Cmax is the
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problem of scheduling tasks with precedence constraints and unit computation costs on two identical processors such that the schedule length is minimized If the two processors are uniformly heterogeneous, this changes to Q2|prec, pi = 1|Cmax Finally, P |tree, cij pmin , dup|Cmax denotes the problem of scheduling a task graph with tree structure, arbitrary computation costs, and communication costs that are smaller than or equal to the minimum computation cost (ie, the graph is coarse grained) on an unlimited number of identical processors Node duplication is allowed and the objective is again to minimize the schedule length Generalizations and Reductions The theory of NP-completeness implies that a general problem cannot be easier, complexity-wise, than any of its special cases Many of the previously de ned sub elds restrict the parameters of the task scheduling in some way, making it a subproblem of a general one In particular, many speci c graph structures are listed, which are also related to each other A fork graph, for instance, is a special form of a tree If it is proved that a particular scheduling problem is NP-complete for tree structured task graphs, then the scheduling problem is also NPcomplete for an arbitrary graph structure In face of this observation, it is interesting to realize the relations of the different graph structures de ned for the eld Figure 613 visualizes these relations Each arrow points in the direction of the more general structure: that is, the graph at its tail is a special case of the graph at its head The above observation is re ected in the way NP-completeness of a problem is usually proved A common step in such proofs is to reduce a known NP-complete problem in polynomial time to that problem, as is done in all NP-completeness proofs of this text This is particularly easy if the known problem is a special case, as, for example, in Theorem 62 Thus, if NP-completeness is shown for a special case, the NP-completeness of all more general cases can be proved using such reductions As a consequence, Figure 613 can also be interpreted as a reduction graph (Brucker and Knust [29]), although not all task graph structures lead necessarily to an NP-complete scheduling problem
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