2 Number of Processors

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Task Characteristics This eld speci es the characteristics of the tasks and it can have various sub elds, separated by a comma, all of which are optional You might nd different orders of these sub elds in other texts, as the order is not consistent across the literature

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1 Precedence Relations This sub eld describes the precedence relations between tasks In other words, it speci es the structure of the task graph G In can take the following values:

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The tasks are independent; that is, E = prec There are precedence constraints between tasks; hence, |E| 1 {outtree, intree, tree} The task graph G is a rooted tree (tree) A rooted tree is a graph with either an in-degree of at most one for each node n V (outtree, eg, Figure 66(a)) or an out-degree of at most one for each node n V (intree, eg, Figure 66(b)) Note that this de nition of tree is general in that a tree can have multiple root nodes, which is usually referred to as forest (Cormen et al [42]) op-forest The task graph G is an opposing forest, consisting of intrees and outtrees { fork, join} The task graph G is a fork or join graph A fork ( join) graph is an outtree (intree) with one source (sink) node nroot , the root node, where all other nodes are successors (predecessors) of nroot , for example, Figure 67 chains The task graph G is a tree, where all nodes have an out-degree and an in-degree of at most one (eg, Figure 68) That is, the task graph consists of disjoint chains of nodes Note that it is denoted by chains not chain, as most scheduling problems are trivial when there is only one chain of nodes

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Figure 66 Examples of (a) outtree and (b) intree

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Figure 67 Examples of (a) fork graph and (b) join graph

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Figure 68 Example of chains graph

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{harpoon, in-harpoon, out-harpoon} The task graph G is a tree, with a harpoon-like structure Such a graph is very similar to a fork or join graph It consists of one root node nroot and x chains In an out-harpoon there is an edge from nroot to each source node of the x chains, while in an in-harpoon there is an edge from each sink node of the x chains to nroot Figure 69 illustrates an out-harpoon, where each chain consists of two nodes fork-join The task graph G is a fork-join graph In a fork-join graph there is one source node ns and one sink node nt All other nodes are successors of ns and predecessors of nt and are independent of each other, for example, in Figure 610 sp-graph The task graph G is a series parallel graph A series parallel graph can be constructed recursively using three basic graphs: (1) a graph consisting of a single node; (2) a single chain graph, that is, V = {n1 , n2 , , nl } and E = l 1 ei,i+1 (the series graph); and (3) a fork-join graph (the parallel i=1 graph), see above Each of these graphs is a series parallel graph in itself Let G1 = (V1 , E1 ) and G2 = (V2 , E2 ) be two series parallel graphs, with ns and

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Figure 69 Example of out-harpoon graph

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Figure 610 Example of fork-join graph

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nt being the sink and source nodes of G2 , respectively A new series parallel graph G can be constructed from G1 by substituting any node ni V1 with the complete G2 That is, ni and all edges that are incident on ni are removed from G1 New edges are created from the predecessors of ni , pred(ni ), to ns and from nt to the successors of ni , succ(ni ) Note that any series parallel graph has exactly one source node and one sink node An example for a series parallel graph is depicted in Figure 611 bipartie The task graph G is a bipartie graph That means V can be partitioned into two subsets V1 and V2 such that eij E implies ni V1 and nj V2 In other words, all edges go from the nodes of V1 to the nodes of V2 (eg, Figure 612) int-ordered The task graph G is interval-ordered Let each node n V be mapped to an interval [l(n), r(n)] on the real line A task graph is said to be interval-ordered if and only if there exists a node-to-interval mapping with the following property for any two nodes ni , nj V: r(ni ) l(nj ) nj desc(ni ) (611)

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This means that the intervals of any two nodes do not overlap if and only if one node is the descendant of the other (El-Rewini et al [65], Kwok and Ahmad [113])

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