44 TASK GRAPH PROPERTIES

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By induction it is clear that node n s DRT is a sum of ancestor node weights As per De nition 417, this is the computation length lenw of a path composed of ancestor nodes of n In particular, for the node with the maximum nish time nlast , that is, sl(Sopt ) = tf (nlast ) (De nition 410), it holds that tf (nlast ) = tdr (nlast ) + w(nlast ) Thus, tf (nlast ) is the length of a path ending with node nlast Per de nition of the computation critical path, this path cannot be longer than the critical path, tf (nlast ) len(cpw ) With Eq (426), it follows that sl(Sopt ) = len(cpw ) The critical path of the sample task graph (eg, in Figure 41) is cp = a, b, f , j, k with length len(cp) = 34, while there are two computation critical paths cpw , a, b, f , j, k and a, d, h, k , with computation length lenw (cpw ) = 14

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442 Node Levels The critical path identi es the nodes of a task graph, whose constrained sequential execution takes at least the execution time of any other path in the task graph This makes the nodes of the critical path important, as a late execution start of any of these nodes directly results in an extended schedule length The scheduling of a less important node, that is, a node not belonging to the critical path, is not that important, as long as it does not delay the execution of any critical path node But how can one differentiate between nodes not belonging to the critical path In general, some nodes are more important than others This problem is tackled by the notion of node levels, which is a natural extention of the critical path concept on a node-speci c basis Given a node n V, in general, paths exist in G that end in n, that is, whose last node is n, and that start with n, that is, whose rst node is n (see also the notion of ancestors and descendants in De nition 34, Section 31) Analogous to the critical path, the longest path in each of the two sets of paths can be distinguished A node level is then the length of the longest path De nition 419 (Bottom and Top Levels) Let G = (V, E, w, c) be a task graph and n V

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Bottom Level The bottom level bl(n) of n is the length of the longest path starting with n: bl(n) = max {len( p(n ni ))} (431)

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ni desc(n) sink(G)

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If desc(n) = , then bl(n) = w(n) A path starting with n of length bl(n) is called a bottom path of n and denoted by pbl(n)

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TASK SCHEDULING

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Top Level The top level tl(n) of n is the length of the longest path ending in n, excluding w(n): tl(n) =

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ni ance(n) source(G)

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{len( p(ni n))} w(n)

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(432)

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If ance(n) = , then tl(n) = 0 A path ending in n of length tl(n) + w(n) is called a top path of n and denoted by ptl(n) Considering the proof of Lemma 43, it is evident that the longest path starting with a node n must always terminate in a sink node (hence, ni desc(n) sink(G) in Eq (431)) and the longest path terminating in n must always start in a source node (hence, ni ance(n) source(G) in Eq (432)) This fact gave name to the top level and the bottom level, as the longest paths start at the top of the task graph and end at its bottom Figure 46 illustrates the top and bottom levels of a node n with the corresponding paths in a task graph with several source and sink nodes It should be noted that the paths corresponding to the computation node levels computation bottom level blw (n) and computation top level tlw (n) are not identical to the paths of the levels de ned earlier In general, pbl(n) = pblw (n) and ptl(n) = ptlw (n) (433)

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The signi cance of the node levels becomes apparent by the following considerations At the time the execution of a node n G is initiated, the minimum remaining time until the execution of G is completed is given by n s bottom level bl(n) (assuming the worst case where all communications are interprocessor communications) The

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