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The computation length of a path p in G is the sum of the weights of its nodes: lenw ( p) =
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Note that the de nition of the path length in a task graph differs from the general De nition 32 in Section 31, where the length of a path equals the number of edges The path length len( p) can be interpreted as the time the path p takes for its execution if all communications between its nodes are interprocessor communications, which happens, for instance, when each node of p is allocated to a different processor Due to the sequential order inherent in the path, none of the nodes is executed concurrently with any other node The computation path length lenw ( p) can be interpreted as the execution time of the path p when all communications between its nodes are local, that is, they have zero costs Consequently, all nodes of p are executed on the same processor In a task graph G = (V, E, w) without communication costs, the path length is necessarily de ned as it is in Eq (421) When the processor allocations of the nodes of p are known, the path length can be determined taking into account that local communications are cost free In a scheduling algorithm it might be desirable to calculate a path length based on a partial schedule: that is, some processor allocations are given and others are not The path length determined for a given (partial) processor allocation A (De nition 41)
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is denoted by len( p, A), the allocated path length Communications between nodes whose processor allocations are unknown are assumed to be remote The allocated path length len( p, A) is thus in between the path length len( p) and the computation path length lenw ( p): len( p) len( p, A) lenw ( p) (422)
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The above scheme for the distinction between the different path lengths is used throughout this text All de nitions based on path lengths will only be formulated for len( p) but are implicitly valid for len( p, A) and lenw (p), too The corresponding de nitions use the same scheme for distinction: that is, the allocated path length is parameterized with A and the subscript w is used for the computation length 441 Critical Path An important concept for scheduling is the critical path the longest path in the task graph De nition 418 (Critical Path (CP)) A critical path cp of a task graph G = (V, E, w, c) is a longest path in G len(cp) = max{len( p)}
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The computation critical path cpw and the allocated critical path cp(A) for a processor allocation A are de ned correspondingly The nodes of a critical path cp, consisting of l nodes, are denoted by ncp,1 , ncp,2 , , ncp,l Clearly, there might be more than one critical path as several paths can have the same maximum length Note that in general cp = cpw = cp(A), from which follows for their path lengths len(cpw ) len(cp) and len(cp(A)) len(cp) (425) (424)
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Equivalent inequalities hold for the computation path length and the allocated path length and the corresponding critical paths Lemma 43 (Critical Path: From Source to Sink) Let G = (V, E, w, c) be a task graph A critical path cp of G always starts in a source node and nishes in a sink node of G
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Proof By contradiction: Suppose cp does not start in a source node Then, per de nition, the rst node of cp, here denoted by n1 , has at least one predecessor n0 pred(n1 ); hence, there is the edge e01 E A new path q can be concatenated from n0 , e01 , and cp, whose length is len(q) = w(n0 ) + c(e01 ) + len(cp) As w(n0 ) > 0, it follows that len(q) > len(cp) a contradiction Likewise for the sink node The critical path gains its importance for scheduling from the fact that its length is a lower bound for the schedule length Lemma 44 (Critical Path Bound on Schedule Length) Let G = (V, E, w, c) be a task graph and cpw a computation critical path of G For any schedule S of G on any system P, sl lenw (cpw ) (426) Proof Due to their precedence constraints (Condition 42), the nodes of cpw can only be executed in sequential order, which takes lenw (cpw ) time, independently of the schedule or the number of processors Thus, the duration of G s execution is at least lenw (cpw ) In the worst case that all communications among nodes are remote, sl len(cp), (427)
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yet Eq (426) is also ful lled For the special case of scheduling without communication costs on an unlimited number of processors (Section 432), the lower bound of the schedule length established by Eq (426) is tight In other words, the length of the optimal schedule is the length of the critical path Lemma 45 (Optimal sl on Unlimited Processors Cost-Free Communication) Let G = (V, E, w) be a task graph, cpw a computation critical path of G, and Pc0, a parallel system, with |Pc0, | |V| For an optimal length schedule Sopt of G on system Pc0, , sl(Sopt ) = lenw (cpw ) (428) Proof Theorem 44 establishes that Algorithm 6 produces an optimal schedule Sopt of G on Pc0, In this algorithm, the start time ts (n) of each node n is set to its DRT ts (n) = tdr (n) n V (429) By De nition 416, node n s DRT is the maximum of the nish times of its predecessor nodes (Eq (417)) Together with tf (n) = ts (n) + w(n) (Eq (42) of De nition 45), it holds that tdr (n) = max {tdr (ni } + w(ni )} (430)
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