Figure 46 Top and bottom levels and the corresponding paths

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44 TASK GRAPH PROPERTIES

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precedence constraints of G do not permit any earlier termination, independent of the number of involved processors The computation bottom level blw (n) re ects the best-case minimum time until the termination of G s execution, since the costs of communication are neglected, so they must all be local Likewise, a node cannot start earlier than at the time given by its top level tl(n) (once more, assuming that all communications are remote; tlw (n) corresponds to the case that all communications are local) Recall that the top level does not include the cost w(n) of n, while the bottom level does The above observations are formalized in the following lemma Lemma 46 (Level Bounds on Start Time) Let S be a schedule for task graph G = (V, E, w, c) on system P For n V, sl ts (n) + blw (n) ts (n) tlw (n) (434) (435)

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Proof Both Eqs (434) and (435) are obvious from the de nition of blw (n) and tlw (n) For Eq (434), due to precedence constraints, all descendant nodes of n must be executed after n (Condition 42) Their execution in precedence order takes at least blw (n) time, including the execution time of n For Eq (435), due to precedence constraints, all ancestors of n must be executed before n (Condition 42) Their execution in precedence order takes at least tlw (n) time Note that Lemma 46 relies on the computation levels, to re ect the best possible case A lemma based on the top and bottom levels needs the additional restriction that all communications among the ancestor and descendent nodes are remote Lemma 46, however, is always true The top path ptl(n) and the bottom path pbl(n) of a node n G together form a path ptb(n) = ptl(n) pbl(n) , (436)

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which is a longest path of G through n, starting in a source node and ending in a sink node This path is called a level path of n and its length is len( ptb(n) ) = tl(n) + bl(n) (437)

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Equation (437) holds for every node ni ptb(n) , ni V For the nodes of a critical path, this lemma follows Lemma 47 (Critical Path Length and Node Levels) Let G = (V, E, w, c) be a task graph For any node ncp,i of a critical path cp len(cp) = tl(ncp,i ) + bl(ncp,i ) (438)

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TASK SCHEDULING

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Proof By contradiction: Divide cp into two subpaths: cpt ending in ncp,i and cpb starting with ncp,i The length of cpt must be len(cpt ) = tl(ncp,i ) + w(ncp,i ): if it was shorter, cpt could be substituted with the top path ptl(ncp,i ) of n and the path concatenated from ptl(ncp,i ) and cpb would be longer than cp; if it was longer, tl(ncp,i ) would not be the top level of ncp,i , but instead len(cpt ) w(ncp,i ) The length of cpb must be len(cpb ) = bl(ncp,i ) with the analogous argumentation Then, len(cp) = len(cpt ) w(ncp,i ) + len(cpb ) = tl(ncp,i ) + bl(ncp,i ) With the argument of the above proof, it can also be seen that cp is a level path of each ncp,i For any source node nsrc G, bl(nsrc ) = len( ptb(nsrc ) ), (439)

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since by de nition tl(nsrc ) = 0 if pred(nsrc ) = Consequently, a source node with the highest bottom level of all nodes is the rst node ncp,1 of a critical path cp of G: bl(ncp,1 ) = len(cp) bl(ni ) ni V (440)

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As-Soon/Late-as-Possible Start Times As a result of Eq (435), the top level tl(n) of a node n is sometimes called the as-soon-as-possible (ASAP) start time of n, ASAP(n) = tl(n) (441)

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This property of the (computation) top level was already used in Theorem 44 It establishes that an optimal schedule on an unlimited number of processors for a task graph without communication costs can be constructed by starting each node as-soon-as-possible, Eq (418); hence at its computation top level The counterpart, the as-late-as-possible (ALAP) start time of a node n, directly correlates to the bottom level of n As stated by the inequality Eq (427), the critical path of a task graph is a lower bound for the schedule length A node n ought to start early enough so that it does not increase the schedule length beyond that bound Together with Eq (434) the ALAP start time of n is thus ALAP(n) = len(cp) bl(n) (442)

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Consequently, arranging nodes in decreasing bottom level order is equivalent to arranging them in increasing ALAP order For consistency, the notation of top and bottom levels will be used as a substitute for ASAP and ALAP, respectively, even when algorithms are described that were proposed with the latter notations

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