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the node ni V, Vcur to be scheduled next on processor procS (ni ) = P, it holds / that tdr,Sinsert,cur (ni , P) tdr,Send,cur (ni , P) tdr,S (ni , P), with the above argumentation With the end technique, ni is scheduled at the end of the last node already scheduled on P But this cannot be later than in S, because the nodes on P in Send,cur are the same nodes that are executed before ni on P in S, due to the schedule order of the nodes, and, according to the assumption, no node starts later than in S With the insertion technique, ni is scheduled in the worst case (ie, there is no idle period between two nodes already scheduled on P complying with Eq (61)) also at the end of the last node already scheduled on P Thus, ts,Sinsert,cur (ni , P) ts,Send,cur (ni , P) ts,S (ni , P) for node ni By induction this is true for all nodes of the schedule, in particular, the last node, which proves the theorem Theorem 51 of Section 51 establishes that an optimal schedule is de ned by the processor allocation and the nodes execution order List scheduling with the end technique can construct an optimal schedule from these inputs While this result is included in Theorem 61, it also establishes that the insertion technique might improve a given nonoptimal schedule Rescheduling a given schedule with list scheduling and the insertion technique, using the processor allocation and the node order of the original schedule, might improve the schedule length In particular, schedules produced with the end technique might be improved What rst sounds like a contradiction to Theorem 51, after all it states that the end technique is optimal given a processor allocation and the nodes execution order, becomes clear when one realizes that rescheduling with the insertion technique can only reduce the length of a schedule by reordering the nodes Inserting a node in an earlier slot changes the node order on the corresponding processor Complexity Regarding the complexity of list scheduling with the insertion technique, the second part of the corresponding Algorithm 9 is examined Determining the data ready time remains O(PE) for all nodes on all processors (see Section 51) What changes is the time complexity of the start time calculation In the worst case, it must be checked for every consecutive node pair on a processor, if the time period between them is large enough to accommodate the node to be scheduled At the time a node is scheduled, there are at most O(V) nodes scheduled on all processors So if the start time is determined for every processor as in the typical case of start time minimization this amortizes to O(max(V, P)), which is of course O(V), because it is meaningless to schedule on more processors than the task graph has nodes The start time is calculated for every node; thus, the nal complexity of the second part of simple list scheduling with start time minimization is O(V2 + PE) For comparison, with the end technique it is O(P( V + E)) (see Section 51) The insertion technique is employed n various scheduling heuristics, for example, ISH (insertion scheduling heuristic) by Kruatrachue [105], (E)CPFD ((economical) critical path fast duplication) by Ahmad and Kwok [4], BSA (bubble scheduling and allocation) by Kwok and Ahmad [114], and MD (mobility directed) by Wu and Gajski [207].

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