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length is 3k (Figure 52(c)) For any C a k with k > 1, k > C can be chosen so that sl(S) = 3k > 3C = C sl(Sopt ) In particular, list scheduling with start time minimization does not even guarantee a schedule length shorter than the sequential time of a task graph Still, numerous experiments have demonstrated that list scheduling produces good schedules for task graphs with low to medium communication (eg, Khan et al [103], Kwok and Ahmad [111], Sinnen and Sousa [177]) 512 With Dynamic Priorities The imple list scheduling outlined in Algorithm 9 establishes the schedule order of the nodes before the actual scheduling process During the node scheduling in the second part, this order remains unaltered, so the node priorities are static To achieve better schedules, it might be bene cial to consider the state of the partial schedule in the schedule order of the nodes Changing the order of the nodes is equivalent to changing their priorities, hence the node priorities are dynamic It remains that the node order must be compatible with the precedence constraints of the task graph, which is obeyed if only free nodes are scheduled A list scheduling algorithm for dynamic node priorities is given in Algorithm 11 Algorithm 11 List Scheduling Dynamic Priorities (G = (V, E, w, c), P) 1: Put source nodes {ni V : pred(ni ) = } into set S 2: while S = do S only contains free nodes 3: Calculate priorities for nodes ni S 4: Choose a node n from S; S S n 5: Choose a processor P for n 6: Schedule n on P 7: S S {ni succ(n) : pred(ni ) Scur } 8: end while Instead of iterating over the node list, Algorithm 11 calculates in each step the priority of all free nodes and only then selects a node and a processor for scheduling After the scheduling of node n, the set S of free nodes is updated with the nodes that became free by the scheduling of n (line 7) Obviously, this approach has higher complexity in general than the simple list scheduling presented in Algorithm 9, because the priority of every node is determined several times Algorithm 11 is a generalization of the simple list scheduling, which becomes apparent by considering line 3 If this line is moved to the beginning of the algorithm and the priorities are calculated for all nodes, not only for the free nodes, the simple list scheduling is obtained This becomes even more distinct in Section 513, where the construction of node lists is explained Choosing Node Processor Pair Some algorithms dynamically compute the priorities of the free nodes by evaluating the start time (see Eq (51)) for every free.

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Dynamic priorities are often a mixture of task graph based metrics and the earliest start time, whereby the task graph characteristics are only quanti ed at the beginning of the algorithm and remain unmodi ed For instance, the DLS algorithm (Sih and Lee [169]) de nes its dynamic level using lw and the node s earliest start time on all processors The ETF algorithm by Hwang et al [94] also selects the node processor pair with the earliest start time, breaking ties with blw 52 SCHEDULING WITH GIVEN PROCESSOR ALLOCATION At the de nition of a schedule (De nition 42) in Section 42, it was stated that scheduling is the spatial and temporal assignment of the tasks to the processors List scheduling does this assignment in an integrated process, that is, at the same time a node is assigned to a processor and attributed a start time In fact, when referring to scheduling, normally both the spatial and temporal assignments are meant Yet, it is a natural idea to split this process into its two fundamental aspects: processor allocation and attribution of start times When doing so, there are two alternatives: (1) rst processor allocation, then attribution of start times; and (2) rst attribution of start times, then processor allocation For a limited number of processors, the second alternative is extremely dif cult, because nodes with overlapping execution times have to be executed on different processors Furthermore, without knowledge of the processor allocation, it is not known which communications will be local and which remote Hence, adhering to the processor and precedence constraints (Conditions 41 and 42) would be very dif cult As a consequence, the two-phase scheduling starts with the allocation (or mapping) of the nodes to the processors Based on these considerations, the generic structure of a two-phase heuristic is shown in Algorithm 13 Algorithm 13 Generic Two-Phase Scheduling Algorithm (G = (V, E, w, c), P) Processor allocation/mapping (1) Assign each node n V to a processor P P Scheduling/ordering nodes (2) Attribute a start time ts (n) to each node n V, adhering to Condition 41 (processor constraint) and Condition 42 (precedence constraint) Next it is analyzed how list scheduling can be utilized to handle the second phase of this algorithm The processor allocation (ie, the rst phase) can be determined in many different ways One possibility is to extract it from another schedule, as is done, for example, in the proof of Theorem 51 This is especially interesting when a more sophisticated scheduling model is used in the second phase, like those discussed in s 7 and 8 With such an approach, the scheduling in the rst phase can be considered an estimate, which is re ned in the second phase A genetic algorithm based heuristic for nding a processor allocation is referenced in Section 842 More on processor allocation heuristics can be found in El-Rewini et al [65] and Rayward-Smith et al [159].

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