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62 NODE DUPLICATION A major challenge in scheduling is to avoid interprocessor communications A node often has to wait for its entering communications, while its executing processor runs idle For instance, in the schedule for the sample task graph visualized by Figure 61(a), nodes c and d are delayed due to the communication from node a and both processors P2 and P3 run idle during that time One is tempted to move node a, for example, to P2 in order to eliminate the cost of communication eac However, this move creates communication costs between nodes a and b, as illustrated by the partial schedule in Figure 63(a) A solution that has been exploited to reduce communication costs, while avoiding the above described problem, is node duplication In this approach, some nodes of a task graph are allocated to more than one processor of the target system For the previous example, node a is scheduled three times, once on each processor as shown in Figure 63(b) The communications from node a are now local on each processor of the target system and nodes b, c, d can start immediately after a nishes Since nodes c and d start earlier, subsequent nodes can also start earlier, potentially leading to a shorter schedule length Figure 63(c) displays a schedule similar to that of Figure 41, yet with the duplication of nodes a and j The schedule length is reduced from 24 to 22 time units But the potential of duplication is even higher, as demonstrated by the schedule in Figure 63(d), where the duplication of a can reduce the schedule length to 18 time units For node duplication, some formal changes must be carried out for the de nitions and conditions of Section 41 The de nition of a schedule (De nition 42) changes as follows
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Figure 63 Partial (a), (b) and full schedules (c), (d) of the sample task graph (eg, in Figure 61); the schedules of (b), (c), and (d) use node duplication
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The function proc(n) for the processor allocation of node n becomes a subset of P, denoted by proc(n), because with duplication a node can be allocated to more than one processor Of course, proc(n) P and |proc(n)| 1 The function ts (n) for the start time assignment to node n is ambiguous with node duplication Only with speci cation of a processor is the start time of a node de ned Thus, it is now obligatory to write ts (n, P) (see Eq (41)) to denote the start time of n on P; ts becomes a function of the nodes and the processors, ts : V P Q+ 0
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Node duplication also has an impact on Condition 42 regarding the precedence constraints of the task graph Condition 62 (Precedence Constraint Node Duplication) For a schedule Sdup with node duplication, Eq (45) of Condition 42 becomes ts (nj , P)
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Given the communication eij , node nj cannot start until at least one instance of the duplicated nodes of ni has provided the communication eij In the schedule of Figure 63(c), for example, the communication ejk is received from the duplicated node j on processor P2 ; the same communication from the instance of node j on P1 does not arrive on time on P2 for the start of k at ts (k, P2 ) = 20 (tf (ejk , P1 , P2 ) = tf ( j, P1 ) + c(ejk ) = 16 + 6 = 22) Following from this altered precedence constraint condition, the de nition of the data ready time (De nition 48) must be adapted De nition 62 (Data Ready Time (DRT) Node Duplication) For a schedule Sdup with node duplication, Eq (46) of De nition 48 becomes tdr (nj , P) =
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The rest of the de nitions and conditions of Section 41 remain unmodi ed, except for the necessary formal adaptation for the altered de nition of a schedule (see above) Even though all forgoing conditions and de nitions were again formulated in a generic form and could be applied to scheduling without communication costs, it makes no sense to use node duplication when communication has no costs Node Order Anomaly A consequence of the modi ed precedence constraint Condition 62 is a possible anomaly in the order of nodes in a schedule Consider the schedule of a small task graph in Figure 64, where node B is duplicated Looking at the node order on P1 , one observes that node B starts after node C, even though C depends on B This is valid according to Condition 62, because node B is a duplicated node and the communication eBC is provided by the instance of B on P2 Hence,
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