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Figure 718 Constructed schedule of nx , n0 , ni1 , ni2 , ni3 and ei1 , ei2 , ei3 on P0 , Pi and L0i
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to 3-PARTITION: let A1 , , Am (triplets) be m disjoint sets such that each Ai , i = 1, , m, contains exactly 3 elements of A, whose sum is B Nodes nx and n0 are allocated to the same processor, which shall be called P0 The remaining nodes n1 , , n3m are allocated in triplets to the processors P1 , , Pm Let ai1 , ai2 , ai3 be the elements of Ai The nodes ni1 , ni2 , ni3 , corresponding to the elements of triplet Ai , are allocated to processor Pi Their entering edges ei1 , ei2 , ei3 , respectively, are scheduled on L0i as early as possible in any order The resulting schedule is illustrated for P0 ,Pi and L0i in Figure 718 What is the length of this schedule The time to execute nx and n0 on P0 is w(nx ) + w(no ) = B + 2 = T On each link L0i the three edges corresponding to the triplet Ai take B time units for their communication (Figure 718) The rst communication starts as early as possible, that is, after 1 time unit when nx nishes After the last communication has nished, only one node remains to be executed on processor Pi ; the other two nodes are already executed during the communication (Figure 718) This is guaranteed by the fact that ai is a positive integer and w(ni ) = 1 for 1 i 3m The execution of this last node takes 1 time unit and, hence, each processor Pi nishes at 1 + B + 1 = T Thus, a feasible schedule S was derived, whose schedule length matches the time bound T ; hence, it is a solution to the constructed C-SCHED instance Conversely, assume that the instance of C-SCHED admits a solution, given by the feasible schedule S with sl(S) T It will now be shown that S is necessarily of the same kind as the schedule constructed previously Nodes nx and n0 must be scheduled on the same processor, say, P0 , since otherwise the communication e0 is remote and takes one time unit As a consequence, the earliest nish time of n0 would be w(nx ) + c(e0 ) + w(n0 ) = B + 3, which is larger than the bound T Since processor P0 is fully occupied with these two nodes until B + 2, all remaining nodes have to be executed on the other processors That means that all corresponding communications are remote
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Between the computation of nx and the computation of the last node on each processor (both take one time unit) exactly B time units are available for the communication on the links, in order to stay below the bound T = B + 2 The total communication time of all edges is 3m c(ei ) = mB Since there are m processors i=1 connected to P0 by m dedicated links, the available communication time B per link must be completely used Hence, each link must be fully occupied with the communication between time instances 1 (when nx nishes) and B + 1 (when the last node must start execution) The condition B/4 < ai < B/2 (ie, B/4 < c(ei ) < B/2) enforces that these B time units are occupied by exactly three edges; any other number of edges cannot have a total communication time of B This distribution of the edges on the links corresponds to a solution of 3-PARTITION | | Notation To characterize the problem of scheduling under the contention model with the | | classi cation (Section 641), the following extensions are proposed for the and elds The distinguishing aspect from the problems discussed in Section 64 is the contention awareness achieved through edge scheduling This is symbolized by the extension -sched in the eld, immediately after the speci cation of the communication costs (eg, cij -sched) Furthermore, the complexity of a problem also depends on the topology graph of the parallel system Hence, the eld is extended with a third sub eld that speci es the communication network, similar to the description of the precedence relations of the task graph:
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The processors of the parallel system are fully connected net The parallel system has a communication network, modeled by a topology graph TG = (N, P, D, H, b) snet The parallel system has a static communication network, modeled as an undirected graph UTG = (P, L) star The parallel system consists of a star network, where each processor is connected to a central switch via two counterdirected links (full duplex) This network corresponds to the one-port model (Beaumont et al [19]), as discussed in Section 711 star-h The parallel system consists of a star network, where each processor is connected to a central switch via an undirected edge (half duplex), as illustrated in Figure 77(b)
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With these extensions, the general problem of scheduling under the contention model is speci ed by P, net|cij -sched|Cmax for homogeneous processors Just before Theorem 71 and its proof, it was mentioned that scheduling with contention is intuitively more complicated than scheduling under the classic model The proof provides evidence It showed the NP-completeness of the special case P|fork, cij -sched|Cmax While the corresponding problem under the classic model, P|fork, cij |Cmax , is also NP-complete (Section 643), the scheduling of the particular
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