Figure 715 Precedence constraint with edge scheduling

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74 CONTENTION AWARE SCHEDULING

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Figure 716 Contention aware scheduling: scheduling order of edges is relevant

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creation of the task graph It only differs for heterogeneous target systems; but this case is not addressed in the classic model, anyway Contention Awareness When an edge is scheduled along its route, it might con ict with other, already scheduled edges on one or more links As a result, the start of the edge is delayed on these links, which eventually results in a delayed total nish time of the edge Thus, the contention is exposed in the extended transfer time of the edge This delays the execution start of the destination node and can in the end result in a longer schedule For instance, in Figure 716, showing the integrated schedule of the depicted task graph and homogeneous target system, edge eAC is delayed on link L1 due to contention, because eAB already occupies L1 This also delays the start of node C on P3 , as the communication arrives later In the classic model, both communications are transferred at the same time so that node C starts at the same time as B A scheduling heuristic sees the contention effect through the node s later DRT on the processors to which communication is affected by contention An interesting consequence of edge scheduling is that the order in which edges are scheduled is relevant For example, the order of the outgoing edges of a node has an in uence on the DRTs of the node s successors This is desired, as it re ects the behavior of real systems If edge eAC in Figure 716 was scheduled before eAB , the start times between node B and C would be swapped that is, node C would start before node B 742 NP-Completeness Intuitively, scheduling is more complicated when considering contention, and in fact, the problem remains NP-complete (Dutot et al [57]) Theorem 71 (NP-Completeness Contention Model) Let G = (V, E, w, c) be a DAG and MTG = ((N, P, D, H, b), ) a parallel system The decision problem CSCHED (G, MTG ) associated with the scheduling problem is as follows Is there a schedule S for G on MTG with length sl(S) T , T Q+ C-SCHED (G, MTG ) is NP-complete in the strong sense

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COMMUNICATION CONTENTION IN SCHEDULING

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Proof First, it is argued that C-SCHED belongs to NP, then it is shown that C-SCHED is NP-hard by reducing the well-known NP-complete problem 3-PARTITION (Garey and Johnson [73]) in polynomial time to C-SCHED 3PARTITION is NP-complete in the strong sense The 3-PARTITION problem is as follows Given is a set A of 3m positive integer numbers ai and a positive integer bound B such that 3m ai = mB with B/4 < ai < i=1 B/2 for i = 1, , 3m Can A be partitioned into m disjoint sets A1 , , Am (triplets) such that each Ai , i = 1, , m, contains exactly 3 elements of A, whose sum is B Clearly, for any given solution S of C-SCHED it can be veri ed in polynomial time that S is feasible and sl(S) T ; hence, C-SCHED NP From an arbitrary instance of 3-PARTITION A = {a1 , a2 , , a3m }, an instance of C-SCHED is constructed in the following way Task Graph G The constructed task graph G is a fork graph as illustrated in Figure 717(a) It consists of one parent node nx and 3m + 1 child nodes n0 , n1 , , n3m There is an edge ei directed from nx to every child node ni , 0 i 3m The weights assigned to the nodes are w(nx ) = 1, w(n0 ) = B + 1, and w(ni ) = 1 for 1 i 3m The edge weights are c(e0 ) = 1 and c(ei ) = ai for 1 i 3m Target System MTG The constructed target system MTG = ((N, P, D, H, b), ), illustrated in Figure 717(b), consists of |P| = m + 1 identical, fully connected processors Each processor Pi is connected to each other processor Pj through a half duplex link Lij , which is represented by an undirected edge (a hyperedge incident on two vertices: Lij = Hij = {Pi , Pj }) Hence formally, N = P, D = , m+1 H= m i=1 j=i+1 Lij , and, as all processors and links are identical, (n, P) = w(n) P P and b(Lij ) = 1 Lij H Time Bound T The time bound is set to T = B + 2 Clearly, the construction of the instance of C-SCHED is polynomial in the size of the instance of 3-PARTITION It is now shown how a schedule S is derived for C-SCHED from an arbitrary instance of 3-PARTITION A = {a1 , a2 , , a3m }, which admits a solution

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Figure 717 The constructed fork DAG (a) and target system (b)

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