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Unformatted text preview: fer data between processors, and assume the processors
are powered down when they are not active.)
There is a set of precedence constraints for the tasks, which is a set of m ordered pairs P ⊆
{1, . . . , n} × {1, . . . , n}. If (i, j ) ∈ P , then task j cannot start until task i ﬁnishes. (This would be
the case, for example, if task j requires data that is computed in task i.) When (i, j ) ∈ P , we refer
to task i as a precedent of task j , since it must precede task j . We assume that the precedence
constraints deﬁne a directed acyclic graph (DAG), with an edge from i to j if (i, j ) ∈ P .
If a task has no precedents, then it starts at time t = 0. Otherwise, each task starts as soon as all
of its precedents have ﬁnished. We let T denote the time for all tasks to be completed. To be sure the precedence constraints are clear, we consider the very small example shown below,
with n = 6 tasks and m = 6 precedence constraints.
P = {(1, 4), (1, 3), (2, 3), (3, 6), (4, 6), (5, 6)}.
5 1 4...
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 Fall '13
 F.Borrelli
 The Aeneid

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