Scheduling Tree-Dags Using FIFO Queues: A Control–Memory Trade-Off

We study a combinatorial problem that is motivated by “client–server” schedulers for parallel computations. Such schedulers are often used, for instance, when computations are being done by a cooperating network of workstations. Our results expose and quantify a control–memory trade-off for such schedulers, when the computation being scheduled has the structure of a binary tree, with all arcs oriented either root-toward-leaves or leaves-toward-root. The combinatorial problem for the root-toward-leaves case takes the following form. (The leaves-toward-root case gives rise to a dual formulation, which yields the same trade-offs.) Consider, for integersk,N> 0, an algorithm that employskFIFO queues in order to schedule anN-leaf binary tree in such a way that each nonleaf node of the tree is executed before its children. We establish a trade-off between the number of queues used by the algorithm—which we view as measuring thecontrol complexityof the algorithm—and thememory requirementsof the algorithm, as embodied in the required capacity of the largest-capacity queue. Specifically, for each integerk∈ {1, 2, ..., log2N}, letQk(N) denote the minimax per-queue capacity for ak-queue algorithm that schedules allN-leaf binary trees; letQ*k(N) denote the analogous quantity forcompletebinary trees. We establish the following bounds: For generalN-leaf binary trees, for allk, [formula]≤Qk(N) ≤ 2N1/k+ 1. For complete binary trees, we derive tighter bounds. We prove that for all constantk, Q*k(N) = Θ[formula]. For generalk, we obtain the following bounds: [formula]≤Q*k(N) ≤ (4k)1−1/k[formula]. Similar trade-offs are readily established for trees of any fixed branching factor.