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2 ,, P P Approximate Scheduling in Polynomial Time For theoretical investigations, an unbounded number of processors may be assumed. Practical studies emphasize bounds on the number of processors. List scheduling techniques dominate the research on heuristics for task scheduling. These techniques order (e.g. assign priorities to) the tasks and then apply a priority queue:. Place all tasks without predecessors in the PQ.. As long as the PQ contains a task do: a. Assign the task with highest priority to an available processor (possibly to reduce communication) at the time when all predecessor tasks must have completed. b. If the scheduled task is the last immediate predecessor scheduled for any tasks, then place those tasks on the priority queue. The approach to assigning priorities is what distinguishes the many variations. The following example introduces some graph-based definitions:

3 node number execution time node sl t-level b-level ALAP sl(i) = maximum execution time of a path including node i through some exit node (static level) t-level(i) = maximum execution + communication time for a path from an entrance node up to node i (top level) b-level(i) = maximum execution + communication time for a path including node i through some exit node (bottom level) ALAP(i) = critical path time - b-level(i), CPT is (As-Late-As-Possible) The following approaches are representative of the many proposed techniques: Communication cost of zero (best to worse) Highest Level First with Estimated Times (HLFET) - highest priority is given to tasks that are the farthest from a exit node (i.e. use sl(i)), but ties are broken based on the task that has the earliest possible start time. Highest Levels First with No Estimated Times (HLFNET) - like HLFET, but assumes tasks have unit time. Assign priorities randomly Smallest Co-levels First with No Estimated Times (SCFNET) - like SCFET, but assumes tasks have unit time. Smallest Co-levels First with Estimated Times (SCFET) - the co-level of each node is the computation time for the longest path from an entry node to the node (i.e. like sl(i), but reversed) With communication

7 initial diffusion tree could be set up artificially before information exchange commences. In this case, some processor acts as the root and all other processors determine an initial parent in the tree. As the information exchange proceeds, a processor is taken as being passive if its load index remains unchanged for a number of exchanges. Of course, if the load index eventually changes as the result of an exchange, then the processor becomes active again (and a parent in the tree must be established). Global termination will be detected when the diffusion tree collapses all the way to the root Problems: (Provided code may be adapted for the last two problems.). Give an optimal schedule for the following task graph assuming that all tasks take unit time, no communication is needed, and there are two processors P P. Consider a -d hypercube that assigns each processor i, 0 < i <, a load index of 000i. Apply averaging dimension exchange (i.e. λ = 0.) to one sweep and observe that the load becomes balanced.. Consider the edge-coloring for the x mesh. Suppose that the rows are numbered top-to-bottom from 0 to. Likewise, suppose that the columns are numbered left-to-right from 0 to. Now suppose that processor (i, j) is assigned load index

8 8 000(i + j). If the color classes are processed cyclically (as indicated in the diagram) by GDE, how many sweeps are needed? , 0,, , P P

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