6/16/2010 DAG Execu>on Model, Work and Depth 1 DAG EXECUTION MODEL, WORK AND DEPTH
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1 6/16/2010 DAG Execu>on Model, Work and Depth 1 DAG EXECUTION MODEL, WORK AND DEPTH
2 6/16/2010 DAG Execu>on Model, Work and Depth 2 Computa>onal Complexity of (Sequen>al) Algorithms Model: Each step takes a unit >me Determine the >me (/space) required by the algorithm as a func>on of input size
3 6/16/2010 DAG Execu>on Model, Work and Depth 3 Sequen>al Sor>ng Example Given an array of size n MergeSort takes O(n. log n) >me BubbleSort takes O(n 2 ) >me
4 6/16/2010 DAG Execu>on Model, Work and Depth 4 Sequen>al Sor>ng Example Given an array of size n MergeSort takes O(n. log n) >me BubbleSort takes O(n 2 ) >me But, a BubbleSort implementa>on can some>mes be faster than a MergeSort implementa>on Why?
5 6/16/2010 DAG Execu>on Model, Work and Depth 5 Sequen>al Sor>ng Example Given an array of size n MergeSort takes O(n. log n) >me BubbleSort takes O(n 2 ) >me But, a BubbleSort implementa>on can some>mes be faster than a MergeSort implementa>on The model is s>ll useful Indicates the scalability of the algorithm for large inputs Lets us prove things like a sor>ng algorithm requires at least O(n. log n) comparisions
6 6/16/2010 DAG Execu>on Model, Work and Depth 6 We need a similar model for parallel algorithms
7 6/16/2010 DAG Execu>on Model, Work and Depth 7 Sequen>al Merge Sort 16MB input (32- bit integers) Time Recurse(le\) Recurse(right) Sequen>al Execu>on Merge to scratch array Copy back to input array
8 6/16/2010 DAG Execu>on Model, Work and Depth 8 Parallel Merge Sort (as Parallel Directed Acyclic Graph) 16MB input (32- bit integers) Time Recurse(le\) Recurse(right) Parallel Execu>on Merge to scratch array Copy back to input array
9 6/16/2010 DAG Execu>on Model, Work and Depth 9 Parallel DAG for Merge Sort (2- core) Sequen>al Sort Merge Sequen>al Sort Time
10 6/16/2010 DAG Execu>on Model, Work and Depth 10 Parallel DAG for Merge Sort (4- core)
11 6/16/2010 DAG Execu>on Model, Work and Depth 11 Parallel DAG for Merge Sort (8- core)
12 6/16/2010 DAG Execu>on Model, Work and Depth 12 The DAG Execu>on Model of a Parallel Computa>on Given an input, dynamically create a DAG Nodes represent sequen>al computa>on Weighted by the amount of work Edges represent dependencies: Node A Node B means that B cannot be scheduled unless A is finished
13 6/16/2010 DAG Execu>on Model, Work and Depth 13 Sor>ng 16 elements in four cores
14 6/16/2010 DAG Execu>on Model, Work and Depth 14 Sor>ng 16 elements in four cores (4 element arrays sorted in constant >me)
15 6/16/2010 DAG Execu>on Model, Work and Depth 15 Performance Measures
16 6/16/2010 DAG Execu>on Model, Work and Depth 16 Work and Depth
17 6/16/2010 DAG Execu>on Model, Work and Depth 17 T (Depth): Cri>cal Path Length (Sequen>al Bobleneck)
18 6/16/2010 DAG Execu>on Model, Work and Depth 18 T 1 (work): Time to Run Sequen>ally
19 6/16/2010 DAG Execu>on Model, Work and Depth 19 Sor>ng 16 elements in four cores (4 element arrays sorted in constant >me) Work = Depth =
20 6/16/2010 DAG Execu>on Model, Work and Depth 20 Some Useful Theorems
21 6/16/2010 DAG Execu>on Model, Work and Depth 21 Work Law You cannot avoid work by parallelizing
22 6/16/2010 DAG Execu>on Model, Work and Depth 22 Work Law
23 6/16/2010 DAG Execu>on Model, Work and Depth 23 Work Law
24 6/16/2010 DAG Execu>on Model, Work and Depth 24 Depth Law
25 6/16/2010 DAG Execu>on Model, Work and Depth 25 Amount of Parallelism
26 6/16/2010 DAG Execu>on Model, Work and Depth 26 Maximum Speedup Possible Speedup Parallelism speedup is bounded above by available parallelism
27 6/16/2010 DAG Execu>on Model, Work and Depth 27 Greedy Scheduler If more than P nodes can be scheduled, pick any subset of size P If less than P nodes can be scheduled, schedule them all
28 6/16/2010 DAG Execu>on Model, Work and Depth 28 Performance of the Greedy Scheduler
29 6/16/2010 DAG Execu>on Model, Work and Depth 29 Performance of the Greedy Scheduler
30 6/16/2010 DAG Execu>on Model, Work and Depth 30 Greedy is op>mal within a factor of 2
31 6/16/2010 DAG Execu>on Model, Work and Depth 31 Work/Depth of Merge Sort (Sequen>al Merge)
32 6/16/2010 DAG Execu>on Model, Work and Depth 32 Main Message Analyze the Work and Depth of your algorithm Parallelism is Work/Depth Try to decrease Depth the cri>cal path a sequen'al bobleneck If you increase Depth beber increase Work by a lot more!
33 6/16/2010 DAG Execu>on Model, Work and Depth 33 Amdahl s law Sor>ng takes 70% of the execu>on >me of a sequen>al program You replace the sor>ng algorithm with one that scales perfectly on mul>- core hardware How many cores do you need to get a 4x speed- up on the program?
34 6/16/2010 DAG Execu>on Model, Work and Depth 34 = the parallel por>on of execu>on = the sequen>al por>on of execu>on = number of cores used
35 6/16/2010 DAG Execu>on Model, Work and Depth 35 Speedup Desired 4x speedup Speedup achieved (perfect scaling on 70%) #cores
36 6/16/2010 DAG Execu>on Model, Work and Depth 36 Speedup Desired 4x speedup Speedup achieved (perfect scaling on 70%) #cores Limit as c = 1/(1- f) = 3.33
37 6/16/2010 DAG Execu>on Model, Work and Depth 37 Speedup Speedup achieved with perfect scaling #cores Amdahl s law limit, just 1.11x
38 6/16/2010 DAG Execu>on Model, Work and Depth Speedup #cores
39 6/16/2010 DAG Execu>on Model, Work and Depth 39 Lesson Speedup is limited by sequen>al code Even a small percentage of sequen>al code can greatly limit poten>al speedup
40 6/16/2010 DAG Execu>on Model, Work and Depth 40 Gustafson s Law Any sufficiently large problem can be parallelized effec>vely = the parallel por>on of execu>on = the sequen>al por>on of execu>on = number of cores used
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