11f6643lec07 - • Boundary value problem • Finding the...

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Unformatted text preview: • Boundary value problem • Finding the maximum • The n-body problem Case Studies Algorithm Design Case Studies Ice water Rod Insulation Boundary Value Problem Foster’s Design Methodology CS6643 F'11 Lec05 2 • Identify communication pattern between primitive tasks • Each interior primitive task has three incoming and three outgoing channels Communication Partitioning • One data item per grid point • Associate one primitive task with each grid point • Two-dimensional domain (data) decomposition Finite Difference Approximation Rod Cools as Time Progresses • p – number of processors • O – message latency • Parallel execution time n(F ªm/pº+2O) Parallel Execution Time Agglomeration Agglomeration and Mapping Reduction F – time to update element m – number of elements n – number of iterations (time steps) Sequential execution time: m n F – – – – Add Multiply And, Or Maximum, Minimum • Given associative operator † • a0 † a1 † a2 † … † an-1 • Examples • • • • Sequential execution time Binomial Trees Parallel Reduction Evolution Subgraph of hypercube Parallel Reduction Evolution Parallel Reduction Evolution -1 3 -3 7 Finding Global Sum 6 10 4 -4 2 9 1 8 -6 -2 5 -3 0 8 2 4 Finding Global Sum 4 1 17 5 7 2 4 8 Finding Global Sum 8 -6 Finding Global Sum sum sum sum sum Agglomeration 25 Binomial Tree Finding Global Sum • Total time: (ªn/pº -1)F + ªlg pº (O+ F) – Each step takes O+ F time • Local reduction time: (ªn/pº -1)F • ªlg pº tree-based reduction steps Parallel Execution Time Agglomeration CS 6643 F’11 Lec07 24 • • • • Gather Partitioning – Get positions of all other particles – Compute new position, velocity Domain (data) partitioning Assume one task per particle (body) Task has particle’s position, velocity vector Iteration The n-body Problem The n-body Problem E ) ( p  1)O  n( p  1) Ep log p § n( p  1) 2i-1 n · ¸ Ologp  ¨O  ¦ ¨ Ep ¸ Ep i 1© ¹ Hypercube ( p  1)(O  n/ p Communication Time Hypercube for All-gather Complete graph Complete Graph for All-gather All-gather ...
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11f6643lec07 - • Boundary value problem • Finding the...

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