n usually on or higher np often o1 or olog2p

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Unformatted text preview: n,p) (i.e. computational complexity usually higher than complexity of communication -- same is often true of σ(n) as well.) ϕ(n) usually O(n) or higher • κ(n,p) often O(1) or O(log2P) • Increasing n allows ϕ(n) to dominate κ(n,p) • Thus, increasing n increases the speedup Ψ for some number of processors • • Another cheat to get good results -- make n large Most benchmarks have standard sized inputs to preclude this Tuesday, February 14, 12 Amdahl Effect n=100000 Speedup n=10000 n=1000 Number of processors Tuesday, February 14, 12 Summary • • • Tuesday, February 14, 12 Allows speedup to be computed for • fixed problem size n • varying number of processes Ignores communication costs Is optimistic, but gives an upper bound Gustafson-Barsis’ Law How does speedup scale with larger problem sizes? Given a fixed amount of time, how much bigger of a problem can we solve by adding more processors? Large problem sizes often correspond to better resolution and precision on the problem being solved. Tuesday, February 14, 12 Basic terms Speedup is Because κ(n,p) > 0 Let s be the fraction of time in a parallel execution of the program that is spent performing sequential operations. Then, (1-s) is the fraction of time spent in a parallel execution of the program performing parallel operations. Tuesday, February 14, 12 Note that Amdahl's Law looks at the sequential and parallel parts of the program for a given problem size, and the value of f is the fraction in a sequential execution that is inherently sequential Or stated differently . . . Tuesday, February 14, 12 Speedup in terms of the serial fraction of a program Given this formulation, the fraction of the program that is serial is simply Speedup can be rewritten in terms of f: This gives us Amdahl’s Law. Note number of processors not mentioned for definition of f because f is for time in a sequential run Tuesday, February 14, 12 Some definitions The sequential part of a parallel computation: The parallel part of a parallel computation: And the speedup Tuesday, February 14, 12 Difference between G-B Law and Amdahl’s Law The serial portion in Amdahl’s law is a fraction of the total execution time of the program. The serial portion in G-B is a fraction of the parallel execution time of the program. To use G-B Law we assume work scales to maintain value of s Tuesday, February 14, 12 Deriving G-B Law substitute for (s + (1 - s)p) First, we show that the formula circled in blue leads to our speedup formula. Multiply through simplify, simply Tuesday, February 14, 12 Deriving G-B Law Second, we show that the formula circled in blue leads (that we just showed is equivalent to speedup) to the G-B Law formula. Tuesday, February 14, 12 An example An application executing on 64 processors requires 220 seconds to run. It is experimentally determined through benchmarking that 5% of the time is spent in the serial code on a single processor. What is the scaled speedup of the application? s = 0.05, thus on 64 processors Ψ = 64 + (1-64)(0.05) = 64 - 3.15 = 60.85 Tuesday, February 14, 12 An example Another way of looking at this: given P processors, P a...
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This note was uploaded on 02/19/2012 for the course ECE 563 taught by Professor Staff during the Spring '08 term at Purdue University-West Lafayette.

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