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Amdahl's law is a model for the relationship between the expected speedup of parallelized implementations of an algorithm relative to the serial algorithm, under the assumption that the problem size remains the same when parallelized. For example, if for a given problem size a parallelized implementation of an algorithm can run 12% of the algorithm's operations arbitrarily fast 70-620 Exam (while the remaining 88% of the operations are not parallelizable), Amdahl's law states that the maximum speedup of the parallelized version is 1/(1 - 0.12) = 1.136 times faster than the non-parallelized implementation. More technically, the law is concerned with the speedup achievable from an improvement to a computation that affects a proportion P of that computation where the improvement has a speedup of S. (For example, if an improvement can speed up 30% of the computation, P will be 0.3; if the improvement makes the portion affected twice as fast, S will be 2.) Amdahl's law states that the overall speedup 70-640 Exam of applying the improvement will be To see how this formula was derived, assume that the running time of the old computation was 1, for some unit of time. The running time of the new computation will be the length of time the unimproved fraction takes, (which is 1 − P), plus the length of time the improved fraction takes. The length of time for the improved part of the computation is the length of the improved part's former running time divided by the speedup, making the length of time of the improved part P/S. The final speedup is computed by 70-647 Exam dividing the old running time by the new running time, which is what the above formula does. Here's another example. We are given a task which is split up into four parts: P1 = 11%, P2 = 18%, P3 = 23%, P4 = 48%, which add up to 100%. Then we say P1 is not sped up, so S1 = 1 or 100%, P2 is sped up 5x, so S2 = 500%, P3 is sped up 20x, so S3 = 2000%, and P4 is sped up 1.6x, so S4 = 160%. By using the formula P1/S1 + P2/S2 + P3/S3 + P4/S4, we find the running time is