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will be ".* Amdahl's law states that the overall speedup of applying the improvement willbe

To see how this formula was derived, assume that the running time of the old computation

was $, for some unit of time. The running time of the new computation will be the length of

time the unimproved fraction takes, which is $ 8 5*, plus the length of time the improvedfraction 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 timeof the improved part 56. The final speedup is computed by dividing the old running time

by the new running time, which is what the above formula does.

+ere's another example. 9e are given a task which is split up into four parts: 5$ 0 $$), 5"0 $-), 51 0 "1), 5; 0 ;-), which add up to $##). Then we say 5$ is not sped up, so 6$

0 $ or $##), 5" is sped up (, so 6" 0 (##), 51 is sped up "#, so 61 0 "###), and 5; is

sped up $.2, so 6; 0 $2#). the original running time which we know is $. Therefore the overall

speed boost is $ #.;(?( 0 ".$-2 or a little more than double the original speed using theformula 5$6$ = 5"6" = 5161 = 5;6;*8$. @otice how the "# and ( speedup don't

have much effect on the overall speed boost and running time when $$) is not sped up,

and ;-) is sped up by $.2.

5aralleli%ationIn the case of paralleli%ation, Amdahl's law states that if 5 is the proportion of a programthat can be made parallel i.e. benefit from paralleli%ation*, and $ 8 5* is the proportion

that cannot be paralleli%ed remains serial*, then the maximum speedup that can be

achieved by using @ processors is

In the limit, as @ tends to infinity, the maximum speedup tends to $ $ 8 5*. In practice,

performance to price ratio falls rapidly as @ is increased once there is even a small

component of $ 8 5*.

As an example, if 5 is ), then $ 8 5* is $#), and the problem can be speed up by amaximum of a factor of $#, no matter how large the value of @ used. !or this reason,

parallel computing is only useful for either small numbers ofprocessors, or problems with

very high values of 5: so3called embarrassingly parallelproblems. A great part of the craft

of parallel programmingconsists of attempting to reduce the component $ / 5* to thesmallest possible value.

5 can be estimated by using the measured speedup 6 on a specific number of processors@5 using

http://en.wikipedia.org/wiki/Extended_real_number_linehttp://en.wikipedia.org/wiki/Central_processing_unithttp://en.wikipedia.org/wiki/Embarrassingly_parallelhttp://en.wikipedia.org/wiki/Embarrassingly_parallelhttp://en.wikipedia.org/wiki/Parallel_programminghttp://en.wikipedia.org/wiki/Extended_real_number_linehttp://en.wikipedia.org/wiki/Central_processing_unithttp://en.wikipedia.org/wiki/Embarrassingly_parallelhttp://en.wikipedia.org/wiki/Parallel_programming

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5 estimated in this way can then be used in Amdahl's law to predict speedup for a differentnumber of processors.

Belation to law of diminishing returnsAmdahl's law is often conflated with the law of diminishing returns,whereas only a special

case of applying Amdahl's law demonstrates 'law of diminishing returns'. If one picks

optimally in terms of the achieved speed3up* what to improve, then one will seemonotonically decreasing improvements as one improves. If, however, one picks non3

optimally, after improving a sub3optimal component and moving on to improve a more

optimal component, one can see an increase in return. Consider, for instance, theillustration. If one picks to work on < then A, one finds an increase in return. If, instead,

one works on improving A then

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If part < is made five times faster p 0 (*, tA 0 1, t< 0 $ and f 0 tA tA = t