# Gustafson's law

Gustafson's Law (also known as Gustafson-Barsis' law) is a law in computer engineering which states that any sufficiently large problem can be efficiently parallelized. Gustafson's Law is closely related to Amdahl's law, which gives a limit to the degree to which a program can be sped up due to parallelization. It was first described by John L. Gustafson in 1988.

$S(P) = P - \alpha\cdot(P-1)$.

where P is the number of processors, S is the speedup, and α the non-parallelizable part of the process.

Gustafson's law addresses the shortcomings of Amdahl's law, which cannot scale to match availability of computing power as the machine size increases. It removes the fixed problem size or fixed computation load on the parallel processors: instead, he proposed a fixed time concept which leads to scaled speed up.

Amdahl's law is based on fixed workload or fixed problem size. It implies that the sequential part of a program does not change with respect to machine size (i.e, the number of processors). However the parallel part is evenly distributed by n processors.

The impact of the law was the shift in research to develop parallelizing compilers and reduction in the serial part of the solution to boost the performance of parallel systems.

## Implementation of Gustafson's Law

Let n be a measure of the problem size.

The execution of the program on a parallel computer is decomposed into:

a(n) + b(n) = 1

where a is the sequential fraction and b is the parallel fraction, ignoring overhead for now.

On a sequential computer, the relative time would be $a(n) + p\cdot b(n)$, where p is the number of processors in the parallel case.

Speedup is therefore:

$(a(n) + p\cdot{}b(n))$ (parallel, relative to sequential a(n) + b(n) = 1)

and thus

$S= a(n) + p\cdot{}(1-a(n))$

where a(n) is the serial function.

Assuming the serial function a(n) diminishes with problem size n, then speedup approaches p as n approaches infinity, as desired.

Thus Gustafson's law seems to rescue parallel processing from Amdahl's law.

Gustafson's law argues that even using massively parallel computer systems does not influence the serial part and regards this part as a constant one. In comparison to that, the hypothesis of Amdahl's law results from the idea that the influence of the serial part grows with the number of processes.

## A Driving Metaphor

Suppose a car is traveling between two cities 60 miles apart, and has already spent one hour traveling half the distance at 30 mph.

Amdahl's Law approximately suggests:

 “ No matter how fast you drive the last half, it is impossible to achieve 90 mph average before reaching the second city. Since it has already taken you 1 hour and you only have a distance of 60 miles total; going infinitely fast you would only achieve 60 mph. ”

Gustafson's Law approximately states:

 “ Given enough time and distance to travel, the car's average speed can always eventually reach 90mph, no matter how long or how slowly it has already traveled. For example, in the two-cities case this could be achieved by driving at 150 mph for an additional hour.[dubious ] ”

## Limitations

Some problems do not have fundamentally larger datasets. As example, processing one data point per world citizen gets larger at only a few percent per year.

Nonlinear algorithms may make it hard to take advantage of parallelism "exposed" by Gustafson's law. Snyder points out an O(N3) algorithm means that double the concurrency gives only about a 9% increase in problem size. Thus, while it may be possible to occupy vast concurrency, doing so may bring little advantage over the original, less concurrent solution.