What is Amadahl’s Law? “Amadahl’s law demonstrates thelaw of diminishing returns”. Justify the statement with ex-ample

Amdahl's Law is a formula that is used to find the maximum improvement of a system when only a part of the system is improved. It is particularly applicable in parallel computing and relates to the speedup of a task that can benefit from parallel execution while considering the portion of the task that cannot be parallelized.

Justification of the Statement with Example

1. Understanding Amdahl's Law:

   • Amdahl's Law can be expressed mathematically as: $S = \frac{1}{(1 - P) + \frac{P}{N}}$ where:
     • $S$ = overall speedup of the system
     • $P$ = proportion of the task that can be parallelized
     • $N$ = number of processors

2. Explaining the Law of Diminishing Returns:

   • The law of diminishing returns states that as you increase one input in the production of a good (in this case, processors), while keeping other inputs constant, the incremental output (speedup) will eventually decrease.

3. Example:

   • Suppose we have a task where 80% (or $P = 0.8$) of the task can be parallelized, and we are using 4 processors ($N = 4$).
   • Using Amdahl's Law: $S = \frac{1}{(1 - 0.8) + \frac{0.8}{4}} = \frac{1}{0.2 + 0.2} = \frac{1}{0.4} = 2.5$
   • This means that the maximum speedup we can achieve by using 4 processors for this task is 2.5 times.

4. Demonstrating Diminishing Returns:

   • If we increase the number of processors to 8: $S = \frac{1}{(1 - 0.8) + \frac{0.8}{8}} = \frac{1}{0.2 + 0.1} = \frac{1}{0.3} \approx 3.33$
   • The improvement is now only an additional speedup of 0.83 (from 2.5 to 3.33) by doubling the processors from 4 to 8.
   • As we continue to increase the number of processors, the increment in speedup will continue to diminish.

Conclusion

Amdahl's Law illustrates the law of diminishing returns as it shows that even with an increase in resources (processors), the overall gain in performance reduces because some parts of the task cannot be parallelized, thereby limiting the maximum achievable speedup.