Question
Answer and Explanation
The theoretical maximum speedup achievable when converting a single-threaded application to a multi-threaded one is governed by Amdahl's Law. Amdahl's Law provides a formula to estimate the maximum speedup based on the portion of the program that can be parallelized.
Here's the formula:
Speedup = 1 / ( (1 - P) + (P / N) )
Where:
- P represents the proportion of the program that can be parallelized (expressed as a decimal between 0 and 1). For instance, if 75% of the code can be parallelized, P would be 0.75.
- N is the number of processing cores or threads that can be used in parallel.
Let's break this down:
- (1 - P) represents the proportion of the program that must be executed serially (i.e., cannot be parallelized). This part is the bottleneck that limits the achievable speedup.
- (P / N) represents the proportion of the program that can be executed in parallel, divided by the number of threads.
For example, consider a scenario:
- If 80% of the code can be parallelized (P = 0.8) and the code is run on a 4-core processor (N=4), then the maximum speedup would be:
Speedup = 1 / ( (1 - 0.8) + (0.8 / 4) )
Speedup = 1 / ( 0.2 + 0.2 )
Speedup = 1 / 0.4
Speedup = 2.5
This suggests the program could theoretically run 2.5 times faster.
Important considerations related to this formula:
- Practical Limits: Amdahl's Law gives theoretical limits. In practice, overheads such as thread creation/management, synchronization, and communication might reduce the achieved speedup.
- Scalability: Amdahl's law shows that the portion of code that cannot be parallelized is the limiting factor for scalability. As N approaches infinity, the maximum speedup approaches 1/(1-P). Thus if P=0.9, the maximum speedup would be 10, regardless of how many threads are used.
- Gustafson's Law: It's worth noting Gustafson's Law presents an alternative perspective that is more optimistic regarding speedup achievable with more cores, especially for large problems. Gustafson's law states that Speedup = S + P N, where S is the serial portion and P is the parallel portion of the program. This model allows for more optimistic speedups when the problem size grows with the number of processors.
In conclusion, Amdahl's Law is crucial for understanding the theoretical limitations of parallelization. While it provides a good estimate of maximum speedup, it doesn't account for all real-world factors. When moving from a single-threaded to multi-threaded approach, both Amdahl’s Law and Gustafson's Law can be helpful to evaluate performance impacts.