How Much Speedup Can Be Attained?
Theoretical Speedup
Amdahl’s Law
Amdahl’s Law provides a theoretical framework for understanding the potential speedup of parallelizing a computation. It is expressed as:
Sp=1(1−P)+PNS_p = \frac{1}{(1 – P) + \frac{P}{N}}Sp=(1−P)+NP1
Where:
- SpS_pSp = Speedup with NNN processors
- PPP = Proportion of the program that can be parallelized
- NNN = Number of processors
Example Calculation:
Suppose 80% of a program can be parallelized (P=0.8P = 0.8P=0.8) and you use 4 processors (N=4N = 4N=4):
Sp=1(1−0.8)+0.84S_p = \frac{1}{(1 – 0.8) + \frac{0.8}{4}}Sp=(1−0.8)+40.81 Sp=10.2+0.2S_p = \frac{1}{0.2 + 0.2}Sp=0.2+0.21 Sp=10.4S_p = \frac{1}{0.4}Sp=0.41 Sp=2.5S_p = 2.5Sp=2.5
This implies that the maximum speedup attainable is 2.5 times faster than the sequential execution.
Gustafson-Barsis’s Law
Gustafson-Barsis’s Law suggests that speedup can be better estimated when scaling the problem size with the number of processors:
Sp=N−α(N−1)S_p = N – \alpha (N – 1)Sp=N−α(N−1)
Where:
- α\alphaα = Fraction of the computation that is serial
- NNN = Number of processors
Example Calculation:
Assuming α=0.2\alpha = 0.2α=0.2 and N=4N = 4N=4:
Sp=4−0.2×(4−1)S_p = 4 – 0.2 \times (4 – 1)Sp=4−0.2×(4−1) Sp=4−0.2×3S_p = 4 – 0.2 \times 3Sp=4−0.2×3 Sp=4−0.6S_p = 4 – 0.6Sp=4−0.6 Sp=3.4S_p = 3.4Sp=3.4
This suggests a speedup of 3.4 times, considering the problem size scales with the number of processors.
Practical Speedup
Overheads and Communication
In practice, the speedup achieved is often less than the theoretical maximum due to various overheads:
- Communication Overhead: Time spent in communication between the master and worker nodes.
- Synchronization Overhead: Time required to synchronize workers or collect results.
- Load Imbalance: Unequal distribution of work among processors.
Example:
If the theoretical speedup is 4x and actual observed speedup is 3x, the difference may be attributed to overheads and inefficiencies.
Scalability Testing
To determine practical speedup, conduct scalability tests:
- Measure Execution Time: Compare execution times for different numbers of processors.
- Analyze Efficiency: Calculate the efficiency EEE as:
E=SpNE = \frac{S_p}{N}E=NSp
Where SpS_pSp is the observed speedup and NNN is the number of processors.
Example:
If using 4 processors results in a speedup of 3, efficiency is:
E=34=0.75E = \frac{3}{4} = 0.75E=43=0.75
This means each processor contributes to 75% of its potential performance.
Factors Affecting Speedup
Nature of the Task
- Embarrassingly Parallel Tasks: Tasks that can be completely separated, like Monte Carlo simulations, generally achieve higher speedup.
- Interdependent Tasks: Tasks that require frequent communication or synchronization will have lower speedup.
Implementation Efficiency
- Algorithmic Overheads: Ensure that parallel algorithms minimize communication and synchronization costs.
- Programming Overheads: Efficient use of parallel constructs and minimization of overheads in snow (e.g., choosing parLapply() over clusterApply() for more straightforward scenarios).
Hardware and System Configuration
- CPU Cores: More cores typically offer better speedup, but diminishing returns may occur as the number of cores increases.
- Network Bandwidth: For distributed clusters, network speed impacts communication overhead and overall speedup.
Measuring and Maximizing Speedup
Benchmarking
Perform benchmarks to measure actual speedup:
library(snow)
# Sequential Execution
start_time_seq <- Sys.time()
result_seq <- sapply(1:1000, function(x) x^2)
end_time_seq <- Sys.time()
time_seq <- end_time_seq - start_time_seq
# Parallel Execution
cl <- makeCluster(4)
start_time_par <- Sys.time()
result_par <- parLapply(cl, 1:1000, function(x) x^2)
end_time_par <- Sys.time()
time_par <- end_time_par - start_time_par
stopCluster(cl)
# Calculate Speedup
speedup <- time_seq / time_par
print(paste("Speedup:", speedup))
Optimize Parallel Code
- Reduce Communication: Minimize data transfer and synchronization.
- Balance Load: Ensure tasks are evenly distributed.
- Profile and Optimize: Use profiling tools to identify bottlenecks and optimize performance.
Conclusion
The speedup achieved with parallel computing using the snow package depends on various factors including theoretical limits, overheads, and the nature of the task. By understanding Amdahl’s Law and Gustafson-Barsis’s Law, conducting scalability tests, and considering practical factors, you can estimate and maximize the speedup for your parallel computations.