Breaking The Law: Amdahl's Law Defied?

Amdahl's Law, presented in 1967, is a formula that predicts the potential speed increase in completing a task with improved system resources while keeping the workload constant. The law is often used to predict the potential performance improvement of a system when adding more processors or improving the speed of individual processors. It is named after Gene Amdahl, who first proposed it. The law assumes that all processors are identical and contribute equally to speedup, and emphasizes the non-parallelizable portion of the task as a bottleneck. Amdahl's Law is popular among developers as it offers a simple and efficient way to determine the maximum potential for improving system performance. However, it has been suggested that Amdahl's Law has been broken by achieving more than a 1000-fold speedup using 1024 processors, which has been justified as massively parallel processing.

Amdahl's Law and the Law of Diminishing Returns

Amdahl's Law, also known as Amdahl's Argument, is a formula that predicts the potential speed increase in completing a task with improved system resources while keeping the workload constant. It was presented by American computer scientist and high-tech entrepreneur Gene Amdahl in 1967 and remains a crucial basis for numerous computing principles and technologies.

The law states that the maximum improvement in speed of a process is limited by the proportion of the program that can be made parallel. In other words, the performance improvement gained by optimizing a single part of a system is limited by the fraction of time spent on the portion of the task that must be run sequentially. This portion of the task that cannot be parallelized is often referred to as the "bottleneck".

Amdahl's Law is often associated with the Law of Diminishing Returns, although only a special case of applying Amdahl's Law demonstrates the Law of Diminishing Returns. If one optimizes a component that is sub-optimal, and then moves on to improve a more optimal component, one can see an increase in returns. However, if one optimizes a component that is already close to optimal, further improvements will result in monotonically decreasing improvements.

While Amdahl's Law is a useful tool, it does have some limitations. It assumes that all processors are identical and contribute equally to speedup, which may not be true in heterogeneous computing environments. Additionally, it assumes that the rest of the system can fully utilize additional processors, which may not always be the case in practice. Despite these assumptions, Amdahl's Law remains a valuable tool for developers to identify bottlenecks and optimize system performance.

Amdahl's Law and Gustafson's Law

Amdahl's Law, presented in 1967, is a formula that predicts the potential speed increase in completing a task with improved system resources while keeping the workload constant. The formula is S = 1 / (1 - p + p/s), where p is the proportion of the overall execution time spent by the part of the task that benefits from parallel processing, and s is the performance improvement, or speedup, of the part of the task that benefits from parallel processing. Amdahl's Law is popular among developers as it offers a simple and efficient way to determine the maximum potential for improving system performance by identifying the bottleneck in a program or system.

However, Amdahl's Law has a significant drawback: it assumes that the problem size remains constant when utilizing more cores to execute an application, i.e., the computing requirements will stay the same, given increased processing power. This limitation was addressed and expanded upon by John L. Gustafson, who proposed a modified perspective now known as Gustafson's Law.

Gustafson's Law, presented in 1988, gives the speedup in the execution time of a task that theoretically gains from parallel computing, using a hypothetical run of the task on a single-core machine as the baseline. It proposes that programmers tend to increase the size of problems to fully exploit the computing power that becomes available as the resources improve. In other words, Gustafson argues that more computing power will lead to more careful and thorough analysis of data.

Gustafson's Law is applicable when an algorithm can dynamically adjust the amount of computation to match the available parallelization. In contrast, Amdahl's Law is more appropriate when the computation load is fixed and cannot be significantly altered by parallelization. These two principles are considered the yin and yang of parallel computing, and they are often used together to obtain estimated speedups as measures of parallel program potential.

Amdahl's Law's use in computer architecture

Amdahl's Law, also known as Amdahl's Argument, is a formula that predicts the speed increase in completing a task with improved system resources while keeping the workload constant. It was presented by computer scientist Gene Amdahl in 1967 and remains a crucial basis for many computing principles and technologies.

In computer architecture, Amdahl's Law is used to determine how much faster a task can be completed when more resources are added to the system. It is often used in parallel computing to predict the theoretical speedup when using multiple processors. The law states that the maximum potential improvement to the performance of a system is limited by the portion of the system that cannot be improved. In other words, the performance improvement of a system as a whole is limited by its bottlenecks.

Amdahl's Law is popular among developers as it offers a simple and efficient way to determine the maximum potential for improving system performance. By identifying the bottleneck in a program or system, developers can focus their efforts on optimizing that particular component instead of wasting time working on parts that will have minimal returns.

The law can be expressed mathematically as S = 1 / (1 - p + p/s), where p is the proportion of overall execution time spent by the part of the task that benefits from parallel processing, and s is the performance improvement, or speedup, of the part of the task that benefits from parallel processing.

It's important to note that Amdahl's Law assumes that all processors are identical and contribute equally to speedup, which may not be the case in heterogeneous computing environments. Additionally, it does not take into account other factors that can affect the performance of parallel programs, such as communication overhead and load balancing. These factors can impact the actual speedup achieved in practice, which may be lower than the theoretical maximum predicted by Amdahl's Law.

Amdahl's Law's base formula

Amdahl's Law is a formula that predicts the potential speed increase in completing a task with improved system resources while keeping the workload constant. It is often used in parallel computing to predict the theoretical speedup when using multiple processors. The law is named after computer scientist Gene Amdahl, who first presented it at the American Federation of Information Processing Societies (AFIPS) Spring Joint Computer Conference in 1967.

The base formula for Amdahl's Law is S = 1 / (1 - p + p/s), where S represents the speedup factor, p is the proportion of the task that can be parallelized, and 1-p is the portion that must be run sequentially. This formula states that the maximum improvement in speed of a process is limited by the proportion of the program that can be made parallel. In other words, it does not matter how many processors are added or how much faster each processor is; the maximum improvement in speed will always be limited by the most significant bottleneck in the system.

For example, consider a program that reads data and then performs an independent calculation on each data item, a common scenario in image processing. Suppose loading the data takes one minute, and processing it takes another three minutes on a single CPU thread. Here, the image processing part is the P in the formula, representing three minutes out of four, so P is 3/4 or 0.75. If we want to double the speed, the desired speedup factor S = 2.

Amdahl's Law assumes a fixed workload and does not account for dynamic or increasing workloads, which can impact the effectiveness of parallel processing. It also neglects overheads associated with concurrency, such as coordination, synchronization, and inter-process communication. Despite these limitations, Amdahl's Law remains a crucial basis for many computing principles and technologies due to its simplicity and ability to identify bottlenecks and guide optimization efforts.

Amdahl's Law's assumptions

Amdahl's Law, presented in 1967, is a formula that predicts the potential speed increase in completing a task with improved system resources while keeping the workload constant. It is often used in parallel computing to predict the theoretical speedup when using multiple processors. Amdahl's Law assumes that:

Fixed Workload

It assumes that the workload remains constant. It does not account for dynamic or increasing workloads, which can impact the effectiveness of parallel processing.

Overhead Ignored

It neglects overheads associated with concurrency, including coordination, synchronization, inter-process communication, and concurrency control. Merging data from multiple threads or processes incurs significant overhead due to conflict resolution, data consistency, versioning, and synchronization.

Neglecting Extrinsic Factors

Amdahl's Law addresses computational parallelism, neglecting extrinsic factors such as data persistence, I/O operations, and memory access overheads, and assumes idealized conditions.

Scalability Issues

While it highlights the limits of parallel speedup, it does not address practical scalability issues, such as the cost and complexity of adding more processors.

Non-Parallelizable Work

Amdahl's Law emphasizes the non-parallelizable portion of the task as a bottleneck but does not provide solutions for reducing or optimizing this portion.

Homogeneous Processors

It assumes that all processors are identical and contribute equally to speedup, which may not be the case in heterogeneous computing environments.

Dividing Parallel Tasks

Amdahl's Law assumes that dividing the parallel part of the task among many nodes does not inflict any sort of runtime penalty. This assumption is reasonable for embarrassingly parallel workloads but may not hold in other cases.

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