Why Don't We Consider the Number of Threads When Calculating Big O Notation?

In the realm of algorithm analysis, Big O notation is a fundamental tool for understanding the efficiency of algorithms. This notation focuses on the growth rate of an algorithm's time or space complexity relative to the size of the input, rather than specific implementation details such as threading or parallelism. Let's explore why the number of threads is not typically considered in standard Big O analysis.

Abstract Complexity

Big O notation is designed to provide a high-level, abstract description of an algorithm's efficiency. It captures the relationship between the input size and the growth of the algorithm's runtime or space requirements. This abstraction allows us to understand how an algorithm scales without getting mired in the specific implementation details of how the computation is carried out.

Threading Variability

The performance gain from using multiple threads can vary widely depending on several factors, such as the hardware, the nature of the task (CPU-bound vs. I/O-bound), and the overhead of managing threads. This variability makes it challenging to represent threading in a consistent manner across different scenarios. For instance, a task that is perfectly divisible might not always benefit from more threads due to overhead, while another task might perform optimally with a specific number of threads.

Focus on Input Size

Big O notation is concerned with the fundamental growth of an algorithm's complexity as the input size increases. For example, an algorithm with a time complexity of O(n^2) will take progressively longer to complete as the input size (n) grows, whether it runs on a single thread or multiple threads. The key interest here is how the algorithm scales with the input, rather than the specific execution details such as threading.

Parallelism Considerations

When analyzing parallel algorithms, different metrics such as work and span are often used, along with specific notations like O_p (parallel complexity) to capture the benefits of threading. These metrics are distinct from the standard Big O notation used for sequential algorithms. The use of these parallel-specific notations emphasizes the importance of understanding parallel execution but keeps them separate from the broader scope of Big O analysis.

Simplicity and Clarity

Introducing the number of threads into Big O notation could add unnecessary complexity to the analysis without providing meaningful insights for most use cases. The primary goal of Big O notation is to provide a simple and clear understanding of an algorithm's efficiency, focusing on how well it scales with the input size rather than the specific mechanisms used to achieve that scalability.

Conclusion

While threading can indeed impact the actual runtime of an algorithm, the role of Big O notation is to abstract away these implementation details to focus on the fundamental growth of the algorithm's complexity as the input size increases. Understanding the growth rate helps developers and analysts make informed decisions about which algorithms to use in different contexts, without the need to delve into the intricacies of threading and parallelism in a general analysis.