Why is Transposing a 513x513 Matrix Faster Than a 512x512 Matrix?

Impact of Matrix Size on Transposition Performance

The phenomenon you observed—slower transposition of a 512x512 matrix compared to a 513x513 matrix—is attributed to cache behavior.

Cache Structure and Access

A cache is a memory structure that helps improve the performance of memory-intensive tasks by storing frequently accessed data close to the processor. It is organized into sets, which contain a number of lines that hold data. Each cache line has a size of several bytes, and a cache set can contain multiple lines.

When a memory address is accessed, the cache checks if the data for that address is present in any of the lines in its corresponding set. If it is, a cache hit occurs and the data is retrieved quickly. If it is not, a cache miss occurs and the data must be fetched from main memory, which is much slower.

Critical Stride and Cache Misses

When working with matrices, a critical stride is the distance between elements that are accessed in a consecutive pattern. In your case, the critical stride is the distance between elements in a single row of the matrix. If the stride matches the cache line size or is a multiple of it, it can lead to cache misses and performance degradation.

Matrix Transposition and Critical Stride

In your matrix transposition code, you are swapping elements along the diagonal. For a 512x512 matrix, the elements in each row are accessed with a stride of 512 bytes, which is equal to the cache line size. This results in numerous cache misses and reduced performance.

Why 513x513 is Faster

In the case of a 513x513 matrix, the critical stride is no longer a multiple of the cache line size. The elements in each row are accessed with a stride of 513 bytes, which ensures that they fall into different cache lines. This reduces the number of cache misses and improves performance.

Practical Implications

Understanding the impact of critical stride on caching is crucial for optimizing memory-intensive tasks. In your case, adjusting the size of your matrix to avoid multiples of critical strides can significantly improve transposition performance.

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