Why is Matrix Transposition Slower for 512x512 Matrices Than for 513x513 Matrices?

Performance Anomaly in Matrix Transposition: 512x512 vs 513x513

Certain performance patterns emerge when working with square matrices of varying sizes, leading to an intriguing phenomenon: transposing matrices with dimensions of 2^n (e.g., 512x512) consistently exhibits slower execution times compared to matrices of dimensions 2^n 1 (e.g., 513x513).

Delving into the Mechanics

The disparity in performance originates from the intricate interplay between data access patterns and cache functionality. Specifically, caches are organized into sets and lines:

• Sets: Cache sections where data is temporarily stored.
• Lines: Units within sets that hold individual portions of data.

Data addresses are mapped to specific sets using a formula. Overlapping address ranges can result in contention for set occupancy, leading to cache misses.

The Critical Stride

A crucial factor in this equation is the "critical stride," which measures the distance between memory locations that effectively compete for cache lines. When data elements are stored at intervals equal to the critical stride, it triggers a cache conflict known as "false alias" or "artificial stride."

The 512x512 Impasse

A matrix of 512x512, occupying a cache with 4 lines per set and a line size of 64 bytes, encounters this pitfall. The critical stride for this configuration is 2048 bytes (4 lines * 64 bytes), coaligned with every fourth row in the matrix.

During transposition, accessing successive elements in a column causes cache lines from the first operation to be evicted. As a result, elements at critical stride intervals in the subsequent row suffer cache misses, degrading performance.

The 513x513 Escape

In contrast, a matrix of 513x513, with an odd dimension, disrupts the critical stride. Elements are no longer spaced at critical stride intervals, reducing the risk of cache conflicts. This leads to improved performance during transposition.

Conclusion

The phenomenon of slower matrix transpositions for dimensions of 2^n compared to 2^n 1 stems from cache memory characteristics. Understanding the critical stride and the impact of data alignment on cache utilization is crucial for optimizing code execution times.

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