How Can SIMD Instructions on Intel CPUs Optimize Prefix Sum Algorithms?

SIMD Prefix Sum on Intel CPU

Prefix sum algorithms are commonly used to compute the cumulative sum of elements in an array. For time-critical applications, optimizing this algorithm is essential. One approach to achieving this is through SIMD (Single Instruction Multiple Data) instructions on Intel CPUs.

Conventional Sequential Approach

A naive implementation involves iterating through the array and recursively summing elements in pairs. While straightforward, this approach is limited by its sequential nature.

SIMD Prefix Sum Algorithm

For faster computation, a parallel prefix sum algorithm can be employed. It consists of two passes:

Pass 1: Calculate partial sums in parallel and store the total sum for each partial sum.

Pass 2: Add the total sum from the preceding partial sum to the next partial sum.

SSE Optimization

The second pass can be optimized using SSE instructions, which perform vector operations in parallel. Instead of iterating sequentially, a constant value is added to multiple elements simultaneously.

Performance Analysis

Assuming n elements in the array, m cores, and a SIMD width of w, the time complexity of the SIMD prefix sum algorithm is:

(n/m) * (1 1/w),

which is notably faster than sequential code.

Example Implementation

The provided code implements the SIMD prefix sum algorithm in C using SSE intrinsics and OpenMP for parallelization.

float scan_SSE(__m128 x) {
    x = _mm_add_ps(x, _mm_castsi128_ps(_mm_slli_si128(_mm_castps_si128(x), 4))); 
    x = _mm_add_ps(x, _mm_shuffle_ps(_mm_setzero_ps(), x, 0x40)); 
    return x;
}

void scan_omp_SSEp2_SSEp1_chunk(float a[], float s[], int n) {
    // ... (code omitted for brevity)
}

Conclusion

This SIMD prefix sum algorithm offers significant performance improvements over the conventional sequential approach. By leveraging parallelism and SSE instructions, it achieves a time complexity close to optimal for the available hardware resources.

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