Multithreaded CPU Design for Valid 1‑D Convolution

Assume valid 1‑D convolution produces output y of length M = N − K + 1 from input x (length N) and kernel h (length K):

• y[i] = Σ_{j=0}^{K−1} x[i + j] · h[j], for i = 0 … M−1.
• Arrays are contiguous in memory; dtype is float32 or float64.
• Target: general-purpose multicore CPU with SIMD (e.g., AVX2/AVX‑512), typical 64‑B cache lines.

Design and implement a multithreaded CPU version. For each case, describe work partitioning, scheduling, synchronization, cache locality, false‑sharing avoidance, vectorization (SIMD), and how to combine partial results. Provide pseudocode or an API‑level design.

Cases:

(a) input length N = 1,000,000; kernel length K = 3.

(b) input length N = 1,000,000; kernel length K = 1,000,000.

(c) maximum worker threads = 100.