1-D Valid Convolution with Multithreading Follow-up

Implement valid 1-D convolution over a numeric input and kernel with bias, then discuss CPU/hardware optimisations and how to parallelise under different input/kernel/thread sizes. Requirements Implement a "valid" 1-D convolution function: compute outputs only for complete input regions covered by the kernel. Inputs include an input vector, a kernel vector, and a scalar bias. For each valid offset, output the dot product between the input slice and kernel, plus bias. Follow-ups ask for optimisation methods on CPU/hardware and how to code multithreading for: input length around 1 million, kernel length 3; input length around 1 million, kernel length 1 million; a maximum of 100 threads. Examples input = [1, 2, 3, 4, 5] kernel = [2, 1, 0.5] bias = 0.5 output[0] = (1 * 2) + (2 * 1) + (3 * 0.5) + 0.5 = 6 output[1] = (2 * 2) + (3 * 1) + (4 * 0.5) + 0.5 = 9.5 output[2] = (3 * 2) + (4 * 1) + (5 * 0.5) + 0.5 = 13 Notes Base output length is len(input) - len(kernel) + 1 when the kernel is no longer than the input. A clean scalar implementation is two loops: output offset outside, kernel index inside. For small kernels and large input, split output indices across worker threads. For a kernel as large as the input, the number of valid output positions is tiny, so thread fan-out over output positions provides little benefit; discuss vectorisation, cache locality, and whether the problem shape justifies parallel overhead. NumPy-style vector slicing is a useful baseline optimisation, but the follow-up explicitly probes hardware-level parallelism. Preparation Implement the scalar valid convolution, then benchmark or reason through the three shapes separately: tiny kernel, huge kernel, and bounded thread count. Practise explaining chunk boundaries by output index, not input index; each output owns a complete kernel window, so threads do not need to write shared output cells.