Papers › Efficient Parallelization of a Ubiquitous Sequential Computation

Efficient Parallelization of a Ubiquitous Sequential Computation

27 Oct 2023arXiv:2311.06281archive 2025-07-28

Franz A. Heinsen

We find a succinct expression for computing the sequence xₜ = aₜ xₜ₋₁ + bₜ in parallel with two prefix sums, given t = (1, 2, …, n), aₜ ∈ℝⁿ, bₜ ∈ℝⁿ, and initial value x₀ ∈ℝ. On n parallel processors, the computation of n elements incurs 𝒪(logn) time and 𝒪(n) space. Sequences of this form are ubiquitous in science and engineering, making efficient parallelization useful for a vast number of applications. We implement our expression in software, test it on parallel hardware, and verify that it executes faster than sequential computation by a factor of n/logn.

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