Hexagon

cs.DS — Data Structures and Algorithms

Diameter-Free Pointwise Gaussian KDE in Near-Additive Query Time

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

Wagner asked whether Gaussian kernel density estimation in $\mathbb R^d$ admits query time $\widetilde O(d+\varepsilon^{-2})$ without dependence on the diameter of the data. We answer this affirmatively for each fixed query: a randomized data structure answers a query to additive error $\varepsilon$ with probability $1-\delta$ in time $O((d+\varepsilon^{-2})\log(d+1)\log(2/\delta))$ , independently of the number of data points and of their diameter. The key ingredient is a variance bound for a Fastfood-type structured Fourier block that holds uniformly in the norm of the displacement.

Primarily AI-generated textHuman understanding: some partsFastfoodGaussian kernel density estimationpointwise approximationrandom Fourier featuresstructured random features

Oct 15, 2026PDF ↗View work →

hexagon:2610.00261v1

cs.DS — Data Structures and Algorithms

Subset Sum in time O(2^{0.4877n})

Contributed by Annanay Kapila

We present a uniform randomized classical algorithm for Subset Sum with running time O(2^{0.4877n}) in the word-RAM model, for inputs whose integer bit lengths are polynomially bounded in the number n of input integers. The algorithm halts on every execution and answers correctly with probability at least 2/3. The construction builds on OpenAI’s “Subset Sum in Time O(2^{0.49n})” (4 October 2026), retaining its framework of isolation, a capped preliminary search, a modular checksum, and alias subtraction. Two components are replaced: an alias procedure that computes phase sums using a smaller modulus, and a family of 43 checksum filters selected according to the input pattern. The analysis combines an averaged small-layer lemma, grouping by reduced and unreduced subset sums, and rigorous numerical certificates. Verification scripts using Arb ball arithmetic check the alias constraints and 430 moment inequalities. The argument relies on results from the earlier paper, which are restated in an appendix; the numerical certificates verify specified inequalities rather than formally verifying the entire proof.

Primarily AI-generated textHuman understanding: some partsSubset Sumcomputational complexityexponential-time algorithmsrandomized algorithmsrigorous numerics

Oct 15, 2026PDF ↗View work →

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Diameter-Free Pointwise Gaussian KDE in Near-Additive Query Time

Subset Sum in time O(2^{0.4877n})

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