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arXiv · 2610.24273

Batch range mode in O(n+Q)^1.395

Filippo di Gravina
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Abstract

We give a deterministic worst-case algorithm for exact Batched Range Mode on an arbitrary array of length n and a batch of Q intervals. The algorithm returns the mode frequency and the smallest mode identifier for each query in time (n+Q)^(tau+o(1)), where tau = 5 - sqrt(13) = 1.394448724536... . The algorithm combines virtual occurrence windows and block-boundary snapping with a query-restricted min-plus product whose first operand is column-monotone and whose second operand is row-monotone. A deterministic modular verifier uses geometric and query potentials to bound the same active regions; false modular matches are corrected by offline orthogonal range updates. The final analysis uses the sharper rectangular exponent curve of Uznanski (2026), together with exact rational parameter balancing and a transfer to prime fields. The analysis covers every frequency scale and arbitrary batch sizes. Exact frequencies and the smallest mode identifiers are recovered without increasing the exponent.

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Filippo di Gravina. 2026-10-16. Batch range mode in O(n+Q)^1.395. https://doi.org/10.5281/zenodo.23300993

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