Repair Entropy in Dynamic Geometric Nearest-Neighbour Structures

Faruk Alpay; Bugra Kilictas

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arXiv · 2606.18314

Repair Entropy in Dynamic Geometric Nearest-Neighbour Structures

Faruk Alpay·Bugra Kilictas

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Abstract

We study dynamic geometric data structures for exact nearest-neighbour maintenance under small motions. For each point we store a certificate consisting of its nearest neighbour and the two smallest neighbour distances, with clearance $c_i=d^i_2-d^i_1$. A triangle-inequality argument gives a sharp validity radius: after a step of maximum displacement $\varepsilon$, every certificate with $c_i>4\varepsilon$ remains valid, so all possible failures are confined to a repair frontier $F_t$. We introduce repair-frontier entropy $H(F_t)$, the normalized Shannon entropy of failed certificates over index cells, as a workload descriptor for choosing between event-driven repair, batched repair, and full rebuild. The resulting maintenance rule repairs only the frontier in $O(|F_t|\log N)$ time under bounded cell occupancy, while a full rebuild costs $\Theta(N)$; moreover, entropy lower-bounds the number of frontier cells touched by event-driven repair and shifts the empirical repair-rebuild crossover. We evaluate ten motion families in $d\in{2,3}$, with $N$ up to $16,000$, using an exact tiled GPU oracle and a GPU grid rebuild as ground truth and competitor. Across $2400$ labelled transitions, the validity rule misses no invalid certificate, low-pressure frontiers are usually cheaper to repair incrementally, and diffuse frontiers of the same size are more expensive for event-driven repair but not for batched repair. The released dataset records frontier geometry, certificate audits, per-strategy times, and best-strategy labels.

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Faruk Alpay, Bugra Kilictas. 2026-06-16. Repair Entropy in Dynamic Geometric Nearest-Neighbour Structures. https://arxiv.org/abs/2606.18314

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