AI中文摘要
我们研究小运动下精确最近邻维护的动态几何数据结构。对每个点，我们存储一个由最近邻和两个最小邻近距离组成的证书，间隙为$c_i=d^i_2-d^i_1$。三角不等式给出一个尖锐的有效性半径：在最大位移为$\varepsilon$的一步后，每个满足$c_i>4\varepsilon$的证书仍然有效，因此所有可能的失效被限制在修复前沿$F_t$内。我们引入修复前沿熵$H(F_t)$，即失效证书在索引单元上的归一化香农熵，作为选择事件驱动修复、批量修复或完全重建的工作负载描述符。由此产生的维护规则在单元占用有界的情况下，仅以$O(|F_t|\log N)$时间修复前沿，而完全重建代价为$\Theta(N)$；此外，熵为事件驱动修复所触及的前沿单元数量提供下界，并改变了经验上的修复-重建交叉点。我们在$d\in\{2,3\}$中评估了十种运动族，$N$高达16,000，使用精确的平铺GPU预言机和GPU网格重建作为真实值和竞争者。在2400个标记的转换中，有效性规则没有遗漏任何无效证书，低压前沿通常通过增量修复更便宜，而相同大小的扩散前沿对于事件驱动修复更昂贵，但对于批量修复则不然。发布的数据集记录了前沿几何、证书审计、每种策略的时间以及最佳策略标签。
英文摘要
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 $Θ(N)$; 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.
