Fréchet距离下多边形曲线的近似最近邻
Approximate Nearest Neighbor for Polygonal Curves under Fréchet Distance
AI总结：
本研究提出了$d$维空间中多边形曲线在Fréchet距离下的$κ$-近似最近邻数据结构，给出了查询、空间及预处理时间复杂度，并在二维、三维场景下实现了复杂度优化。
摘要：
我们提出了$\u211d^d$空间中$n$条多边形曲线在Fréchet距离下的$κ$-近似最近邻（ANN）数据结构，其中$κ\in \{1+\varepsilon,3+\varepsilon\}$且$d \geq 2$。我们假设每条输入曲线最多有$m$个顶点，每条查询曲线最多有$k$个顶点，满足$k \ll m$，且$k$在预处理阶段给定。$(1+\varepsilon)$-ANN的查询时间为$\tilde{O}(k(mn)^{0.5+\varepsilon}/\varepsilon^d+ k(d/\varepsilon)^{O(dk)})$，$(3+\varepsilon)$-ANN的查询时间为$\tilde{O}(k(mn)^{0.5+\varepsilon}/\varepsilon^d)$。两种情况的空间复杂度和期望预处理时间均为$\tilde{O}(k(mnd^d/\varepsilon^d)^{O(k+1/\varepsilon^2)})$。在二维和三维空间中，我们将$(1+\varepsilon)$-ANN的查询时间优化为$O(1/\varepsilon)^{O(k)} \cdot \tilde{O}(k)$，$(3+\varepsilon)$-ANN的查询时间优化为$\tilde{O}(k)$；两种情况的空间复杂度和期望预处理时间均优化为$O(mn/\varepsilon)^{O(k)} \cdot \tilde{O}(k)$。为便于表述，我们将界中仅依赖$d$的因子视为$O(1)$。大-$\tilde{O}$符号中隐藏的多对数因子的幂次依赖于$d$。
英文摘要：
We propose $κ$-approximate nearest neighbor (ANN) data structures for $n$ polygonal curves under the Fréchet distance in $\mathbb{R}^d$, where $κ\in \{1+\varepsilon,3+\varepsilon\}$ and $d \geq 2$. We assume that every input curve has at most $m$ vertices, every query curve has at most $k$ vertices, $k \ll m$, and $k$ is given for preprocessing. The query times are $\tilde{O}(k(mn)^{0.5+\varepsilon}/\varepsilon^d+ k(d/\varepsilon)^{O(dk)})$ for $(1+\varepsilon)$-ANN and $\tilde{O}(k(mn)^{0.5+\varepsilon}/\varepsilon^d)$ for $(3+\varepsilon)$-ANN. The space and expected preprocessing time are $\tilde{O}(k(mnd^d/\varepsilon^d)^{O(k+1/\varepsilon^2)})$ in both cases. In two and three dimensions, we improve the query times to $O(1/\varepsilon)^{O(k)} \cdot \tilde{O}(k)$ for $(1+\varepsilon)$-ANN and $\tilde{O}(k)$ for $(3+\varepsilon)$-ANN. The space and expected preprocessing time improve to $O(mn/\varepsilon)^{O(k)} \cdot \tilde{O}(k)$ in both cases. For ease of presentation, we treat factors in our bounds that depend purely on $d$ as~$O(1)$. The hidden polylog factors in the big-$\tilde{O}$ notation have powers dependent on $d$.
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