AI 中文总结
针对平面含h个凸多边形障碍物、总顶点数n的多边形域，提出预处理算法生成数据结构，可快速查询两点间满足(1+ε)乘性与13ℓ加性伸展的近似最短路径。
AI 中文摘要
给定由h个互不相交的凸多边形障碍物构成、总顶点数为n的多边形域P，以及取值在(0,0.6)内的正实数ε，本文提出一种算法，可在O(n + (h/ε)(h + 1/√ε)lg(h/√ε))时间内预处理P，生成大小为O(n + (h/√ε)(h + 1/ε))的数据结构；对自由空间中任意两点s和t，该算法能在O((1/√ε)lg(h/√ε) + (h/ε^2.5)lg lg(h/√ε))时间内输出满足(1+ε)乘性伸展和13ℓ加性伸展的路径，其中ℓ的上界为√(2ε)乘以P中任意多边形P_i内两点间的最大距离。
英文摘要
Given a polygonal domain $\cal P$ consisting $h$ pairwise disjoint convex polygonal obstacles together defined with $n$ vertices and a positive real number $ε$ in $(0, 0.6)$, this paper presents an algorithm to preprocess $\cal P$ in $O(n+\frac{h}ε(h+\frac{1}{\sqrtε})\lg(\frac{h}{\sqrtε}))$ time to compute data structures of size $O(n+\frac{h}{\sqrtε} (h+\frac{1}ε))$ so that given any two points $s$ and $t$ in the free space defined by $\cal P$, a path between $s$ and $t$ with a $(1+ε)$ multiplicative stretch and $13\ell$ additive stretch is output in $O(\frac{1}{\sqrtε}(\lg{\frac{h}{\sqrtε}})+\frac{h}{ε^{2.5}}(\lg{\lg(\frac{h}{\sqrtε})}))$ time. Here, $\ell$ is upper bounded by $(\sqrt{2ε}) (\max_{P_i \in \cal P} \max_{p, q \in P_i} |pq|)$.
