Shortest Paths Among Obstacles in the Plane Revisited
Haitao Wang
备注：
Published in SODA 2021. Observation 2 in the previous version (and also in SODA proceedings) is not correct. The issue is addressed in this version with slightly different analysis
Given a set of pairwise disjoint polygonal obstacles in the plane, finding an obstacle-avoiding Euclidean shortest path between two points is a classical problem in computational geometry and has been studied extensively. The previous best algorithm was given by Hershberger and Suri [FOCS 1993, SIAM J. Comput. 1999] and the algorithm runs in $O(n\log n)$ time and $O(n\log n)$ space, where $n$ is the total number of vertices of all obstacles. The algorithm is time-optimal because $Ω(n\log n)$ is a lower bound. It has been an open problem for over two decades whether the space can be reduced to $O(n)$. In this paper, we settle it by solving the problem in $O(n\log n)$ time and $O(n)$ space, which is optimal in both time and space; we achieve this by modifying the algorithm of Hershberger and Suri. Like their original algorithm, our new algorithm can build a shortest path map for a source point $s$ in $O(n\log n)$ time and $O(n)$ space, such that given any query point $t$, the length of a shortest path from $s$ to $t$ can be computed in $O(\log n)$ time and a shortest path can be produced in additional time linear in the number of edges of the path.
Algorithms for the Shortest Vector Problem in $2$-dimensional Lattices, Revisited
College of Computer Science and Technology, Qingdao University（青岛大学计算机科学与技术学院）；State Key Laboratory of Cryptography and Digital Economy Security, Shandong University（山东大学密码与数字经济安全国家重点实验室）；School of Cyberspace Security, Beijing University of Posts and Telecommunications（北京邮电大学网络空间安全学院）；School of Cyber Science and Technology, Shandong University (Qingdao)（山东大学（青岛）网络科学学院）
Computing a rectilinear shortest path amid splinegons in plane
我们把样条域\calS中两个给定点s和t之间的直线最短路径的计算问题归结为多边形域中两个点之间的直线最短路径的计算问题。作为其中的一部分，我们定义了一个多边形域\calP从\calS和转换的直线最短路径计算\calP之间的s和t之间的路径之间的样条障碍\calS。当\calS由h个两两不相交的样条函数组成，总共有n个顶点时，不包括在\calP中计算多边形之间的直线最短路径的时间，我们的约简算法需要O（n + h \lg{n}）时间。对于\calS由凹进样条函数组成的特殊情况，我们设计了另一种算法，在该算法中，简化过程不依赖于算法中所使用的结构来计算多边形域中的直线最短路径。作为其中的一部分，我们的特点是一些性质的直线最短路径之间的样条，这可能是独立的利益。
Multiple Shooting Approach for Finding Approximately Shortest Paths for Autonomous Robots in Unknown Environments in 2D
Institute of Mathematical and Computational Sciences, Ho Chi Minh City University of Technology (HCMUT)（胡志明市理工大学数学与计算科学研究所）；Vietnam National University Ho Chi Minh City（越南国家大学胡志明市分校）；Institute of Mathematics, Vietnam Academy of Science and Technology (VAST)（越南科学技术翰林院数学研究所）
Dubins路径是具有有界曲率的最短路径。非完整运动规划的开创性结果是（在没有障碍物的情况下）Dubins路径由一个圆弧、一个线段和另一个圆弧组成，或者由三个圆弧组成[Dubins，1957]。Dubins最初的证明使用了高等微积分;后来，Dubins的结果被使用控制理论技术重新证明[Reeds and Shepp，1990]，[Sussmann and Tang，1991]，[Boissonnat，Cérézo，and Leblond，1994]。我们介绍和研究的曲率约束运动的离散模拟。我们发现，最短的“有界曲率”的多边形路径具有相同的结构Dubins路径。Dubins路径的性质遵循我们的结果作为一个限制的情况下-这给出了一个新的，“离散”的证明Dubins结果。
