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arXiv 2403.05697cs.CGcs.DS
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简单路径的动态凸包

Dynamic Convex Hulls for Simple Paths

Bruce Brewer, Gerth Stølting Brodal, Haitao Wang

备注：
To appear in SoCG 2024
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AI总结：

本文研究平面动态凸包的两种受限场景，提出基于双端队列的最优方案，实现最坏情况常数时间更新与对数时间查询，并证明匹配下界，改进了此前的摊还对数时间更新结果。

摘要：

我们研究平面动态凸包问题。现有文献中的解决方案能够以多对数时间支持点的插入与删除，并以对数时间响应对当前点集凸包的各类查询。若允许任意点的插入与删除，已被证明无法实现常数时间更新与快速查询。本文研究两种可实现最坏情况常数时间更新与对数时间查询的受限场景：我们假设所有更新操作都在点的双端队列（deque）上执行。第一种是单调路径场景，所有点按给定方向（例如水平从左到右）排序，仅可对最左和最右点进行插入与删除。第二种场景假设双端队列中的点构成一条简单路径，注意单调场景是简单路径场景的特例。针对这两种场景，我们提出的方案可在最坏情况常数时间内支持双端队列的插入与删除操作，并以$O(\n)$时间支持对凸包的标准查询，其中$n$为当前点集的点数。当前点集的凸包可在$O(h+\log n)$时间内输出，其中$h$为凸包的边数。对于仅允许在单侧进行更新的单侧单调路径场景，凸包输出时间可降至$O(h)$，且凸包查询可在$O(\log h)$时间内完成。我们所有的时间界均为最坏情况界。此外，我们证明了与这些时间界匹配的下界，因此我们的结果是最优的。作为快速对比，此前简单路径问题的最优更新时间界为Friedman、Hershberger与Snoeyink[SoCG 1989]提出的摊还$O(\log n)$时间。

英文摘要：

We consider the planar dynamic convex hull problem. In the literature, solutions exist supporting the insertion and deletion of points in poly-logarithmic time and various queries on the convex hull of the current set of points in logarithmic time. If arbitrary insertion and deletion of points are allowed, constant time updates and fast queries are known to be impossible. This paper considers two restricted cases where worst-case constant time updates and logarithmic time queries are possible. We assume all updates are performed on a deque (double-ended queue) of points. The first case considers the monotonic path case, where all points are sorted in a given direction, say horizontally left-to-right, and only the leftmost and rightmost points can be inserted and deleted. The second case assumes that the points in the deque constitute a simple path. Note that the monotone case is a special case of the simple path case. For both cases, we present solutions supporting deque insertions and deletions in worst-case constant time and standard queries on the convex hull of the points in $O(\log n)$ time, where $n$ is the number of points in the current point set. The convex hull of the current point set can be reported in $O(h+\log n)$ time, where $h$ is the number of edges of the convex hull. For the 1-sided monotone path case, where updates are only allowed on one side, the reporting time can be reduced to $O(h)$, and queries on the convex hull are supported in $O(\log h)$ time. All our time bounds are worst case. In addition, we prove lower bounds that match these time bounds, and thus our results are optimal. For a quick comparison, the previous best update bounds for the simple path problem were amortized $O(\log n)$ time by Friedman, Hershberger, and Snoeyink [SoCG 1989].

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