AI 中文总结
研究给定正整数\(k\)时，寻找包围\(\mathbf{Z}^2\)中\(k\)个点的最小周长凸多边形问题，利用结构界限给出确定性算法，可在\(O(k^{29/18 + o(1)})\)时间内计算出最优多边形，改进了之前算法。
AI 中文摘要
给定正整数\(k\)，我们研究寻找包围\(\mathbf{Z}^2\)中恰好\(k\)个点的最小周长凸多边形的问题。我们证明最优多边形包含在宽度为\(O(k^{1/6})\)的圆环内，有\(\Theta(k^{1/3})\)个边界网格点，最长边长度为\(\Theta(k^{1/4})\)。利用这些结构界限，我们给出一种确定性算法，能在\(O(k^{29/18 + o(1)})\)时间内计算出最优多边形，改进了之前\(O(k^3)\)时间的算法。
英文摘要
Given a positive integer $k$, we study the problem of finding a convex polygon of minimum perimeter that encloses exactly $k$ points of $\mathbf{Z}^2$. We show that an optimal polygon is contained in a circular annulus of width $O(k^{1/6})$, has $Θ(k^{1/3})$ boundary grid points, and its longest edge has length $Θ(k^{1/4})$. Using these structural bounds, we present a deterministic algorithm that computes an optimal polygon in $O(k^{29/18+o(1)})$ time, improving over the previous $O(k^3)$-time algorithm.
