How to Catch $k$ Grid Points

Sariel Har-Peled; Elfarouk Harb; Qizheng He

How to Catch $k$ Grid Points

计算几何 2026-07-12 v1

作者: Sariel Har-Peled , Elfarouk Harb , Qizheng He

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 $\Theta(k^{1/3})$ boundary grid points, and its longest edge has length $\Theta(k^{1/4})$ . 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.

引用

@article{arxiv.2607.10824,
  title  = {How to Catch $k$ Grid Points},
  author = {Sariel Har-Peled and Elfarouk Harb and Qizheng He},
  journal= {arXiv preprint arXiv:2607.10824},
  year   = {2026}
}