[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86048-en":3,"doc-seo-86048-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},86048,1099514067438,"River Wang","https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542",8,"Research & Report","How to Catch k Grid Points","Given a positive integer k, the document addresses the minimum-perimeter problem of constructing a closed convex polygon that contains exactly k points of the integer grid Z2. It proves structural properties of an optimal polygon: it lies inside a circular annulus of width O(k^1/6), has Θ(k^1/3) boundary grid points, and has a longest edge of length Θ(k^1/4). Leveraging these bounds, a deterministic algorithm computes an optimal polygon in O(k^29/18+o(1)) time, improving upon a prior O(k^3) approach.","arXiv :2607 . 10824v1 [ cs .CG] 12 Jul 2026  \nHow to Catch k Grid Points  \nSariel Har-Peled∗ Elfarouk Harb† Qizheng He‡  \nAbstract  \nGiven a positive integer k, we study the problem of finding a convex polygon of minimum perimeter that encloses exactly k points of Z2 . We show that an optimal polygon is contained in a circular annulus of width O(k1/6), has Θ(k1/3) boundary grid points, and its longest edge has length Θ(k1/4) . Using these structural bounds, we present a deterministic algorithm that computes an optimal polygon in O (k29/18+o(1)) time, improving over the previous O(k3 )-time algorithm.  \nThe minimum perimeter polygon covering k points, for k = 3 , . . . , 12. More examples are at  , ,  , , and  .  \n∗ School of Computing and Data Science; University of Illinois; 201 N. Goodwin Avenue; Urbana, IL, 61801, USA; ;  .  \n†School of Computing and Data Science; University of Illinois; 201 N. Goodwin Avenue; Urbana, IL, 61801, USA; ;  .  \n‡  \n.  \n1. Introduction  \nWe study the following problem.  \nProblem 1.1 . Given a positive integer k, find a (closed) convex polygon P of minimum perimeter, such that P contains exactly k points of the integer grid Z2 .  \nBy viewing the boundary of P as a rubber band trying to shrink to its minimum length, it is clear that such an optimal polygon has vertices only at points of the grid Z2 . Let k denote such an optimal polygon, and let kdenote its perimeter. Observe that 0 = 1 \u003C 2 \u003C ··· is a strictly increasing monotone sequence, since one can always snip a corner of k to obtain a shorter-perimeter polygon containing k − 1 points.  \nWhy is this problem interesting? Coming up with a DP for this problem that is relatively efficient is not obvious. We describe below the known algorithm of Eppstein and Erickson [] modified for this case, which has running time O(k3 ), which is our baseline. It is natural to ask if one can do better. First, because this is a natural problem. Secondly, as k increases, the optimal solution converges to a disk. There is a lot of related work (see below), some of it recent, trying to understand the behavior of disks on a grid (such as the Gauss circle problem) .  \nIndeed, it is tempting to conjecture that maybe an output sensitive linear-time algorithm exists for this problem (i.e., this would be running time O (k1/3) in this case), since the related discrete hull-problem (compute the convex-hull of the grid points inside a given disk of radius √k  ) has such an algorithm [] . And yet, to even beat the baseline requires non-trivial ideas and work. One needs to use tools from the geometry of numbers (such as Farey’s sequences), and rather intricate exchange arguments to argue about the shape of the optimal solution. Indeed, the DP uses the structure to improve its performance.  \nAnd even if the question is not interesting beyond discrete geometry, we believe the answer is (arguably) quite interesting. As our work hopefully demonstrates, this problem has a rather intricate and fascinating mathematical structure. Beyond that, scaling our implementation to work on huge inputs required many interesting ideas on the applied side, which should be useful for other DP problems whose instances are too large to fit in memory.  \nThe starting point: An O (k3 )-time algorithm. The algorithm of Eppstein and Erickson [], building on Dobkin et al. [], implies  \n|  |  |\n| --- | --- |\n|  |  |\n|  | p  |\n|  |  |\n|  |  |\n|  |  |\n|  |  |\n|  |  |\n|  |  |\n|  |  |\n|  |  |\n|  |  |\n\nFigure 1.1: Left: An optimal solution for 100 points and its origin-anchored triangulation. Right: A DP state consists of an edge from the origin together with the number of grid points covered by the minimum-perimeter partial polygon ending at this edge—in this case (p,52) .  \nan O (k4 )-time algorithm for this problem. We sketch how to improve it further to O(k3 ), following the DP of Eppstein et al. [] . The optimal solution has perimeter at most 4 √k , so one can restrict the solution to the grid G = J−2 √k ","cbCaih2DPVQfbsLW","https://ap.wps.com/l/cbCaih2DPVQfbsLW","pdf",2000361,1,62,"English","en",105,"# Introduction\n## Problem 1.1 and motivation\n## O(k^3)-time baseline via dynamic programming\n## Improved algorithm and state structure\n## Related work","[{\"question\":\"What is the optimization goal for a given k in the problem?\",\"answer\":\"For a positive integer k, the task is to find a closed convex polygon with minimum perimeter that contains exactly k integer-grid points from Z2.\"},{\"question\":\"Why can an optimal polygon be assumed to have vertices only on Z2 points?\",\"answer\":\"The document explains that boundary shrinking behavior implies an optimal solution can be realized with vertices at grid points, since a corner can be removed to obtain a shorter-perimeter polygon while reducing the number of covered points.\"},{\"question\":\"What algorithmic improvement is achieved over the previous O(k^3) method?\",\"answer\":\"Using structural bounds on the optimal polygon’s location, boundary size, and edge length, the deterministic algorithm computes an optimal polygon in O(k^29/18+o(1)) time, improving the baseline O(k^3) 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is the optimization goal for a given k in the problem?","Question",{"text":75,"@type":76},"For a positive integer k, the task is to find a closed convex polygon with minimum perimeter that contains exactly k integer-grid points from Z2.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why can an optimal polygon be assumed to have vertices only on Z2 points?",{"text":80,"@type":76},"The document explains that boundary shrinking behavior implies an optimal solution can be realized with vertices at grid points, since a corner can be removed to obtain a shorter-perimeter polygon while reducing the number of covered points.",{"name":82,"@type":73,"acceptedAnswer":83},"What algorithmic improvement is achieved over the previous O(k^3) method?",{"text":84,"@type":76},"Using structural bounds on the optimal polygon’s location, boundary size, and edge length, the deterministic algorithm computes an optimal polygon in O(k^29/18+o(1)) time, improving the baseline O(k^3) 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