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P07 · 极值构型 · 经典问题

圆内的散点分离 · n = 20

在半径 1 的圆内放置 n 个点，使两两之间的最小距离尽可能大。

子题n = 20
目标最大化 最小两点距离

严格定义

  • 容器半径 1 的圆，圆心在 (1, 1)，两个坐标都在 0 到 2 之间
  • 提交恰好 n 个点 points，两两不重合
  • 约束每个点都在圆内或圆周上
  • 目标让最小的两点距离尽可能大。内部以其平方精确比较
放大来摆，然后提交 ↗
2y0
0x2
已验证构造最近的一对点

帮助理解

哪里有优化空间

散点分离就是装等圆：以每个点为圆心、最小距离一半为半径的圆必须互不重叠。最优构形因此也是卡死的接触结构，容器的形状决定一切。

前沿在哪里

圆内散点与等圆装圆互为对偶：最小间距 d 的点集就是半径 d/2 的装圆。cci 表的证明经换算适用，n = 4 在本站已证；其余按对偶随 cci 的进度开放。

查看来源
当前第一名

0.421738225

最小两点距离

纪录保持者NUE_13
解题方式人工
挑战这个纪录 ↗
历史纪录（1 次易主）
  1. NUE_13人工
    0.156434464→0.421738225
ANSWER FORMAT

答案怎么写

容器是半径 1 的圆，圆心在 (1, 1)，所以坐标范围是 0 到 2。坐标直接写成小数，例如 "1.5"，最多九位小数。

提交 points。每个坐标写成十进制字符串，例如 "1.5"。

当前第一名的答案

{
  "points": [
    [
      "1.833912990",
      "0.495652174"
    ],
    [
      "1.684223909",
      "0.924943332"
    ],
    [
      "1.926247627",
      "1.290414285"
    ],
    [
      "1.624327143",
      "1.613204150"
    ],
    [
      "1.258856323",
      "1.885963014"
    ],
    [
      "0.770434571",
      "1.896521792"
    ],
    [
      "0.372448078",
      "1.677267362"
    ],
    [
      "0.104368402",
      "1.307117219"
    ],
    [
      "0.080708788",
      "0.869544773"
    ],
    [
      "0.235776249",
      "0.469290947"
    ],
    [
      "0.555652386",
      "0.144347879"
    ],
    [
      "1.000993375",
      "0.048969579"
    ],
    [
      "1.506796010",
      "0.173933620"
    ],
    [
      "1.363498732",
      "0.637230660"
    ],
    [
      "1.388100411",
      "1.234037146"
    ],
    [
      "1.027825928",
      "1.501739236"
    ],
    [
      "0.650160671",
      "1.314037119"
    ],
    [
      "0.532153177",
      "0.866795824"
    ],
    [
      "0.889725781",
      "0.518281498"
    ],
    [
      "1.026832553",
      "0.995056959"
    ]
  ]
}
提交格式与技术细节需要编写程序或准备 JSON 答案时再查看＋

子题参数

{
  "n": 20
}

当前第一名的答案

{
  "points": [
    [
      "1.833912990",
      "0.495652174"
    ],
    [
      "1.684223909",
      "0.924943332"
    ],
    [
      "1.926247627",
      "1.290414285"
    ],
    [
      "1.624327143",
      "1.613204150"
    ],
    [
      "1.258856323",
      "1.885963014"
    ],
    [
      "0.770434571",
      "1.896521792"
    ],
    [
      "0.372448078",
      "1.677267362"
    ],
    [
      "0.104368402",
      "1.307117219"
    ],
    [
      "0.080708788",
      "0.869544773"
    ],
    [
      "0.235776249",
      "0.469290947"
    ],
    [
      "0.555652386",
      "0.144347879"
    ],
    [
      "1.000993375",
      "0.048969579"
    ],
    [
      "1.506796010",
      "0.173933620"
    ],
    [
      "1.363498732",
      "0.637230660"
    ],
    [
      "1.388100411",
      "1.234037146"
    ],
    [
      "1.027825928",
      "1.501739236"
    ],
    [
      "0.650160671",
      "1.314037119"
    ],
    [
      "0.532153177",
      "0.866795824"
    ],
    [
      "0.889725781",
      "0.518281498"
    ],
    [
      "1.026832553",
      "0.995056959"
    ]
  ]
}

提交 points。每个坐标写成十进制字符串，例如 "1.5"。 · 验证器 v1.0.0