P52 · 极值构型 · 经典问题 · 难

最远与最近距离之比 · n = 24

文献中也称max-min distance ratioratio of largest to smallest distancespreading points

在平面上放置 n 个点，记最远点对距离为 D、最近点对距离为 d，使比值 D/d 尽可能小。

子题n = 24

目标最小化最远与最近距离之比

纪录对比

最远与最近距离之比 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	4.574735319	见来源署名	Erich Friedman ↗
本站纪录	暂无选手纪录	—	—

严格定义

容器无容器几何约束：距离比值对平移和缩放不变，提交时将点集缩放到单位正方形 [0, 1]² 内即可
提交恰好 n 个点 points，两两不重合
极端距离d(P) 为最近点对距离，D(P) 为最远点对距离 $d (P) = \min_{i < j} ‖ p_{i} - p_{j} ‖, D (P) = \max_{i < j} ‖ p_{i} - p_{j} ‖$
分数ρ(P) 对平移、旋转与非零缩放不变；验证器在内部按距离平方精确比较，并将 ρ(P) 向上取整到九位小数 $ρ (P) = \frac{D (P)}{d (P)}$
目标在所有 n 点构形 P 中最小化 ρ(P) $\min_{P} ρ (P)$

放大编辑并提交 ↗

当前展示：起步布局

1y0

123456789101112131415161718192021222324

0x1

已验证构造最远与最近的一对点

帮助理解

结构特征

同时压低最远距离并推高最近距离，会迫使点集整体趋向圆形、内部趋向正三角晶格（蜂窝排布）。难点在于如何裁剪外圈边界，在晶格规整性与外接直径之间取得平衡。

研究进展

Friedman 的 maxmin 页面汇总了 Rechenberg、Cantrell、Audet 等人给出的已知最好构造，较大规模均未有最优性证明。

查看来源

构型与历史

起步布局

查看构型、求解笔记与纪录历史。

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

ANSWER FORMAT

答案怎么写

距离比值对平移与非零缩放不变，而任意点集的宽和高均不超过其直径，因此总能平移缩放到单位正方形 [0, 1]² 内。左下角为原点 (0, 0)，右上角为 (1, 1)，坐标写成最多九位小数的十进制字符串。

提交 points，每个坐标写成十进制字符串，例如 "0.5"。距离比值对平移与缩放不变，将点集缩放到单位正方形 [0, 1]² 内提交即可。

起步布局答案

{
  "points": [
    [
      "0.299306709",
      "0.972647780"
    ],
    [
      "0.685748370",
      "0.979583741"
    ],
    [
      "0.494529779",
      "0.864559662"
    ],
    [
      "0.880971440",
      "0.871495623"
    ],
    [
      "0.108088118",
      "0.857623701"
    ],
    [
      "0.303311188",
      "0.749535583"
    ],
    [
      "0.689752849",
      "0.756471544"
    ],
    [
      "0.000000000",
      "0.662400631"
    ],
    [
      "0.995995519",
      "0.680277032"
    ],
    [
      "0.498534258",
      "0.641447465"
    ],
    [
      "0.195223070",
      "0.554312513"
    ],
    [
      "0.804776928",
      "0.565252953"
    ],
    [
      "0.004004479",
      "0.439288434"
    ],
    [
      "0.999999998",
      "0.457164835"
    ],
    [
      "0.390446140",
      "0.446224395"
    ],
    [
      "0.613558337",
      "0.450228874"
    ],
    [
      "0.199227549",
      "0.331200316"
    ],
    [
      "0.808781407",
      "0.342140756"
    ],
    [
      "0.394450619",
      "0.223112198"
    ],
    [
      "0.617562816",
      "0.227116677"
    ],
    [
      "0.203232028",
      "0.108088119"
    ],
    [
      "0.812785886",
      "0.119028559"
    ],
    [
      "0.398455098",
      "0.000000001"
    ],
    [
      "0.621567295",
      "0.004004480"
    ]
  ]
}

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": 24
}

起步布局答案

{
  "points": [
    [
      "0.299306709",
      "0.972647780"
    ],
    [
      "0.685748370",
      "0.979583741"
    ],
    [
      "0.494529779",
      "0.864559662"
    ],
    [
      "0.880971440",
      "0.871495623"
    ],
    [
      "0.108088118",
      "0.857623701"
    ],
    [
      "0.303311188",
      "0.749535583"
    ],
    [
      "0.689752849",
      "0.756471544"
    ],
    [
      "0.000000000",
      "0.662400631"
    ],
    [
      "0.995995519",
      "0.680277032"
    ],
    [
      "0.498534258",
      "0.641447465"
    ],
    [
      "0.195223070",
      "0.554312513"
    ],
    [
      "0.804776928",
      "0.565252953"
    ],
    [
      "0.004004479",
      "0.439288434"
    ],
    [
      "0.999999998",
      "0.457164835"
    ],
    [
      "0.390446140",
      "0.446224395"
    ],
    [
      "0.613558337",
      "0.450228874"
    ],
    [
      "0.199227549",
      "0.331200316"
    ],
    [
      "0.808781407",
      "0.342140756"
    ],
    [
      "0.394450619",
      "0.223112198"
    ],
    [
      "0.617562816",
      "0.227116677"
    ],
    [
      "0.203232028",
      "0.108088119"
    ],
    [
      "0.812785886",
      "0.119028559"
    ],
    [
      "0.398455098",
      "0.000000001"
    ],
    [
      "0.621567295",
      "0.004004480"
    ]
  ]
}

提交 points，每个坐标写成十进制字符串，例如 "0.5"。距离比值对平移与缩放不变，将点集缩放到单位正方形 [0, 1]² 内提交即可。 · 验证器 v1.0.0

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