flowchart TB
    classDef start fill:#D1FAE5,stroke:#10B981,stroke-width:2px,color:#064E3B
    classDef step fill:#DBEAFE,stroke:#2563EB,stroke-width:2px,color:#1E3A8A
    classDef decide fill:#FEF3C7,stroke:#D97706,stroke-width:2px,color:#78350F
    classDef guard fill:#FFE4E6,stroke:#BE123C,stroke-width:2px,color:#881337
    classDef done fill:#FFEDD5,stroke:#EA580C,stroke-width:2px,color:#7C2D12

    input["输入 Y = { yᵢ } ⊂ ℝ³², i = 1…N&nbsp;&nbsp;（可先随机抽子集拟合：fit_max_samples）<br/>目标 &nbsp; min_{C,μ} J(C,μ) = Σᵢ ‖yᵢ − μ_{c(i)}‖²&nbsp;&nbsp;— NP-hard，只求局部最优"]:::start

    subgraph INIT["① k-means++ 初始化&nbsp;&nbsp;（期望近似比 E[J] ≤ 8(ln k + 2)·J_opt）"]
        direction LR
        pick1["均匀随机取 μ₁"]:::step
        dist["D(y) = min_{j≤t} ‖y − μⱼ‖"]:::step
        sample["按 p(y) ∝ D(y)² 采样 μ_{t+1}<br/>离已有质心越远越易被选中"]:::step
        kdone{"够 k 个？"}:::decide
        pick1 --> dist --> sample --> kdone
        kdone -.->|"否，继续挑"| dist
    end

    subgraph LLOYD["② Lloyd 迭代&nbsp;&nbsp;（J 单调不增且有下界，故必收敛）"]
        direction LR
        assign["指派步<br/>c(i) = argminⱼ ‖yᵢ − μⱼ‖²<br/>一次 cdist 算完 N×k 距离"]:::step
        update["更新步<br/>μⱼ ← mean{ yᵢ : c(i) = j }<br/>均值是簇内平方距离的唯一最小点"]:::step
        empty{"有空簇？"}:::decide
        steal["把离自身质心最远的点<br/>提为该空簇的新质心<br/>保证簇数不缩水"]:::guard
        conv{"|J_prev − J| ≤ tol·J_prev<br/>或迭代满 100 次？"}:::decide
        assign --> update --> empty
        empty -->|"否"| conv
        empty -->|"是"| steal --> conv
        conv -.->|"否，回到指派步"| assign
    end

    repeat["③ 以上 ①②&nbsp;整体重复 n_init = 10 次<br/>每次换随机种子 seed + r"]:::step
    best["④ 取 10 次里 J 最小的一次 (C*, μ*)<br/>缓解 Lloyd 只保证局部最优"]:::done
    final["⑤ 全量指派&nbsp;&nbsp;cᵢ = argminⱼ ‖yᵢ − μ*ⱼ‖，i = 1…N<br/>并记录到质心的距离 dᵢ，供排序与离群检查"]:::done

    input --> INIT
    kdone ==>|"是，得到初值 μ⁽⁰⁾"| assign
    conv ==>|"是，本次收敛"| repeat
    repeat --> best --> final