> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/advanced-math/binary-lifting.md).

# 倍增法

### 思想

倍增通过“每次跳 2 的幂次”把 O(n) 的线性操作降到 O(log n)。预处理 up\[k]\[i] 表示从 i 往上跳 2^k 步到达的结点。

### 预处理

```cpp
const int LOG = 20;
int up[LOG][N], depth[N];

// 先 dfs 求每个点的父亲 up[0][u] 和深度
for (int k = 1; k < LOG; k++)
    for (int u = 1; u <= n; u++)
        up[k][u] = up[k-1][ up[k-1][u] ];
```

### 求最近公共祖先（LCA）

```cpp
int lca(int u, int v) {
    if (depth[u] < depth[v]) swap(u, v);
    int diff = depth[u] - depth[v];
    for (int k = 0; k < LOG; k++)
        if (diff >> k & 1) u = up[k][u];      // 对齐深度
    if (u == v) return u;
    for (int k = LOG - 1; k >= 0; k--)
        if (up[k][u] != up[k][v]) { u = up[k][u]; v = up[k][v]; }
    return up[0][u];
}
```

### 应用

树上 LCA、快速幂、稀疏表 ST 表（区间最值）、快速跳表。


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