zkw线段树

介绍

非递归线段树实现方法，码量较短。

zkw 线段树的构造原理：

普通线段树采用堆存储，zkw线段树 本质上是满二叉树（若没有该区间则为空点）

但根据实际情况，原区间不一定构成满二叉树，据查询方式限制，空间开到最接近的 $2^n$（据性质树值域 = 底层节点数），即不存在的点有虚点填充。

既然不存在区间也需用空点填充，zkw线段树 对比 普通线段树 空间较大？

相比于普通线段树的结构混乱，一般开到 4 倍空间，而 zkw线段树 则只大约开到 3 倍空间

普通线段树本质上是从上往下搜索，从根节点向下操作。

zkw 线段树不同于普通线段树，本质上是从下往上搜索，从叶子节点向上操作。

zkw 线段树的基本操作

建树操作：

while (m <= n) {
    m <<= 1;
}
for (int i = m + 1; i <= m + n; ++ i) {
    tr[i] = read ();
}
m -= 1;
while (m --) {
    solve ();
}

维护区间和：$\mathtt{t}\mathtt{r}\mathtt{[}\mathtt{]}\to \mathtt{s}\mathtt{u}\mathtt{m}\mathtt{[}\mathtt{]}$

solve () : sum[i] = sum[i << 1] + sum[i << 1 | 1];

维护区间最小值：$\mathtt{t}\mathtt{r}\mathtt{[}\mathtt{]}\to \mathtt{m}\mathtt{i}\mathtt{n}\mathtt{n}\mathtt{[}\mathtt{]}$

solve () : minn[i] = min (minn[i << 1], minn[i << 1 | 1]); // 不支持修改

solve () : minn[i] = min (minn[i << 1], minn[i << 1 | 1]);
minn[i << 1] -= minn[i];
minn[i << 1 | 1] -= minn[i];

维护区间最大值：$\mathtt{t}\mathtt{r}\mathtt{[}\mathtt{]}\to \mathtt{m}\mathtt{a}\mathtt{x}\mathtt{x}\mathtt{[}\mathtt{]}$

solve () : maxx[i] = max (maxx[i << 1], maxx[i << 1 | 1]); // 不支持修改

solve () : maxx[i] = max (maxx[i << 1], maxx[i << 1 | 1]);
maxx[i << 1] -= maxx[i];
maxx[i << 1 | 1] -= maxx[i];

单点查询：

沿根节点向叶子节点累计加和

ll total = 0;
for (int i = id + m; i; i >>= 1) {
    total += minn[i]; 
}

单点修改：

修改当前节点并更新父亲节点

tr[id = id + m] += v;
while (m) {
    solve ();
    m >>= 1;
}

维护区间和：$\mathtt{t}\mathtt{r}\mathtt{[}\mathtt{]}\to \mathtt{s}\mathtt{u}\mathtt{m}\mathtt{[}\mathtt{]}$

solve () : sum[i] = sum[i << 1] + sum[i << 1 | 1];

维护区间最小值：$\mathtt{t}\mathtt{r}\mathtt{[}\mathtt{]}\to \mathtt{m}\mathtt{i}\mathtt{n}\mathtt{n}\mathtt{[}\mathtt{]}$

solve () : minn[i] = min (minn[i << 1], minn[i << 1 | 1]); // 不支持修改

solve () : minn[i] = min (minn[i << 1], minn[i << 1 | 1]);
minn[i << 1] -= minn[i];
minn[i << 1 | 1] -= minn[i];

维护区间最大值：$\mathtt{t}\mathtt{r}\mathtt{[}\mathtt{]}\to \mathtt{m}\mathtt{a}\mathtt{x}\mathtt{x}\mathtt{[}\mathtt{]}$

solve () : maxx[i] = max (maxx[i << 1], maxx[i << 1 | 1]); // 不支持修改

solve () : maxx[i] = max (maxx[i << 1], maxx[i << 1 | 1]);
maxx[i << 1] -= maxx[i];
maxx[i << 1 | 1] -= maxx[i];

区间查询

图片来源于网络（https://csacademy.com/app/graph_editor/）

假定先需查询区间 $[1,4]$ 那么操作如下：

1. 闭区间改开区间：$[1,4]\to (0,5)$ 扩增至 $(M+0,M+5)=(8,13)$
2. 判断：左端点 $1000$ 为 左儿子，其兄弟节点必在区间内，累加 $total+=D[{1001}_B]$；判断：右端点 $1101$ 为右儿子，其兄弟节点必在区间内，累加 $total+=D[{1101}_B]$
3. 缩小区间：$\mathtt{l}\mathtt{>>=}\mathtt{1}$ $\to$ ${\mathtt{1000}}_{\mathtt{B}}\mathtt{>>}\mathtt{1}\mathtt{=}{\mathtt{0100}}_{\mathtt{B}}\mathtt{,}$ $\mathtt{r}\mathtt{>>=}\mathtt{1}$ $\to$ ${\mathtt{1101}}_{\mathtt{B}}\mathtt{>>}\mathtt{1}\mathtt{=}{\mathtt{0110}}_{\mathtt{B}}$
4. 判断：左端点 $0100$ 为 左儿子，其兄弟节点必在区间内，累加 $total+=D[{0101}_B]$；判断：右端点 $1101$ 为左儿子，不操作
5. 缩小区间：$\mathtt{l}\mathtt{>>=}\mathtt{1}$ $\to$ ${\mathtt{0100}}_{\mathtt{B}}\mathtt{>>}\mathtt{1}\mathtt{=}{\mathtt{0010}}_{\mathtt{B}}\mathtt{,}$ $\mathtt{r}\mathtt{>>=}\mathtt{1}$ $\to$ ${\mathtt{0110}}_{\mathtt{B}}\mathtt{>>}\mathtt{1}\mathtt{=}{\mathtt{0011}}_{\mathtt{B}}$
6. 此时 节点 $\mathtt{l}$ 和 节点 $\mathtt{r}$ 为兄弟节点，停止操作。

伪代码：

for (int l = 开区间左端点, r = 开区间右端点; l, r 不是兄弟节点; 缩小区间) {
    if (l 为左儿子) {
        total += c[l 的兄弟节点];
    }
    if (r 为右儿子) {
        total += c[r 的兄弟节点];
    }
}

维护区间和：

for (int l = l + m - 1, r = r + m - 1, l ^ r ^ 1; l >>= 1, r >>= 1) {
    if (~ l & 1) {
        total += c[l ^ 1];
    }
    if (r & 1) {
        total += c[r ^ 1];
    }
}

维护区间最小值：

ll L = 0, R = 0;
for (int l = l + m - 1, r = r + m - 1; l ^ r ^ 1; l >>= 1, r >>= 1) {
    L += minn[l], R += minn[r];
    if (~ l & 1) {
        L = min (L, minn[l ^ 1]);
    }
    if (r & 1) {
        R = min (R, minn[r ^ 1]);
    }
}
ll res = min (L, R);
while (l) {
    res += minn[l >>= 1];
}

维护区间最大值：

ll L = 0, R = 0;
for (int l = l + m - 1, r = r + m - 1; l ^ r ^ 1; l >>= 1, r >>= 1) {
    L += maxx[l], R += maxx[r];
    if (~ l & 1) {
        L = max (L, maxx[l ^ 1]);
    }
    if (r & 1) {
        R = max (R, maxx[r ^ 1]);
    }
}
ll res = max (L, r);
while (l) {
    res += maxx[l >> 1];
}

区间修改

咕了。