QOJ.ac

QOJ

Type: Editorial

Status: Open

Posted by: incent

Posted at: 2026-05-15 11:52:55

Last updated: 2026-05-15 14:32:16

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只有三种操作:新增点;横扫;竖扫。 查询单点坐标

扫除以遇到 直角三角形 边界为限。

上文图片为分治结构,每个cell为正方形,以下事实:

对每个点,完整周期内,经过的正方形折半,只有log个

每次扫除的时候,将会经过log个正方形。

还有一个均摊:每次修改某个点的x or y的时候,坐标的颜色数要减少的。


在正方形内,在单点的视角,扫除操作总是可以看作对 x 或 对 y 的chkmax,

每次操作一定是平等施加到每个点上的,(这是因为如若不然,此操作将会使此正方形内扫到的点进入另一个正方形,根据上文标黑性质)。

框架:

正方形内:浓缩管辖内的操作的总和(以处理查询),对将点移出正方形的扫除操作,枚举,并加入到新的正方形内

操作时:枚举log个正方形,施加操作


正方形内:

所有点 x,y 构成两个集合,分别维护,假设用 动态开点线段树(只保留虚树,额外记录最小值位置)

操作:chkmax x:(即修改x坐标,y不变)

检查此次操作是否直接将点移出正方形:

是:遍历y的set并将前缀取出 否:则操作是全局操作,遍历x的set并将某个前缀合并为单点


上文 “是” 的分支还需要插入。直接插入到对应的正方形中即可。


复杂度:

取前缀,并修改:O(1) 插入:合并线段树,套线段树合并理论所以总复杂度O(n log n)(视n,V同阶),注意第一个操作不会导致线段树总点数增加,但是可能会修改log个线段树节点成为另外log个节点,所以用虚树就避免了这里的复杂度开销。

因此最终复杂度我想是 $O(n\log n)$ 的。


使用set而非动态开点虚线段树的实现

这个思路目前处于过耦合的状态,而且做到单log做的并不漂亮,也难写,用ds维护点也有点奇怪。

现在还有另外一个做法,来自超级天才easontao。(如果它允许我写出来的话):

上文结构直观的理解是把点看作区间,然后挂到线段树最后一次包含这个区间的线段树区间上,修改想想可以发现两种,一种是chkmax chkmin一种是有交就弹出。

分别用stack维护这两个东西。查询直接二分。但是可能会在某个时刻之后 “跌落” 到下面的区间。不过这个次数是log次。所以查询复杂度两个log修改只有一个!此做法在这里分道扬镳我要维护点它维护的是操作。它给我讲还有别的处理方法,它觉得一定可以做到一个log,希望它能教教我()


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