SPOJ COT2 Count on a tree II

题目大意

给定一棵n个点的树,每个点有点权。有m次询问,每次查询u,v点间的链上有几种不同颜色的点。
n<=40000 m<=100000

Solution

如果这棵树是一条链,该怎么做呢?最方便的算法肯定是莫队。可以参考PhoenixGS的博客《莫队算法》Orz。它的一个小常数优化,也可以在他的博客中找到,我就不讲了。
但是树上可不可以实现呢?答案是可以的。考虑整棵树的括号序列,每个点入栈时记下它的时间戳,出栈时也记下时间戳。这样,我们可以对每组询问进行转化。
如:
其中,黑色代表入栈时间戳,红色代表出时间戳。
如果统计每组询问的两个入时间戳之内的所有数。在上图中,询问E,C点间的链,相当于询问EEFFBC

如果统计每组询问的两个入时间戳间的所有数,就会发现,有些点被算了一次,有些点被算了两次,还有些点一次都没被计算。
被计算一次的肯定在这条链上,记录下只被计算一次的点的答案。其他点呢?我们需要分类讨论。
假设下面的uv中u肯定是入时间戳小的。
如果$lca(u,v)=u$,即u,v链是一条父子链,只被算了一次的点可以构成整条链(脑补一下)。
否则,u这个点被算了两次,lca(u,v)则一次都没算,把答案加上这两个点的贡献就可以。

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