MEX Tree Manipulation

5000ms

524288K

Description:

Given a rooted tree, define the value of vertex $$$u$$$ in the tree recursively as the MEX$$$^\dagger$$$ of the values of its children. Note that it is only the children, not all of its descendants. In particular, the value of a leaf is $$$0$$$.

Pak Chanek has a rooted tree that initially only contains a single vertex with index $$$1$$$, which is the root. Pak Chanek is going to do $$$q$$$ queries. In the $$$i$$$-th query, Pak Chanek is given an integer $$$x_i$$$. Pak Chanek needs to add a new vertex with index $$$i+1$$$ as the child of vertex $$$x_i$$$. After adding the new vertex, Pak Chanek needs to recalculate the values of all vertices and report the sum of the values of all vertices in the current tree.

$$$^\dagger$$$ The MEX (minimum excluded) of an array is the smallest non-negative integer that does not belong to the array. For example, the MEX of $$$[0,1,1,2,6,7]$$$ is $$$3$$$ and the MEX of $$$[6,9]$$$ is $$$0$$$.

Input:

The first line contains a single integer $$$q$$$ ($$$1 \le q \le 3 \cdot 10^5$$$) — the number of operations.

Each of the next $$$q$$$ lines contains a single integer $$$x_i$$$ ($$$1 \leq x_i \leq i$$$) — the description of the $$$i$$$-th query.

Output:

For each query, print a line containing an integer representing the sum of the new values of all vertices in the tree after adding the vertex.

Sample Input:

Sample Output:

Sample Input:

Sample Output:

Note:

In the first example, the tree after the $$$6$$$-th query will look like this.

$\text{[math]}$

Vertex $$$7$$$ is a leaf, so its value is $$$0$$$.
Vertex $$$6$$$ is a leaf, so its value is $$$0$$$.
Vertex $$$5$$$ only has a child with value $$$0$$$, so its value is $$$1$$$.
Vertex $$$4$$$ is a leaf, so its value is $$$0$$$.
Vertex $$$3$$$ only has a child with value $$$0$$$, so its value is $$$1$$$.
Vertex $$$2$$$ has children with values $$$0$$$ and $$$1$$$, so its value is $$$2$$$.
Vertex $$$1$$$ has children with values $$$1$$$ and $$$2$$$, so its value is $$$0$$$.

The sum of the values of all vertices is $$$0+0+1+0+1+2+0=4$$$.

Informação

Provedor Codeforces

Código CF1740H