102742: [AtCoder]ABC274 C - Ameba

Problem Statement

You observed amoebae and kept some records.

Initially, there was one amoeba, numbered $1$.

You made $N$ records. According to the $i$-th record, the amoeba numbered $A_i$ disappeared by dividing itself into two new amoebae, which were then numbered $2i$ and $2i+1$.
Here, amoeba $A_i$ is said to be the parent of amoebae $2i$ and $2i+1$.

For each $k=1,\ldots,2N+1$, how many generations away is amoeba $k$ from amoeba $1$?

Constraints

• $1 \leq N \leq 2\times 10^5$
• The records are consistent. That is:
  • $1\leq A_i \leq 2i-1$.
  • $A_i$ are distinct integers.

Input

The input is given from Standard Input in the following format:

$N$
$A_1$ $A_2$ $\ldots$ $A_N$

Output

Print $2N+1$ lines. The $k$-th line should contain the generation distance between amoeba $1$ and amoeba $k$.

Sample Input 1

2
1 2

Sample Output 1

0
1
1
2
2

From amoeba $1$, amoebae $2$ and $3$ were born. From amoeba $2$, amoebae $4$ and $5$ were born.

• Amoeba $1$ is zero generations away from amoeba $1$.
• Amoeba $2$ is one generation away from amoeba $1$.
• Amoeba $3$ is one generation away from amoeba $1$.
• Amoeba $4$ is one generation away from amoeba $2$, and two generations away from amoeba $1$.
• Amoeba $5$ is one generation away from amoeba $2$, and two generations away from amoeba $1$.

Sample Input 2

4
1 3 5 2

Sample Output 2

0
1
1
2
2
3
3
2
2

问题描述

你观察了变形虫并记录了一些信息。

最初，有一个变形虫，编号为1。

你记录了N次。根据第i次记录，编号为A_i的变形虫通过分裂成两个新的变形虫消失了，编号分别为2i和2i+1。

在这里，变形虫A_i被称为变形虫2i和2i+1的父代。

对于每个k=1,...,2N+1，变形虫k离变形虫1有多少代远？

约束条件

• 1≤N≤2×10^5
• 记录是连贯的。即：
  • 1≤A_i≤2i-1。
  • A_i是不同的整数。

输入

输入从标准输入以以下格式给出：

$N$
$A_1$ $A_2$ $\ldots$ $A_N$

输出

打印2N+1行。第k行应包含变形虫1和变形虫k之间的代数距离。

样例输入1

2
1 2

样例输出1

0
1
1
2
2

从变形虫1，变形虫2和3诞生。从变形虫2，变形虫4和5诞生。

• 变形虫1离变形虫1有0代远。
• 变形虫2离变形虫1有1代远。
• 变形虫3离变形虫1有1代远。
• 变形虫4离变形虫2有1代远，离变形虫1有2代远。
• 变形虫5离变形虫2有1代远，离变形虫1有2代远。

样例输入2

4
1 3 5 2

样例输出2

0
1
1
2
2
3
3
2
2