Diverse Coloring

题意翻译

对于一颗二叉树，一个合法的染色方案定义为将每个点染成白色或蓝色，所有蓝色点有白色点邻居，白色点有蓝色点邻居。一种合法染色方案的平衡值定义为蓝色点与白色点数量差的绝对值。 有一颗给定的 $n$ 个点的二叉树，第 $i$ 个点的父亲是 $p_i(p_i < i)$，请对于 $\forall 1 < i \leqslant n$，求出只保留树上编号在 $[1, i]$ 的点时，所有合法染色方案的平衡值最小值。同时对整棵树求出一个合法染色方案取到这个最小值。 共 $T$ 组测试数据，$T \leqslant 10^4, n > 1, \sum n \leqslant 2 \cdot 10^5$。

题目描述

In this problem, we will be working with rooted binary trees. A tree is called a rooted binary tree if it has a fixed root and every vertex has at most two children. Let's assign a color — white or blue — to each vertex of the tree, and call this assignment a coloring of the tree. Let's call a coloring diverse if every vertex has a neighbor (a parent or a child) colored into an opposite color compared to this vertex. It can be shown that any tree with at least two vertices allows a diverse coloring. Let's define the disbalance of a coloring as the absolute value of the difference between the number of white vertices and the number of blue vertices. Now to the problem. Initially, the tree consists of a single vertex with the number $ 1 $ which is its root. Then, for each $ i $ from $ 2 $ to $ n $ , a new vertex $ i $ appears in the tree, and it becomes a child of vertex $ p_i $ . It is guaranteed that after each step the tree will keep being a binary tree rooted at vertex $ 1 $ , that is, each vertex will have at most two children. After every new vertex is added, print the smallest value of disbalance over all possible diverse colorings of the current tree. Moreover, after adding the last vertex with the number $ n $ , also print a diverse coloring with the smallest possible disbalance as well.

输入输出格式

输入格式

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 10^4 $ ). The description of the test cases follows. The first line of each test case contains a single integer $ n $ ( $ 2 \le n \le 2 \cdot 10^5 $ ) — the number of vertices in the final tree. The second line contains $ n-1 $ integers $ p_2, p_3, \ldots, p_n $ ( $ 1 \le p_i \le i - 1 $ ) — the numbers of parents of vertices $ 2, 3, \ldots, n $ . No integer appears more than twice among $ p_2, p_3, \ldots, p_n $ . It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

输出格式

For each test case, print $ n-1 $ integers — the smallest value of disbalance over all possible diverse colorings of the tree after adding vertices $ 2, 3, \ldots, n $ . Then print a string of $ n $ characters 'w' and 'b', describing a diverse coloring with the smallest possible disbalance for the whole tree after adding vertex $ n $ : the $ i $ -th character must be equal to 'w' if vertex $ i $ is colored white, and 'b' if it's colored blue. The absolute value of the difference between the numbers of 'w' and 'b' characters must be equal to the last printed integer. Each vertex must have a parent or a child colored into the color opposite to the vertex's color.

输入输出样例

输入样例 #1

2
7
1 2 2 1 5 5
8
1 2 3 4 5 6 7

输出样例 #1

0
1
2
1
0
1
wbwwwbb
0
1
0
1
0
1
0
wbwbwbwb

说明

In the first test case, examples of diverse colorings with the smallest possible disbalances for all intermediate trees are illustrated below: