201434: [AtCoder]ARC143 E - Reversi
Description
Score : $700$ points
Problem Statement
We have a tree with $N$ vertices.
The vertices are numbered $1$ to $N$,
and the $i$-th edge connects Vertex $A_i$ and Vertex $B_i$.
Additionally, each vertex has a reversi piece on it.
The status of the piece on each vertex is given by a string $S$:
if the $i$-th character of $S$ is B, the piece on Vertex $i$ is placed with the black side up;
if the $i$-th character of $S$ is W, the piece on Vertex $i$ is placed with the white side up.
Determine whether it is possible to perform the operation below $N$ times to remove the pieces from all vertices. If it is possible, find the lexicographically smallest possible sequence $P_1, P_2, \ldots, P_N$ such that Vertices $P_1, P_2, \ldots, P_N$ can be chosen in this order during the process.
- Choose a vertex containing a piece with the white side up, and remove the piece from that vertex. Then, flip all pieces on the vertices adjacent to that vertex.
Notes on reversi pieces
A reversi piece has a black side and a white side, and flipping it changes which side faces up.What is the lexicographical order on sequences?
The following is an algorithm to determine the lexicographical order between different sequences $S$ and $T$.
Below, let $S_i$ denote the $i$-th element of $S$. Also, if $S$ is lexicographically smaller than $T$, we will denote that fact as $S \lt T$; if $S$ is lexicographically larger than $T$, we will denote that fact as $S \gt T$.
- Let $L$ be the smaller of the lengths of $S$ and $T$. For each $i=1,2,\dots,L$, we check whether $S_i$ and $T_i$ are the same.
- If there is an $i$ such that $S_i \neq T_i$, let $j$ be the smallest such $i$. Then, we compare $S_j$ and $T_j$. If $S_j$ is less than $T_j$ (as a number), we determine that $S \lt T$ and quit; if $S_j$ is greater than $T_j$, we determine that $S \gt T$ and quit.
- If there is no $i$ such that $S_i \neq T_i$, we compare the lengths of $S$ and $T$. If $S$ is shorter than $T$, we determine that $S \lt T$ and quit; if $S$ is longer than $T$, we determine that $S \gt T$ and quit.
Constraints
- $1 \leq N \leq 2\times 10^5$
- $1 \leq A_i, B_i \leq N$
- The given graph is a tree.
- $S$ is a string of length $N$ consisting of the characters
BandW.
Input
Input is given from Standard Input in the following format:
$N$
$A_1$ $B_1$
$\vdots$
$A_{N-1}$ $B_{N-1}$
$S$
Output
If the objective is unachievable, print -1. If it is achievable, print the answer in the following format:
$P_1$ $P_2$ $\cdots$ $P_N$
Sample Input 1
4 1 2 2 3 3 4 WBWW
Sample Output 1
1 2 4 3
Sample Input 2
4 1 2 2 3 3 4 BBBB
Sample Output 2
-1
In this case, you cannot perform the operation at all.