A14883 | Words on Tree

You are given a tree consisting of $n$ vertices, and $q$ triples $(x_i, y_i, s_i)$ , where $x_i$ and $y_i$ are integers from $1$ to $n$ , and $s_i$ is a string with length equal to the number of vertices on the simple path from $x_i$ to $y_i$ .

You want to write a lowercase Latin letter on each vertex in such a way that, for each of $q$ given triples, at least one of the following conditions holds:

- if you write out the letters on the vertices on the simple path from $x_i$ to $y_i$ in the order they appear on this path, you get the string $s_i$ ;
- if you write out the letters on the vertices on the simple path from $y_i$ to $x_i$ in the order they appear on this path, you get the string $s_i$ .

Find any possible way to write a letter on each vertex to meet these constraints, or report that it is impossible.

The first line contains two integers $n$ and $q$ ( $2 \le n \le 4 \cdot 10^5$ ; $1 \le q \le 4 \cdot 10^5$ ) — the number of vertices in the tree and the number of triples, respectively.

Then $n - 1$ lines follow; the $i$ -th of them contains two integers $u_i$ and $v_i$ ( $1 \le u_i, v_i \le n$ ; $u_i \ne v_i$ ) — the endpoints of the $i$ -th edge. These edges form a tree.

Then $q$ lines follow; the $j$ -th of them contains two integers $x_j$ and $y_j$ , and a string $s_j$ consisting of lowercase Latin letters. The length of $s_j$ is equal to the number of vertices on the simple path between $x_j$ and $y_j$ .

Additional constraint on the input: $\sum \limits_{j=1}^{q} |s_j| \le 4 \cdot 10^5$ .

If there is no way to meet the conditions on all triples, print NO. Otherwise, print YES in the first line, and a string of $n$ lowercase Latin letters in the second line; the $i$ -th character of the string should be the letter you write on the $i$ -th vertex. If there are multiple answers, print any of them.