102994: [Atcoder]ABC299 E - Nearest Black Vertex

Problem Statement

You are given a simple connected undirected graph with $N$ vertices and $M$ edges (a simple graph contains no self-loop and no multi-edges).
For $i = 1, 2, \ldots, M$, the $i$-th edge connects vertex $u_i$ and vertex $v_i$ bidirectionally.

Determine whether there is a way to paint each vertex black or white to satisfy both of the following conditions, and show one such way if it exists.

• At least one vertex is painted black.
• For every $i = 1, 2, \ldots, K$, the following holds:
  • the minimum distance between vertex $p_i$ and a vertex painted black is exactly $d_i$.

Here, the distance between vertex $u$ and vertex $v$ is the minimum number of edges in a path connecting $u$ and $v$.

Constraints

• $1 \leq N \leq 2000$
• $N-1 \leq M \leq \min\lbrace N(N-1)/2, 2000 \rbrace$
• $1 \leq u_i, v_i \leq N$
• $0 \leq K \leq N$
• $1 \leq p_1 \lt p_2 \lt \cdots \lt p_K \leq N$
• $0 \leq d_i \leq N$
• The given graph is simple and connected.
• All values in the input are integers.

Input

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

$N$ $M$
$u_1$ $v_1$
$u_2$ $v_2$
$\vdots$
$u_M$ $v_M$
$K$
$p_1$ $d_1$
$p_2$ $d_2$
$\vdots$
$p_K$ $d_K$

Output

If there is no way to paint each vertex black or white to satisfy the conditions, print No.
Otherwise, print Yes in the first line, and a string $S$ representing a coloring of the vertices in the second line, as shown below.
Here, $S$ is a string of length $N$ such that, for each $i = 1, 2, \ldots, N$, the $i$-th character of $S$ is $1$ if vertex $i$ is painted black and $0$ if white.

Yes
$S$

If multiple solutions exist, you may print any of them.

Sample Input 1

5 5
1 2
2 3
3 1
3 4
4 5
2
1 0
5 2

Sample Output 1

Yes
10100

One way to satisfy the conditions is to paint vertices $1, 3$ black and vertices $2, 4, 5$ white.
Indeed, for each $i = 1, 2, 3, 4, 5$, let $A_i$ denote the minimum distance between vertex $i$ and a vertex painted black, and we have $(A_1, A_2, A_3, A_4, A_5) = (0, 1, 0, 1, 2)$, where $A_1 = 0, A_5 = 2$.

Sample Input 2

5 5
1 2
2 3
3 1
3 4
4 5
5
1 1
2 1
3 1
4 1
5 1

Sample Output 2

No

There is no way to satisfy the conditions by painting each vertex black or white, so you should print No.

Sample Input 3

1 0
0

Sample Output 3

Yes
1