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Ex - Constrained Topological Sort

Score : $625$ points

Problem Statement

You are given a directed graph with $N$ vertices and $M$ edges. For $i = 1, 2, \ldots, M$, the $i$-th edge is directed from vertex $s_i$ to vertex $t_i$.

Determine whether there is a permutation $P = (P_1, P_2, \ldots, P_N)$ of $(1, 2, \ldots, N)$ that satisfies both of the following conditions, and if there is, provide an example.

• $P_{s_i} \lt P_{t_i}$ for all $i = 1, 2, \ldots, M$.
• $L_i \leq P_i \leq R_i$ for all $i = 1, 2, \ldots, N$.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $0 \leq M \leq \min\lbrace 4 \times 10^5, N(N-1) \rbrace$
• $1 \leq s_i, t_i \leq N$
• $s_i \neq t_i$
• $i \neq j \implies (s_i, t_i) \neq (s_j, t_j)$
• $1 \leq L_i \leq R_i \leq N$
• All input values are integers.

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Input

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

$N$ $M$

$s_1$ $t_1$

$s_2$ $t_2$

$\vdots$

$s_M$ $t_M$

$L_1$ $R_1$

$L_2$ $R_2$

$\vdots$

$L_N$ $R_N$

Output

If there is no $P$ that satisfies the conditions, print No. If there is a $P$ that satisfies the conditions, print Yes in the first line, and the elements of $P$ separated by spaces in the second line, in the following format. If multiple $P$'s satisfy the conditions, any of them will be accepted.

``` Yes $P_1$ $P_2$ $\ldots$ $P_N$ ```

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

Yes
3 1 4 2 5

$P = (3, 1, 4, 2, 5)$ satisfies the conditions. In fact,

• for the first edge $(s_1, t_1) = (1, 5)$, we have $P_1 = 3 \lt 5 = P_5$;
• for the second edge $(s_2, t_2) = (2, 1)$, we have $P_2 = 1 \lt 3 = P_1$;
• for the third edge $(s_3, t_3) = (2, 5)$, we have $P_2 = 1 \lt 5 = P_5$;
• for the fourth edge $(s_4, t_4) = (4, 3)$, we have $P_4 = 2 \lt 4 = P_3$.

Also,

• $L_1 = 1 \leq P_1 = 3 \leq R_1 = 5$,
• $L_2 = 1 \leq P_2 = 1 \leq R_2 = 3$,
• $L_3 = 3 \leq P_3 = 4 \leq R_3 = 4$,
• $L_4 = 1 \leq P_4 = 2 \leq R_4 = 3$,
• $L_5 = 4 \leq P_5 = 5 \leq R_5 = 5$.

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2 2
1 2
2 1
1 2
1 2

No

No $P$ satisfies the conditions, so print No.