102777: [AtCoder]ABC277 Ex - Constrained Sums

Problem Statement

Determine whether there is a sequence of $N$ integers $X = (X_1, X_2, \ldots ,X_N)$ that satisfies all of the following conditions, and construct one such sequence if it exists.

• $0 \leq X_i \leq M$ for every $1 \leq i \leq N$.
  
• $L_i \leq X_{A_i} + X_{B_i} \leq R_i$ for every $1 \leq i \leq Q$.

Constraints

• $1 \leq N \leq 10000$
• $1 \leq M \leq 100$
• $1 \leq Q \leq 10000$
• $1 \leq A_i, B_i \leq N$
• $0 \leq L_i \leq R_i \leq 2 \times M$
• All values in the input are integers.

Input

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

$N$ $M$ $Q$
$A_1$ $B_1$ $L_1$ $R_1$
$A_2$ $B_2$ $L_2$ $R_2$
$\vdots$
$A_Q$ $B_Q$ $L_Q$ $R_Q$

Output

If there is an integer sequence that satisfies all of the conditions in the Problem Statement, print the elements $X_1, X_2, \ldots, X_N$ of one such sequence, separated by spaces. Otherwise, print -1.

Sample Input 1

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

Sample Output 1

2 4 3 0

For $X = (2,4,3,0)$, we have $X_1 + X_3 = 5$, $X_1 + X_4 = 2$, and $X_2 + X_2 = 8$, so all conditions are satisfied. There are other sequences, such as $X = (0,2,5,2)$ and $X = (1,3,4,1)$, that satisfy all conditions, and those will also be accepted.

Sample Input 2

3 7 3
1 2 3 4
3 1 9 12
2 3 2 4

Sample Output 2

-1

No sequence $X$ satisfies all conditions.

问题描述

确定是否存在一个满足以下所有条件的包含$N$个整数的序列$X = (X_1, X_2, \ldots ,X_N)$，如果存在，则构造一个这样的序列。

• 对于每个$1 \leq i \leq N$，有$0 \leq X_i \leq M$。
• 对于每个$1 \leq i \leq Q$，有$L_i \leq X_{A_i} + X_{B_i} \leq R_i$。

约束条件

• $1 \leq N \leq 10000$
• $1 \leq M \leq 100$
• $1 \leq Q \leq 10000$
• $1 \leq A_i, B_i \leq N$
• $0 \leq L_i \leq R_i \leq 2 \times M$
• 输入中的所有值都是整数。

输入

输入从标准输入按以下格式给出：

$N$ $M$ $Q$
$A_1$ $B_1$ $L_1$ $R_1$
$A_2$ $B_2$ $L_2$ $R_2$
$\vdots$
$A_Q$ $B_Q$ $L_Q$ $R_Q$

输出

如果存在一个满足问题描述中所有条件的整数序列，用空格分隔地打印该序列的元素$X_1, X_2, \ldots, X_N$。否则，打印-1。

样例输入1

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

样例输出1

2 4 3 0

对于$X = (2,4,3,0)$，我们有$X_1 + X_3 = 5$，$X_1 + X_4 = 2$，以及$X_2 + X_2 = 8$，因此所有条件都得到满足。存在其他满足所有条件的序列，如$X = (0,2,5,2)$和$X = (1,3,4,1)$，这些也将被接受。

样例输入2

3 7 3
1 2 3 4
3 1 9 12
2 3 2 4

样例输出2

-1

不存在满足所有条件的序列$X$。