Contest Duration: - (local time) (100 minutes) Back to Home
Ex - Constrained Sums /

Time Limit: 4 sec / Memory Limit: 1024 MB

### 問題文

• すべての 1 \leq i \leq N に対して 0 \leq X_i \leq M
• すべての 1 \leq i \leq Q に対して L_i \leq X_{A_i} + X_{B_i} \leq R_i

### 制約

• 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
• 入力はすべて整数

### 入力

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


### 入力例 1

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


### 出力例 1

2 4 3 0


X = (2,4,3,0) のとき X_1 + X_3 = 5, X_1 + X_4 = 2, X_2 + X_2 = 8 よりすべての条件を満たします。この他、X = (0,2,5,2), X = (1,3,4,1) などもすべての条件を満たすため、これらを出力しても正解となります。

### 入力例 2

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


### 出力例 2

-1


すべての条件を満たす数列 X は存在しません。

Score : 600 points

### 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.