2035: [USACO 2025 January Contest, Silver] Problem 3. Table Recovery
题目描述
Bessie has an () addition table where the integer in the cell at row and column is , for all . For example, for , the table would look like this:
2 3 4 3 4 5 4 5 6
Unfortunately, Elsie got ahold of the table and permuted it by performing the following three types of operations as many times as she wanted.
- Swap two rows
- Swap two columns
- Select two values and that are both present in the table, then simultaneously change every occurrence of to and every occurrence of to .
Elsie will always perform operations in increasing order of type; that is, she performs as many operations as she likes (possibly none) first of type , then of type , and finally of type .
Help Bessie recover a possible state of the table after Elsie finished applying all of her operations of types and , but before applying any operations of type . There may be multiple possible answers, in which case you should output the lexicographically smallest one.
To compare two tables lexicographically, compare the first entries at which they differ, when reading both tables in the natural order (rows from top to bottom, left to right within a row).
输入
The next lines each contain integers, representing Bessie's addition table after Elsie has permuted it.
输出
样例输入
1
2样例输出
2提示
Regardless of what operations Elsie performs, the table won't change.
3
3 4 2
5 2 3
6 3 5
4 2 3
5 3 4
6 4 5
Here is a possible sequence of operations Elsie might have performed.
2 3 4
3 4 5
4 5 6
-> (op 1: swap columns 2 and 3)
2 4 3
3 5 4
4 6 5
-> (op 1: swap columns 1 and 2)
4 2 3
5 3 4
6 4 5
-> (op 3: swap values 2 and 3)
4 3 2
5 2 4
6 4 5
-> (op 3: swap values 3 and 4)
3 4 2
5 2 3
6 3 5
Note: the following is also a possible state of the table after operations of types 1 and 2, but it is not the lexicographically smallest because the second entry of the first row is larger than in the correct answer.
4 6 5
3 5 4
2 4 3
SAMPLE INPUT:
6
8 10 5 6 7 4
12 11 10 4 8 2
5 4 6 7 9 8
10 2 4 8 5 12
6 8 7 9 3 5
4 12 8 5 6 10
SAMPLE OUTPUT:
7 5 8 9 10 6
4 2 5 6 7 3
8 6 9 10 11 7
5 3 6 7 8 4
9 7 10 11 12 8
6 4 7 8 9 5
SAMPLE INPUT:
SAMPLE OUTPUT:
SCORING: