Alice is the teacher of a class of students. She is going to divide the students into groups. She has the following requirements:
- Alice knows that some pair of students are very good friends and they must not be assigned to the same group.
- Alice also wants to minimize the difference between the sizes of the two groups.
Let's write a program to help Alice to do the grouping, in which the program should:
-1if it is not possible to do the grouping.
- Report the group assigned to each student that fulfills the requirements.
- The first line contains two integers and , where denotes the number of students, and denotes the number of known relationships between students.
- Following are lines, and each line contains two integers and , which means student and are very good friends and they must not be assigned to the same group. Each relationship will only be provided once.
- In our test cases, we have and .
- In of the test cases, .
- In another of the test cases, the number of possible ways to group students .
- If there is no way to do the grouping, output
- Otherwise, output a string containing characters representing the group assigned to each of the students. If the th student is assigned to group , then the th character in the string is
1. Otherwise, the th character in the string is
- The string should represent the grouping with minimum difference between the sizes of the two groups. (That is, the difference between the number of 1's and the number of 2's in the output should be minimized.) If there is a tie, output the lexicographically smallest solution.
Sample Input 1
4 3 1 2 3 4 1 3
Output for Sample Input 1
Sample Input 2
3 3 1 2 1 3 2 3
Output for Sample Input 2
Sample Input 3
6 4 1 2 1 3 4 6 4 5
Output for Sample Input 3
In Sample 1, since , and cannot be assigned into the same group, to make the difference minimized, we want and be in the same group, and and be in the other. Among the outputs, is lexicographically smallest. That is, and are in group , and and are in group .
In Sample 2, there is no way to do the grouping. The reason is that cannot be in the same group of or . But and cannot be in the same group, either.
In Sample 3, cannot be with and , and cannot be with and . The minimized difference is , by group and together.