Problem Statement

You are given a tree with N vertices and N-1 edges.
The vertices are numbered 1,2,\ldots,N. The i-th edge connects Vertex a_i and Vertex b_i.

Determine whether this tree is a star.

Here, a star is a tree where there is a vertex directly connected to all other vertices.

Notes

For the definition of a tree, see Tree (graph theory) - Wikipedia.

Constraints

• 3 \leq N \leq 10^5
• 1 \leq a_i \lt b_i \leq N
• The given graph is a tree.

Sample Input 1

5
1 4
2 4
3 4
4 5

Sample Output 1

Yes

The given graph is a star.

Sample Input 2

4
2 4
1 4
2 3

Sample Output 2

No

The given graph is not a star.

Sample Input 3

10
9 10
3 10
4 10
8 10
1 10
2 10
7 10
6 10
5 10

Sample Output 3

Yes

Problem Statement

Takahashi aggregated usage records from ticket gates. However, he accidentally erased some records of entering and exiting stations. He is trying to restore the erased records.

You are given a string S consisting of i and o. We want to insert zero or more characters at arbitrary positions in S so that the resulting string satisfies the following conditions:

• Its length is even, and every odd-numbered (1st, 3rd, ...) character is i while every even-numbered (2nd, 4th, ...) character is o.

Find the minimum number of characters that need to be inserted. It can be proved under the constraints of this problem that by inserting an appropriate finite number of characters, S can be made to satisfy the conditions.

Constraints

• S is a string of length between 1 and 100, consisting of i and o.

Sample Input 1

ioi

Sample Output 1

1

We can insert o after the 3rd character to form ioio to satisfy the conditions. The conditions cannot be satisfied by inserting zero or fewer characters.

Sample Input 2

iioo

Sample Output 2

2

We can insert o after the 1st character and i after the 3rd character to satisfy the conditions. The conditions cannot be satisfied by inserting one or fewer characters.

Sample Input 3

io

Sample Output 3

0

S already satisfies the conditions.

Problem Statement

You are given an undirected graph with N vertices and M edges, where the vertices are numbered 1 through N and the edges are numbered 1 through M. Edge i connects vertices u_i and v_i.
To make the graph simple by removing edges, what is the minimum number of edges that must be removed?
Here, a graph is called simple if and only if it does not contain self-loops or multi-edges.

Constraints

• 1 \leq N \leq 2 \times 10^5
• 0 \leq M \leq 5 \times 10^5
• 1 \leq u_i \leq N
• 1 \leq v_i \leq N
• All input values are integers.

Sample Input 1

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

Sample Output 1

2

By removing edges 3 and 5, the graph becomes simple. This is one of the ways to remove the minimum number of edges, so the answer is 2.

Sample Input 2

1 0

Sample Output 2

0

Sample Input 3

6 10
6 2
4 1
5 1
6 6
5 3
5 1
1 4
6 4
4 2
5 6

Sample Output 3

3

Problem Statement

You are given six integers: h_1, h_2, h_3, w_1, w_2, and w_3.
Consider writing a positive integer on each square of a 3 \times 3 grid so that all of the following conditions are satisfied:

• For i=1,2,3, the sum of numbers written in the i-th row from the top is h_i.
• For j=1,2,3, the sum of numbers written in the j-th column from the left is w_i.

For example, if (h_1, h_2, h_3) = (5, 13, 10) and (w_1, w_2, w_3) = (6, 13, 9), then all of the following three ways satisfy the conditions. (There are other ways to satisfy the conditions.)

How many ways are there to write numbers to satisfy the conditions?

Constraints

• 3 \leq h_1, h_2, h_3, w_1, w_2, w_3 \leq 30
• All values in input are integers.

Sample Input 1

3 4 6 3 3 7

Sample Output 1

1

The following is the only way to satisfy the conditions. Thus, 1 should be printed.

Sample Input 2

3 4 5 6 7 8

Sample Output 2

0

There may not be a way to satisfy the conditions.

Sample Input 3

5 13 10 6 13 9

Sample Output 3

120

Sample Input 4

20 25 30 22 29 24

Sample Output 4

30613

Problem Statement

There is one server and N PCs. The server and each PC each hold one string, and initially all strings are empty.

Q queries are given. Each query is in one of the following formats:

• 1 p: Replace the string of PC p with the string of the server.
• 2 p s: Append string s to the end of the string of PC p.
• 3 p: Replace the string of the server with the string of PC p.

Find the final string of the server after processing all queries in the given order.

Constraints

• N,Q are integers
• 1\leq N,Q \leq 2\times 10^5
• For every query, p is an integer and 1 \leq p\leq N.
• For every query of type 2, s is a string of length at least 1 consisting of lowercase English letters.
• The sum of the lengths of s over all queries of type 2 is at most 10^6.

Sample Input 1

2 6
2 1 at
3 1
2 2 on
1 2
2 2 coder
3 2

Sample Output 1

atcoder

• Initially, the strings of the server and PCs 1,2 are all empty.
• 1st query: Append at to the end of the string of PC 1. At this time, the strings of the server, PC 1,2 are empty, at, empty, respectively.
• 2nd query: Replace the string of the server with the string of PC 1. At this time, the strings of the server, PC 1,2 are at, at, empty, respectively.
• 3rd query: Append on to the end of the string of PC 2. At this time, the strings of the server, PC 1,2 are at, at, on, respectively.
• 4th query: Replace the string of PC 2 with the string of the server. At this time, the strings of the server, PC 1,2 are at, at, at, respectively.
• 5th query: Append coder to the end of the string of PC 2. At this time, the strings of the server, PC 1,2 are at, at, atcoder, respectively.
• 6th query: Replace the string of the server with the string of PC 2. At this time, the strings of the server, PC 1,2 are atcoder, at, atcoder, respectively.

Thus, the final string of the server is atcoder.

Sample Input 2

100000 3
1 100
2 300 abc
3 200

Sample Output 2

The final string of the server is empty.

Sample Input 3

10 10
2 7 ladxf
2 7 zz
2 7 kfm
3 7
1 5
2 5 irur
3 5
1 6
2 6 ptilun
3 6

Sample Output 3

ladxfzzkfmirurptilun