Problem Statement

Given a length-3 string S consisting of uppercase English letters, print Yes if S equals one of ACE, BDF, CEG, DFA, EGB, FAC, and GBD; print No otherwise.

Constraints

• S is a length-3 string consisting of uppercase English letters.

Sample Input 1

ABC

Sample Output 1

No

When S = ABC, S does not equal any of ACE, BDF, CEG, DFA, EGB, FAC, and GBD, so No should be printed.

Sample Input 2

FAC

Sample Output 2

Yes

Sample Input 3

XYX

Sample Output 3

No

Problem Statement

Takahashi and Aoki played N games. You are given a string S of length N, representing the results of these games. Takahashi won the i-th game if the i-th character of S is T, and Aoki won that game if it is A.

The overall winner between Takahashi and Aoki is the one who won more games than the other. If they had the same number of wins, the overall winner is the one who reached that number of wins first. Find the overall winner: Takahashi or Aoki.

Constraints

• 1\leq N \leq 100
• N is an integer.
• S is a string of length N consisting of T and A.

Sample Input 1

5
TTAAT

Sample Output 1

T

Takahashi won three games, and Aoki won two. Thus, the overall winner is Takahashi, who won more games.

Sample Input 2

6
ATTATA

Sample Output 2

T

Both Takahashi and Aoki won three games. Takahashi reached three wins in the fifth game, and Aoki in the sixth game. Thus, the overall winner is Takahashi, who reached three wins first.

Sample Input 3

1
A

Sample Output 3

A

Problem Statement

There are N people numbered Person 1, Person 2, \dots, and Person N. Person i has a family name s_i and a given name t_i.

Consider giving a nickname to each of the N people. Person i's nickname a_i should satisfy all the conditions below.

• a_i coincides with Person i's family name or given name. In other words, a_i = s_i and/or a_i = t_i holds.
• a_i does not coincide with the family name and the given name of any other person. In other words, for all integer j such that 1 \leq j \leq N and i \neq j, it holds that a_i \neq s_j and a_i \neq t_j.

Is it possible to give nicknames to all the N people? If it is possible, print Yes; otherwise, print No.

Constraints

• 2 \leq N \leq 100
• N is an integer.
• s_i and t_i are strings of lengths between 1 and 10 (inclusive) consisting of lowercase English alphabets.

Sample Input 1

3
tanaka taro
tanaka jiro
suzuki hanako

Sample Output 1

Yes

The following assignment satisfies the conditions of nicknames described in the Problem Statement: a_1 = taro, a_2 = jiro, a_3 = hanako. (a_3 may be suzuki, too.)
However, note that we cannot let a_1 = tanaka, which violates the second condition of nicknames, since Person 2's family name s_2 is tanaka too.

Sample Input 2

3
aaa bbb
xxx aaa
bbb yyy

Sample Output 2

No

There is no way to give nicknames satisfying the conditions in the Problem Statement.

Sample Input 3

2
tanaka taro
tanaka taro

Sample Output 3

No

There may be a pair of people with the same family name and the same given name.

Sample Input 4

3
takahashi chokudai
aoki kensho
snu ke

Sample Output 4

Yes

We can let a_1 = chokudai, a_2 = kensho, and a_3 = ke.

Problem Statement

On the xy-plane, there are N points with ID numbers from 1 to N. Point i is located at coordinates (X_i, Y_i), and no two points have the same coordinates.

From each point, find the farthest point and print its ID number. If multiple points are the farthest, print the smallest of the ID numbers of those points.

Here, we use the Euclidean distance: for two points (x_1,y_1) and (x_2,y_2), the distance between them is \sqrt{(x_1-x_2)^{2}+(y_1-y_2)^{2}}.

Constraints

• 2 \leq N \leq 100
• -1000 \leq X_i, Y_i \leq 1000
• (X_i, Y_i) \neq (X_j, Y_j) if i \neq j.
• All input values are integers.

Sample Input 1

4
0 0
2 4
5 0
3 4

Sample Output 1

3
3
1
1

The following figure shows the arrangement of the points. Here, P_i represents point i. The farthest point from point 1 are points 3 and 4, and point 3 has the smaller ID number.

The farthest point from point 2 is point 3.

The farthest point from point 3 are points 1 and 2, and point 1 has the smaller ID number.

The farthest point from point 4 is point 1.

Sample Input 2

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

Sample Output 2

6
6
6
6
6
4

Problem Statement

Find the K-th lexicographically smallest string among the strings that are permutations of a string S.

Constraints

• 1 \le |S| \le 8
• S consists of lowercase English letters.
• There are at least K distinct strings that are permutations of S.

Sample Input 1

aab 2

Sample Output 1

aba

There are three permutations of a string aab: \{ aab, aba, baa \}. The 2-nd lexicographically smallest of them is aba.

Sample Input 2

baba 4

Sample Output 2

baab

Sample Input 3

ydxwacbz 40320

Sample Output 3

zyxwdcba

Problem Statement

2N players, with ID numbers 1 through 2N, will participate in a rock-scissors-paper contest.

The contest has M rounds. Each round has N one-on-one matches, where each player plays in one of them.

For each i=0, 1, \ldots, M, the players' ranks at the end of the i-th round are determined as follows.

• A player with more wins in the first i rounds ranks higher.
• Ties are broken by ID numbers: a player with a smaller ID number ranks higher.

Additionally, for each i=1, \ldots, M, the matches in the i-th round are arranged as follows.

• For each k=1, 2, \ldots, N, a match is played between the players who rank (2k-1)-th and 2k-th at the end of the (i-1)-th round.

In each match, the two players play a hand just once, resulting in one player's win and the other's loss, or a draw.

Takahashi, who can foresee the future, knows that Player i will play A_{i, j} in their match in the j-th round, where A_{i,j} is G, C, or P.
Here, G stands for rock, C stands for scissors, and P stands for paper. (These derive from Japanese.)

Find the players' ranks at the end of the M-th round.

Constraints

• 1 \leq N \leq 50
• 1 \leq M \leq 100
• A_{i,j} is G, C, or P.

Sample Input 1

2 3
GCP
PPP
CCC
PPC

Sample Output 1

3
1
2
4

In the first round, matches are played between Players 1 and 2, and between Players 3 and 4. Player 2 wins the former, and Player 3 wins the latter.
In the second round, matches are played between Players 2 and 3, and between Players 1 and 4. Player 3 wins the former, and Player 1 wins the latter.
In the third round, matches are played between Players 3 and 1, and between Players 2 and 4. Player 3 wins the former, and Player 4 wins the latter.
Therefore, in the end, the players are ranked in the following order: 3,1,2,4, from highest to lowest.

Sample Input 2

2 2
GC
PG
CG
PP

Sample Output 2

1
2
3
4

In the first round, matches are played between Players 1 and 2, and between Players 3 and 4. Player 2 wins the former, and Player 3 wins the latter.
In the second round, matches are played between Players 2 and 3, and between Players 1 and 4. The former is drawn, and Player 1 wins the latter.
Therefore, in the end, the players are ranked in the following order: 1,2,3,4, from highest to lowest.

Problem Statement

You are given a simple connected undirected graph with N vertices and M edges. Each vertex i\,(1\leq i \leq N) has a weight A_i. Each edge j\,(1\leq j \leq M) connects vertices U_j and V_j bidirectionally and has a weight B_j.

The weight of a path in this graph is defined as the sum of the weights of the vertices and edges that appear on the path.

For each i=2,3,\dots,N, solve the following problem:

• Find the minimum weight of a path from vertex 1 to vertex i.

Constraints

• 2 \leq N \leq 2 \times 10^5
• N-1 \leq M \leq 2 \times 10^5
• 1 \leq U_j < V_j \leq N
• (U_i, V_i) \neq (U_j, V_j) if i \neq j.
• The graph is connected.
• 0 \leq A_i \leq 10^9
• 0 \leq B_j \leq 10^9
• All input values are integers.

Sample Input 1

3 3
1 2 3
1 2 1
1 3 6
2 3 2

Sample Output 1

4 9

Consider the paths from vertex 1 to vertex 2. The weight of the path 1 \to 2 is A_1 + B_1 + A_2 = 1 + 1 + 2 = 4, and the weight of the path 1 \to 3 \to 2 is A_1 + B_2 + A_3 + B_3 + A_2 = 1 + 6 + 3 + 2 + 2 = 14. The minimum weight is 4.

Consider the paths from vertex 1 to vertex 3. The weight of the path 1 \to 3 is A_1 + B_2 + A_3 = 1 + 6 + 3 = 10, and the weight of the path 1 \to 2 \to 3 is A_1 + B_1 + A_2 + B_3 + A_3 = 1 + 1 + 2 + 2 + 3 = 9. The minimum weight is 9.

Sample Input 2

2 1
0 1
1 2 3

Sample Output 2

4

Sample Input 3

5 8
928448202 994752369 906965437 942744902 907560126
2 5 975090662
1 2 908843627
1 5 969061140
3 4 964249326
2 3 957690728
2 4 942986477
4 5 948404113
1 3 988716403

Sample Output 3

2832044198 2824130042 4696218483 2805069468

Note that the answers may not fit in a 32-bit integer.

Problem Statement

There are N monsters on a two-dimensional plane. The monsters are numbered from 1 to N, and the coordinates of monster i are (X_i,Y_i). Here, (X_i,Y_i) \neq (0,0). (Each monster can be regarded as a stationary point. That is, monsters have no size.)

Takahashi is standing at the origin of this plane. A powerful laser is always emitted from his eyes, instantly destroying monsters in the direction he is facing. If multiple monsters exist in the direction he is facing, all of them are instantly destroyed.

Aoki is conducting Q independent thought experiments. The j-th thought experiment is as follows:

• Initially, Takahashi is facing the direction toward monster A_j. From now on, Takahashi will rotate clockwise and stop the moment he faces the direction toward monster B_j. Here, how many monsters in total (including monsters A_j and B_j) will be destroyed? If monsters A_j and B_j exist in the same direction from the origin, Takahashi does not rotate at all.

Find the answer to each thought experiment.

Constraints

• 2\leq N \leq 2\times 10^5
• 1\leq Q \leq 2\times 10^5
• -10^9\leq X_i,Y_i \leq 10^9
• (X_i,Y_i)\neq (0,0)
• 1\leq A_j,B_j\leq N
• A_j\neq B_j
• All input values are integers.

Sample Input 1

5 4
0 1
1 -2
1 0
-2 0
3 0
4 1
1 4
5 4
3 5

Sample Output 1

2
5
4
2

• 1-st thought experiment: Initially, Takahashi is facing the direction toward monster 4 (at this time, monster 4 is destroyed). From here, he continues to rotate clockwise and stops the moment he faces the direction toward monster 1 (at this time, monster 1 is destroyed). Since he does not face the direction toward any other monsters, the answer is 2.
• 2-nd thought experiment: Initially, Takahashi is facing the direction toward monster 1 (at this time, monster 1 is destroyed). From here, as he continues to rotate clockwise, he faces the direction toward monsters 3 and 5 along the way, so they are destroyed. As he continues to rotate further, he faces the direction toward monster 2 along the way, so it is destroyed. Finally, he stops the moment he faces the direction toward monster 4 (at this time, monster 4 is destroyed). Therefore, the answer is 5.
• 3-rd thought experiment: Monsters 3, 5, 2, 4 are destroyed, so the answer is 4.
• 4-th thought experiment: Monsters 3, 5 are destroyed, so the answer is 2. Note that since monsters 3 and 5 exist in the same direction from the origin, Takahashi does not rotate at all.

Sample Input 2

2 1
1 2
1 2
1 2

Sample Output 2

2

Multiple monsters may exist at the same coordinates.

Sample Input 3

8 10
-84 -60
-100 8
77 55
-14 -10
50 -4
-63 -45
26 -17
-7 -5
3 7
2 4
8 4
8 4
7 1
1 7
6 3
4 7
4 5
2 6

Sample Output 3

3
8
4
4
5
8
6
8
7
8

Problem Statement

There is a graph G with N vertices and 0 edges on a 2-dimensional plane. The vertices are numbered from 1 to N, and vertex i is located at coordinates (x_i,y_i).

For vertices u and v of G, the distance d(u,v) between u and v is defined as the Manhattan distance d(u,v)=|x_u-x_v|+|y_u-y_v|.

Also, for two connected components A and B of G, let V(A) and V(B) be the vertex sets of A and B, respectively. The distance d(A,B) between A and B is defined as d(A,B)=\min\lbrace d(u,v)\mid u \in V(A), v\in V(B)\rbrace.

Process Q queries as described below. Each query is one of the following three types:

• 1 a b : Let n be the number of vertices in G. Add vertex n+1 to G with coordinates (x_{n+1},y_{n+1})=(a,b).
• 2 : Let n be the number of vertices in G and m be the number of connected components.
  • If m=1, output -1.
  • If m\geq 2, merge all connected components with the minimum distance and output the value of that minimum distance. Formally, let the connected components of G be A_1,A_2,\ldots,A_m and let \displaystyle k=\min_{1\leq i\lt j\leq m} d(A_i,A_j). For all pairs of vertices (u,v)\ (1\leq u\lt v\leq n) that are not in the same connected component and satisfy d(u,v)=k, add an edge between vertices u and v. Then, output k.
• 3 u v : If vertices u and v are in the same connected component, output Yes; otherwise, output No.

Constraints

• 2\leq N\leq 1500
• 1\leq Q\leq 1500
• 0\leq x_i,y_i\leq 10^9
• For queries of type 1, 0\leq a,b\leq 10^9.
• For queries of type 3, let n be the number of vertices in G just before processing that query, then 1\leq u\lt v\leq n.
• All input values are integers.

Sample Input 1

4 11
3 4
3 3
7 3
2 2
3 1 2
2
3 1 2
1 6 4
2
3 2 5
2
3 2 5
2
1 2 2
2

Sample Output 1

No
1
Yes
2
No
3
Yes
-1
0

Initially, vertices 1,2,3,4 are located at coordinates (3,4),(3,3),(7,3),(2,2), respectively.

• For the 1st query, vertices 1 and 2 are not connected, so output No.
• For the 2nd query, there are 4 connected components, and the vertex set of each connected component is \lbrace 1\rbrace, \lbrace 2\rbrace, \lbrace 3\rbrace, \lbrace 4\rbrace. The minimum distance between different connected components is 1, and an edge is added between vertices 1 and 2. Output 1.
• For the 3rd query, vertices 1 and 2 are connected, so output Yes.
• For the 4th query, add vertex 5 at coordinates (6,4).
• For the 5th query, there are 4 connected components, and the vertex set of each connected component is \lbrace 1,2\rbrace,\lbrace 3\rbrace,\lbrace 4\rbrace,\lbrace 5\rbrace. The minimum distance between different connected components is 2, and edges are added between vertices 2 and 4 and between vertices 3 and 5. Output 2.
• For the 6th query, vertices 2 and 5 are not connected, so output No.
• For the 7th query, there are 2 connected components, and the vertex set of each connected component is \lbrace 1,2,4\rbrace,\lbrace 3,5\rbrace. The minimum distance between different connected components is 3, and an edge is added between vertices 1 and 5. Output 3.
• For the 8th query, vertices 2 and 5 are connected, so output Yes.
• For the 9th query, there is 1 connected component, so output -1.
• For the 10th query, add vertex 6 at coordinates (2,2).
• For the 11th query, there are 2 connected components, and the vertex set of each connected component is \lbrace 1,2,3,4,5\rbrace,\lbrace 6\rbrace. The minimum distance between different connected components is 0, and an edge is added between vertices 4 and 6. Output 0.