Problem Statement

You are given a sequence A = (A_1, A_2, \ldots, A_N) of length N. For each K = 0, 1, 2, \ldots, N-1, solve the following problem.

Find the number of integers i between 1 and N (inclusive) such that:

• A contains exactly K distinct integers greater than A_i.

Constraints

• 1 \leq N \leq 2 \times 10^5
• 1 \leq A_i \leq 10^9
• All values in the input are integers.

Sample Input 1

6
2 7 1 8 2 8

Sample Output 1

2
1
2
1
0
0

For example, we will find the answer for K=2.

• Regarding A_1 = 2, A contains 2 distinct integers greater than A_1: 7 and 8.
• Regarding A_2 = 7, A contains 1 distinct integer greater than A_2: 8.
• Regarding A_3 = 1, A contains 3 distinct integers greater than A_3: 2, 7, and 8.
• Regarding A_4 = 8, A contains 0 distinct integers greater than A_4 (there is no such integer).
• Regarding A_5 = 2, A contains 2 distinct integers greater than A_5: 7 and 8.
• Regarding A_6 = 8, A contains 0 distinct integers greater than A_6 (there is no such integer).

Thus, there are two i's, i = 1 and i = 5, such that A contains exactly K = 2 distinct integers greater than A_i. Therefore, the answer for K = 2 is 2.

Sample Input 2

1
1

Sample Output 2

1

Sample Input 3

10
979861204 57882493 979861204 447672230 644706927 710511029 763027379 710511029 447672230 136397527

Sample Output 3

2
1
2
1
2
1
1
0
0
0

Problem Statement

You are given simple undirected graphs G and H, each with N vertices: vertices 1, 2, \ldots, N. Graph G has M_G edges, and its i-th edge (1\leq i\leq M_G) connects vertices u_i and v_i. Graph H has M_H edges, and its i-th edge (1\leq i\leq M_H) connects vertices a_i and b_i.

You can perform the following operation on graph H any number of times, possibly zero.

• Choose a pair of integers (i,j) satisfying 1\leq i<j\leq N. Pay A_{i,j} yen, and if there is no edge between vertices i and j in H, add one; if there is, remove it.

Find the minimum total cost required to make G and H isomorphic.

Constraints

• 1\leq N\leq8
• 0\leq M _ G\leq\dfrac{N(N-1)}2
• 0\leq M _ H\leq\dfrac{N(N-1)}2
• 1\leq u _ i\lt v _ i\leq N\ (1\leq i\leq M _ G)
• (u _ i,v _ i)\neq(u _ j,v _ j)\ (1\leq i\lt j\leq M _ G)
• 1\leq a _ i\lt b _ i\leq N\ (1\leq i\leq M _ H)
• (a _ i,b _ i)\neq(a _ j,b _ j)\ (1\leq i\lt j\leq M _ H)
• 1\leq A _ {i,j}\leq 10 ^ 6\ (1\leq i\lt j\leq N)
• All input values are integers.

Sample Input 1

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

Sample Output 1

9

The given graphs are as follows:

For example, you can perform the following four operations on H to make it isomorphic to G at a cost of 9 yen.

• Choose (i,j)=(1,3). There is an edge between vertices 1 and 3 in H, so pay 1 yen to remove it.
• Choose (i,j)=(2,5). There is no edge between vertices 2 and 5 in H, so pay 2 yen to add it.
• Choose (i,j)=(1,5). There is an edge between vertices 1 and 5 in H, so pay 1 yen to remove it.
• Choose (i,j)=(3,5). There is no edge between vertices 3 and 5 in H, so pay 5 yen to add it.

After these operations, H becomes:

You cannot make G and H isomorphic at a cost less than 9 yen, so print 9.

Sample Input 2

5
3
1 2
2 3
3 4
4
1 2
2 3
3 4
4 5
9 1 1 1
1 1 1
1 1
9

Sample Output 2

3

For example, performing the operations (i,j)=(2,3),(2,4),(3,4) on H will make it isomorphic to G.

Sample Input 3

5
3
1 2
2 3
3 4
4
1 2
2 3
3 4
4 5
5 4 4 4
4 4 4
4 4
5

Sample Output 3

5

For example, performing the operation (i,j)=(4,5) once will make G and H isomorphic.

Sample Input 4

2
0
0
371

Sample Output 4

0

Note that G and H may have no edges.

Also, it is possible that no operations are needed.

Sample Input 5

8
13
1 8
5 7
4 6
1 5
7 8
1 6
1 2
5 8
2 6
5 6
6 7
3 7
4 8
15
3 5
1 7
4 6
3 8
7 8
1 2
5 6
1 6
1 5
1 4
2 8
2 6
2 4
4 7
1 3
7483 1694 5868 3296 9723 5299 4326
5195 4088 5871 1384 2491 6562
1149 6326 2996 9845 7557
4041 7720 1554 5060
8329 8541 3530
4652 3874
3748

Sample Output 5

21214

Problem Statement

You are given a directed graph with N vertices and M edges. The j-th directed edge goes from vertex u_j to vertex v_j and has a weight of w_j.

Find one way to write an integer between -10^{18} and 10^{18}, inclusive, to each vertex such that the following condition is satisfied.

• Let x_i be the value written on vertex i. For all edges j=1,2,\dots,M, it holds that x_{v_j} - x_{u_j} = w_j.

It is guaranteed that at least one such assignment exists for the given input.

Constraints

• 2 \leq N \leq 2 \times 10^5
• 1 \leq M \leq \min\{2 \times 10^5,N(N-1)/2\}
• 1 \leq u_j, v_j \leq N
• u_j \neq v_j
• If i \neq j, then (u_i, v_i) \neq (u_j, v_j) and (u_i, v_i) \neq (v_j, u_j)
• |w_j| \leq 10^9
• All input values are integers.
• There exists at least one assignment satisfying the conditions.

Sample Input 1

3 3
1 2 2
3 2 3
1 3 -1

Sample Output 1

3 5 2

By setting x = (3, 5, 2), we have x_2 - x_1 = w_1 = 2, x_2 - x_3 = w_2 = 3, x_3 - x_1 = w_3 = -1, satisfying the conditions.

For example, x = (-1, 1, -2) is also a valid answer.

Sample Input 2

4 2
2 1 5
3 4 -3

Sample Output 2

5 0 6 3

For example, x = (0, -5, 4, 1) and x = (5, 0, 4, 1) are also valid answers.

Sample Input 3

5 7
2 1 18169343
3 1 307110901
4 1 130955934
2 3 -288941558
2 5 96267410
5 3 -385208968
4 3 -176154967

Sample Output 3

200401298 182231955 -106709603 69445364 278499365

Problem Statement

You are given N points (x_1, y_1), (x_2, y_2), \dots, (x_N, y_N) on a two-dimensional plane, and a non-negative integer D.

Find the number of integer pairs (x, y) such that \displaystyle \sum_{i=1}^N (|x-x_i|+|y-y_i|) \leq D.

Constraints

• 1 \leq N \leq 2 \times 10^5
• 0 \leq D \leq 10^6
• -10^6 \leq x_i, y_i \leq 10^6
• (x_i, y_i) \neq (x_j, y_j) for i \neq j.
• All input values are integers.

Sample Input 1

2 3
0 0
1 0

Sample Output 1

8

The following figure visualizes the input and the answer for Sample 1. The blue points represent the input. The blue and red points, eight in total, satisfy the condition in the statement.

Sample Input 2

2 0
0 0
2 0

Sample Output 2

0

Sample Input 3

6 100
9 -6
10 -1
2 10
-1 7
-7 5
-1 -4

Sample Output 3

419

Problem Statement

Given are N distinct points in a two-dimensional plane. Point i (1 \leq i \leq N) has the coordinates (x_i,y_i).

Let us define the distance between two points i and j be \mathrm{min} (|x_i-x_j|,|y_i-y_j|): the smaller of the difference in the x-coordinates and the difference in the y-coordinates.

Find the maximum distance between two different points.

Constraints

• 2 \leq N \leq 200000
• 0 \leq x_i,y_i \leq 10^9
• (x_i,y_i) \neq (x_j,y_j) (i \neq j)
• All values in input are integers.

Sample Input 1

3
0 3
3 1
4 10

Sample Output 1

4

The distances between Points 1 and 2, between Points 1 and 3, and between Points 2 and 3 are 2, 4, and 1, respectively, so your output should be 4.

Sample Input 2

4
0 1
0 4
0 10
0 6

Sample Output 2

0

Sample Input 3

8
897 729
802 969
765 184
992 887
1 104
521 641
220 909
380 378

Sample Output 3

801