Problem Statement

You are given a permutation P = (P_1, \dots, P_N) of (1, \dots, N), where (P_1, \dots, P_N) \neq (1, \dots, N).

Assume that P is the K-th lexicographically smallest among all permutations of (1 \dots, N). Find the (K-1)-th lexicographically smallest permutation.

Constraints

• 2 \leq N \leq 100
• 1 \leq P_i \leq N \, (1 \leq i \leq N)
• P_i \neq P_j \, (i \neq j)
• (P_1, \dots, P_N) \neq (1, \dots, N)
• All values in the input are integers.

Sample Input 1

3
3 1 2

Sample Output 1

2 3 1

Here are the permutations of (1, 2, 3) in ascending lexicographical order.

• (1, 2, 3)
• (1, 3, 2)
• (2, 1, 3)
• (2, 3, 1)
• (3, 1, 2)
• (3, 2, 1)

Therefore, P = (3, 1, 2) is the fifth smallest, so the sought permutation, which is the fourth smallest (5 - 1 = 4), is (2, 3, 1).

Sample Input 2

10
9 8 6 5 10 3 1 2 4 7

Sample Output 2

9 8 6 5 10 2 7 4 3 1

Problem Statement

Takahashi has N pairs of socks, and the i-th pair consists of two socks of color i. One day, after organizing his chest of drawers, Takahashi realized that he had lost one sock each of colors A_1, A_2, \dots, A_K, so he decided to use the remaining 2N-K socks to make \lfloor\frac{2N-K}{2}\rfloor new pairs of socks, each pair consisting of two socks. The weirdness of a pair of a sock of color i and a sock of color j is defined as |i-j|, and Takahashi wants to minimize the total weirdness.

Find the minimum possible total weirdness when making \lfloor\frac{2N-K}{2}\rfloor pairs from the remaining socks. Note that if 2N-K is odd, there will be one sock that is not included in any pair.

Constraints

• 1\leq K\leq N \leq 2\times 10^5
• 1\leq A_1 < A_2 < \dots < A_K \leq N
• All input values are integers.

Sample Input 1

4 2
1 3

Sample Output 1

2

Below, let (i,j) denote a pair of a sock of color i and a sock of color j.

There are 1, 2, 1, 2 socks of colors 1, 2, 3, 4, respectively. Creating the pairs (1,2),(2,3),(4,4) results in a total weirdness of |1-2|+|2-3|+|4-4|=2, which is the minimum.

Sample Input 2

5 1
2

Sample Output 2

0

The optimal solution is to make the pairs (1,1),(3,3),(4,4),(5,5) and leave one sock of color 2 as a surplus (not included in any pair).

Sample Input 3

8 5
1 2 4 7 8

Sample Output 3

2

Problem Statement

You are given an integer W.
You are going to prepare some weights so that all of the conditions below are satisfied.

• The number of weights is between 1 and 300, inclusive.
• Each weight has a mass of positive integer not exceeding 10^6.
• Every integer between 1 and W, inclusive, is a good integer. Here, a positive integer n is said to be a good integer if the following condition is satisfied:
  • We can choose at most 3 different weights from the prepared weights with a total mass of n.

Print a combination of weights that satisfies the conditions.

Constraints

• 1 \leq W \leq 10^6
• W is an integer.

Sample Input 1

6

Sample Output 1

3
1 2 3

In the output above, 3 weights with masses 1, 2, and 3 are prepared.
This output satisfies the conditions. Especially, regarding the 3-rd condition, we can confirm that every integer between 1 and W, inclusive, is a good integer.

• If we choose only the 1-st weight, it has a total mass of 1.
• If we choose only the 2-nd weight, it has a total mass of 2.
• If we choose only the 3-rd weight, it has a total mass of 3.
• If we choose the 1-st and the 3-rd weights, they have a total mass of 4.
• If we choose the 2-nd and the 3-rd weights, they have a total mass of 5.
• If we choose the 1-st, the 2-nd, and the 3-rd weights, they have a total mass of 6.

Sample Input 2

12

Sample Output 2

6
2 5 1 2 5 1

You may prepare multiple weights with the same mass.

Problem Statement

You are given a simple undirected graph with N vertices and M edges. The i-th edge connects Vertices U_i and V_i. Vertex i has a positive integer A_i written on it.

You will repeat the following operation N times.

• Choose a Vertex x that is not removed yet, and remove Vertex x and all edges incident to Vertex x. The cost of this operation is the sum of the integers written on the vertices directly connected by an edge with Vertex x that are not removed yet.

We define the cost of the entire N operations as the maximum of the costs of the individual operations. Find the minimum possible cost of the entire operations.

Constraints

• 1 \le N \le 2 \times 10^5
• 0 \le M \le 2 \times 10^5
• 1 \le A_i \le 10^9
• 1 \le U_i,V_i \le N
• The given graph is simple.
• All values in input are integers.

Sample Input 1

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

Sample Output 1

3

By performing the operations as follows, the maximum of the costs of the N operations can be 3.

• Choose Vertex 3. The cost is A_1=3.
• Choose Vertex 1. The cost is A_2+A_4=3.
• Choose Vertex 2. The cost is 0.
• Choose Vertex 4. The cost is 0.

The maximum of the costs of the N operations cannot be 2 or less, so the solution is 3.

Sample Input 2

7 13
464 661 847 514 74 200 188
5 1
7 1
5 7
4 1
4 5
2 4
5 2
1 3
1 6
3 5
1 2
4 6
2 7

Sample Output 2

1199

Problem Statement

Consider a binary tree with N vertices numbered 1, 2, \ldots, N. Here, a binary tree is a rooted tree where each vertex has at most two children. Specifically, each vertex in a binary tree has at most one left child and at most one right child.

Determine whether there exists a binary tree rooted at Vertex 1 satisfying the conditions below, and present such a tree if it exists.

• The depth-first traversal of the tree in pre-order is (P_1, P_2, \ldots, P_N).
• The depth-first traversal of the tree in in-order is (I_1, I_2, \ldots, I_N).

Constraints

• 2 \leq N \leq 2 \times 10^5
• N is an integer.
• (P_1, P_2, \ldots, P_N) is a permutation of (1, 2, \ldots, N).
• (I_1, I_2, \ldots, I_N) is a permutation of (1, 2, \ldots, N).

Sample Input 1

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

Sample Output 1

3 6
0 0
0 5
0 0
0 0
4 2

The binary tree rooted at Vertex 1 shown in the following image satisfies the conditions.

Sample Input 2

2
2 1
1 2

Sample Output 2

-1

No binary tree rooted at Vertex 1 satisfies the conditions, so -1 should be printed.