Problem Statement

We have a grid of $H$ rows and $W$ columns. Initially, there is a stone in the top left cell. Shik is trying to move the stone to the bottom right cell. In each step, he can move the stone one cell to its left, up, right, or down (if such cell exists). It is possible that the stone visits a cell multiple times (including the bottom right and the top left cell).

You are given a matrix of characters $a_{ij}$ ($1 \leq i \leq H$, $1 \leq j \leq W$). After Shik completes all moving actions, $a_{ij}$ is # if the stone had ever located at the $i$-th row and the $j$-th column during the process of moving. Otherwise, $a_{ij}$ is .. Please determine whether it is possible that Shik only uses right and down moves in all steps.

Constraints

• $2 \leq H, W \leq 8$
• $a_{i,j}$ is either # or ..
• There exists a valid sequence of moves for Shik to generate the map $a$.

Sample Input 1

4 5
##...
.##..
..##.
...##

Sample Output 1

Possible

The matrix can be generated by a $7$-move sequence: right, down, right, down, right, down, and right.

Sample Input 2

5 3
###
..#
###
#..
###

Sample Output 2

Impossible

Sample Input 3

4 5
##...
.###.
.###.
...##

Sample Output 3

Impossible

Problem Statement

You are given a permutation $p$ of the set {$1, 2, ..., N$}. Please construct two sequences of positive integers $a_1$, $a_2$, ..., $a_N$ and $b_1$, $b_2$, ..., $b_N$ satisfying the following conditions:

• $1 \leq a_i, b_i \leq 10^9$ for all $i$
• $a_1 < a_2 < ... < a_N$
• $b_1 > b_2 > ... > b_N$
• $a_{p_1}+b_{p_1} < a_{p_2}+b_{p_2} < ... < a_{p_N}+b_{p_N}$

Constraints

• $2 \leq N \leq 20,000$
• $p$ is a permutation of the set {$1, 2, ..., N$}

Sample Input 1

2
1 2

Sample Output 1

1 4
5 4

$a_1 + b_1 = 6$ and $a_2 + b_2 = 8$. So this output satisfies all conditions.

Sample Input 2

3
3 2 1

Sample Output 2

1 2 3
5 3 1

Sample Input 3

3
2 3 1

Sample Output 3

5 10 100
100 10 1

Problem Statement

There are $N$ balls and $N+1$ holes in a line. Balls are numbered $1$ through $N$ from left to right. Holes are numbered $1$ through $N+1$ from left to right. The $i$-th ball is located between the $i$-th hole and $(i+1)$-th hole. We denote the distance between neighboring items (one ball and one hole) from left to right as $d_i$ ($1 \leq i \leq 2 \times N$). You are given two parameters $d_1$ and $x$. $d_i - d_{i-1}$ is equal to $x$ for all $i$ ($2 \leq i \leq 2 \times N$).

We want to push all $N$ balls into holes one by one. When a ball rolls over a hole, the ball will drop into the hole if there is no ball in the hole yet. Otherwise, the ball will pass this hole and continue to roll. (In any scenario considered in this problem, balls will never collide.)

In each step, we will choose one of the remaining balls uniformly at random and then choose a direction (either left or right) uniformly at random and push the ball in this direction. Please calculate the expected total distance rolled by all balls during this process.

For example, when $N = 3$, $d_1 = 1$, and $x = 1$, the following is one possible scenario:

• first step: push the ball numbered $2$ to its left, it will drop into the hole numbered $2$. The distance rolled is $3$.
• second step: push the ball numbered $1$ to its right, it will pass the hole numbered $2$ and drop into the hole numbered $3$. The distance rolled is $9$.
• third step: push the ball numbered $3$ to its right, it will drop into the hole numbered $4$. The distance rolled is $6$.

So the total distance in this scenario is $18$.

Note that in all scenarios every ball will drop into some hole and there will be a hole containing no ball in the end.

Constraints

• $1 \leq N \leq 200,000$
• $1 \leq d_1 \leq 100$
• $0 \leq x \leq 100$
• All input values are integers.

Sample Input 1

1 3 3

Sample Output 1

4.500000000000000

The distance between the only ball and the left hole is $3$ and distance between the only ball and the right hole is $6$. There are only two scenarios (i.e. push left and push right). The only ball will roll $3$ and $6$ unit distance, respectively. So the answer for this example is $(3+6)/2 = 4.5$.

Sample Input 2

2 1 0

Sample Output 2

2.500000000000000

Sample Input 3

1000 100 100

Sample Output 3

649620280.957660079002380

Problem Statement

Imagine a game played on a line. Initially, the player is located at position $0$ with $N$ candies in his possession, and the exit is at position $E$. There are also $N$ bears in the game. The $i$-th bear is located at $x_i$. The maximum moving speed of the player is $1$ while the bears do not move at all.

When the player gives a candy to a bear, it will provide a coin after $T$ units of time. More specifically, if the $i$-th bear is given a candy at time $t$, it will put a coin at its position at time $t+T$. The purpose of this game is to give candies to all the bears, pick up all the coins, and go to the exit. Note that the player can only give a candy to a bear if the player is at the exact same position of the bear. Also, each bear will only produce a coin once. If the player visits the position of a coin after or at the exact same time that the coin is put down, the player can pick up the coin. Coins do not disappear until collected by the player.

Shik is an expert of this game. He can give candies to bears and pick up coins instantly. You are given the configuration of the game. Please calculate the minimum time Shik needs to collect all the coins and go to the exit.

Constraints

• $1 \leq N \leq 100,000$
• $1 \leq T, E \leq 10^9$
• $0 < x_i < E$
• $x_i < x_{i+1}$ for $1 \leq i < N$
• All input values are integers.

Partial Scores

• In test cases worth $600$ points, $N \leq 2,000$.

Sample Input 1

3 9 1
1 3 8

Sample Output 1

12

The optimal strategy is to wait for the coin after treating each bear. The total time spent on waiting is $3$ and moving is $9$. So the answer is $3 + 9 = 12$.

Sample Input 2

3 9 3
1 3 8

Sample Output 2

16

Sample Input 3

2 1000000000 1000000000
1 999999999

Sample Output 3

2999999996

Problem Statement

In the country there are $N$ cities numbered $1$ through $N$, which are connected by $N-1$ bidirectional roads. In terms of graph theory, there is a unique simple path connecting every pair of cities. That is, the $N$ cities form a tree. Besides, if we view city $1$ as the root of this tree, the tree will be a full binary tree (a tree is a full binary tree if and only if every non-leaf vertex in the tree has exactly two children). Roads are numbered $1$ through $N-1$. The $i$-th road connects city $i+1$ and city $a_i$. When you pass through road $i$, the toll you should pay is $v_i$ (if $v_i$ is $0$, it indicates the road does not require a toll).

A company in city $1$ will give each employee a family travel and cover a large part of the tolls incurred. Each employee can design the travel by themselves. That is, which cities he/she wants to visit in each day and which city to stay at each night. However, there are some constraints. More details are as follows:

• The travel must start and end at city $1$. If there are $m$ leaves in the tree formed by the cities, then the number of days in the travel must be $m+1$. At the end of each day, except for the last day, the employee must stay at some hotel in a leaf city. During the whole travel, the employee must stay at all leaf cities exactly once.

• During the whole travel, all roads of the country must be passed through exactly twice.

• The amount that the employee must pay for tolls by him/herself is the maximum total toll incurred in a single day during the travel, except the first day and the last day. The remaining tolls will be covered by the company.

Shik, as an employee of this company, has only one hope for his travel. He hopes that the amount he must pay for tolls by himself will be as small as possible. Please help Shik to design the travel which satisfies his hope.

Constraints

• $2 < N < 131,072$
• $1 \leq a_i \leq i$ for all $i$
• $0 \leq v_i \leq 131,072$
• $v_i$ is an integer
• The given tree is a full binary tree

Sample Input 1

7
1 1
1 1
2 1
2 1
3 1
3 1

Sample Output 1

4

There are $4$ leaves in the tree formed by the cities (cities $4$, $5$, $6$, and $7$), so the travel must be $5$ days long. There are $4! = 24$ possible arrangements of the leaf cities to stay during the travel, but some of them are invalid. For example, $(4,5,6,7)$ and $(6,7,5,4)$ are valid orders, but $(5,6,7,4)$ is invalid because the route passes $4$ times through the road connecting city $1$ and city $2$. The figure below shows the routes of the travel for these arrangements.

For all valid orders, day $3$ is the day with the maximum total incurred toll of $4$, passing through $4$ roads.

Sample Input 2

9
1 2
1 2
3 2
3 2
5 2
5 2
7 2
7 2

Sample Output 2

6

The following figure shows the route of the travel for one possible arrangement of the stayed cities for the solution. Note that the tolls incurred in the first day and the last day are always covered by the company.

Sample Input 3

15
1 2
1 5
3 3
4 3
4 3
6 5
5 4
5 3
6 3
3 2
11 5
11 3
13 2
13 1

Sample Output 3

15

Sample Input 4

3
1 0
1 0

Sample Output 4

0

Problem Statement

Shik's job is very boring. At day $0$, his boss gives him a string $S_0$ of length $N$ which consists of only lowercase English letters. In the $i$-th day after day $0$, Shik's job is to copy the string $S_{i-1}$ to a string $S_i$. We denote the $j$-th letter of $S_i$ as $S_i[j]$.

Shik is inexperienced in this job. In each day, when he is copying letters one by one from the first letter to the last letter, he would make mistakes. That is, he sometimes accidentally writes down the same letter that he wrote previously instead of the correct one. More specifically, $S_i[j]$ is equal to either $S_{i-1}[j]$ or $S_{i}[j-1]$. (Note that $S_i[1]$ always equals to $S_{i-1}[1]$.)

You are given the string $S_0$ and another string $T$. Please determine the smallest integer $i$ such that $S_i$ could be equal to $T$. If no such $i$ exists, please print -1.

Constraints

• $1 \leq N \leq 1,000,000$
• The lengths of $S_0$ and $T$ are both $N$.
• Both $S_0$ and $T$ consist of lowercase English letters.

Sample Input 1

5
abcde
aaacc

Sample Output 1

2

$S_0$ = abcde, $S_1$ = aaccc and $S_2$ = aaacc is a possible sequence such that $S_2 = T$.

Sample Input 2

5
abcde
abcde

Sample Output 2

0

Sample Input 3

4
acaa
aaca

Sample Output 3

2

Sample Input 4

5
abcde
bbbbb

Sample Output 4

-1