Problem p03203 - Generation 2

Orig Description

Score : 800 points
Problem StatementTakahashi and Aoki will play a game using a grid with H rows and W columns of square cells.
There are N obstacles on this grid; the i-th obstacle is at (X_i,Y_i).
Here, we represent the cell at the i-th row and j-th column (1 \leq i \leq H, 1 \leq j \leq W) by (i,j).
There is no obstacle at (1,1), and there is a piece placed there at (1,1).
Starting from Takahashi, he and Aoki alternately perform one of the following actions:
Move the piece to an adjacent cell.
    Here, let the position of the piece be (x,y). Then Takahashi can only move the piece to (x+1,y), and Aoki can only move the piece to (x,y+1).
    If the destination cell does not exist or it is occupied by an obstacle, this action cannot be taken.
Do not move the piece, and end his turn without affecting the grid.
The game ends when the piece does not move twice in a row.
Takahashi would like to perform as many actions (including not moving the piece) as possible before the game ends, while Aoki would like to perform as few actions as possible before the game ends.
How many actions will Takahashi end up performing?
Constraints
1 \leq H,W \leq 2\times 10^5
0 \leq N \leq 2\times 10^5
1 \leq X_i \leq H
1 \leq Y_i \leq W
If i \neq j, (X_i,Y_i) \neq (X_j,Y_j)
(X_i,Y_i) \neq (1,1)
X_i and Y_i are integers.
InputInput is given from Standard Input in the following format:
H W N
X_1 Y_1
:
X_N Y_N
OutputPrint the number of actions Takahashi will end up performing.
Sample Input 13 3 1
3 2
Sample Output 12
For example, the game proceeds as follows:
Takahashi moves the piece to (2,1).
Aoki does not move the piece.
Takahashi moves the piece to (3,1).
Aoki does not move the piece.
Takahashi does not move the piece.
Takahashi performs three actions in this case, but if both players play optimally, Takahashi will perform only two actions before the game ends.
Sample Input 210 10 14
4 3
2 2
7 3
9 10
7 7
8 1
10 10
5 4
3 4
2 8
6 4
4 4
5 8
9 2
Sample Output 26
Sample Input 3100000 100000 0
Sample Output 3100000

Extracted Specification

An integer H (1 ≤ H ≤ 2 * 10^5), representing some dimension.
An integer W (1 ≤ W ≤ 2 * 10^5), representing another dimension.
An integer N (0 ≤ N ≤ 2 * 10^5), representing some quantity.
A list of N pairs of integers (X_i, Y_i), where each integer is between (1 ≤ X_i ≤ H) and (1 ≤ Y_i ≤ W).

### Example Input:

```
3 3 1
3 2
```

### Function Signature:
Write a function f(H, W, N, obstacles) that takes in the input.
```python
def f(H: int, W: int, N: int, obstacles: List[Tuple[int, int]]):
    '''
    H: an integer
    W: an integer
    N: an integer
    obstacles: a list of N pairs of integers
    '''
```

Test Case Generator

Sure, let's create a test case generator for the given problem statement. The generator will produce random values for \( H \), \( W \), \( N \), and the obstacle coordinates \( X_i \), \( Y_i \). We'll ensure that the constraints are respected: no obstacles at (1,1) and no duplicate obstacle coordinates.

Here is the Python code for the test case generator:

```python
import random

def tcgen():
    # Generate random H and W within given constraints
    H = random.randint(1, 2 * 10**5)
    W = random.randint(1, 2 * 10**5)
    
    # Generate random N within given constraints
    N = random.randint(0, 2 * 10**5)
    
    # Create a set to ensure no duplicate obstacles
    obstacles = set()
    
    # Generate N obstacles, ensuring no duplicates and not at (1,1)
    for _ in range(N):
        while True:
            X = random.randint(1, H)
            Y = random.randint(1, W)
            if (X, Y) != (1, 1) and (X, Y) not in obstacles:
                obstacles.add((X, Y))
                break
    
    # Convert the set to a list for easier handling
    obstacles = list(obstacles)

    # Format the output
    output = f"{H} {W} {N}\n"
    for X, Y in obstacles:
        output += f"{X} {Y}\n"
    
    return output.strip()

# Example usage
print(tcgen())
```

This code snippet defines the `tcgen` function, which generates and returns a random test case in the specified format. It ensures that the coordinates for the obstacles are unique and not at (1,1). The function randomly selects values for \(H\), \(W\), and \(N\) within the given constraints and then generates the required number of obstacles accordingly.

Extract Arguments

def extract_arguments(fh):
    H, W, N = map(int, fh.readline().strip().split())
    obstacles = []
    for _ in range(N):
        X, Y = map(int, fh.readline().strip().split())
        obstacles.append((X, Y))
    return H, W, N, obstacles

# if __name__ == "__main__":
#     input_path = sys.argv[1]
#     with open(input_path, 'r') as fh: 
#     H, W, N, obstacles = extract_arguments(fh)
#     f(H, W, N, obstacles)