Canonicalized Input Description:

An integer H (1 ≤ H ≤ 10^3), representing some quantity or size.
An integer W (1 ≤ W ≤ 10^3), representing some quantity or size.
An integer N (1 ≤ N ≤ 10^3), representing some quantity or size.
A list of N integers L, where each integer (1 ≤ L[i] ≤ 10^3) represents some quantity or size.
A list of N lists of tuples, where each tuple contains two integers (1 ≤ x, y ≤ 10^3).

### Example Input:

```
5
5
3
2 3 4
[(1, 2), (3, 4)]
[(1, 1), (2, 2), (3, 3)]
[(2, 3), (3, 4), (4, 5), (5, 6)]
```

### Function Signature:
Write a function f(H, W, N, L, paths) that takes in the input.
```python
def f(H: int, W: int, N: int, L: List[int], paths: List[List[Tuple[int, int]]]):
    '''
    H: an integer
    W: an integer
    N: an integer
    L: a list of integers
    paths: a list of lists of tuples, where each tuple contains two integers
    '''
```