Sure! Below is the Python code for a test case generator for the "Sliding Minimum Element" problem:

```python
import random

def tcgen():
    # Generate a random value for N within the constraints
    N = random.randint(1, 10**6)
    
    # Generate a random value for L such that 1 <= L <= N
    L = random.randint(1, N)
    
    # Generate the array A with N elements, each element in the range [1, 10**9]
    A = [random.randint(1, 10**9) for _ in range(N)]
    
    return N, L, A

# Example usage
N, L, A = tcgen()
print(N, L)
print(" ".join(map(str, A)))
```

### Explanation:

1. **Generating \( N \)**:
   - A random integer \( N \) is generated between 1 and \( 10^6 \), which specifies the number of elements in the array.

2. **Generating \( L \)**:
   - A random integer \( L \) is generated between 1 and \( N \), ensuring that the length of the sub-arrays is valid.

3. **Generating Array \( A \)**:
   - An array \( A \) of size \( N \) is generated, where each element is a random integer between 1 and \( 10^9 \).

4. **Returning the test case**:
   - The function returns \( N \), \( L \), and the array \( A \).

5. **Example Usage**:
   - The example usage shows how to call the `tcgen` function and output the generated test case in the required format.

This code ensures that all constraints are respected, and it provides a random but valid test case for the "Sliding Minimum Element" problem.