Generate Valley Permutations in Python

Given a non-negative integer n, return all the valley permutations of length n.

A valley permutation is defined as a permutation that is a concatenation of two parts: a monotonically decreasing sequence and then a monotonically increasing sequence.

E.g. valley permutations follow this format: [p[0] < p[1] < p[2] < ... < p[k] > p[k+1] > p[k + 2] > ... > p[n]].

Example:

Input: n = 4

Output:[
    [1, 2, 3, 4],
    [2, 1, 3, 4],
    [3, 1, 2, 4],
    [3, 2, 1, 4],
    [4, 1, 2, 3],
    [4, 2, 1, 3],
    [4, 3, 1, 2],
    [4, 3, 2, 1]
]

Explanation:
Permutations suchs as [1, 4, 2, 3] are not valley because 1 < 4 > 2 < 3

Notes: n <= 10

Understanding the Problem

The core challenge of this problem is to generate permutations that follow a specific pattern: a monotonically decreasing sequence followed by a monotonically increasing sequence. This pattern is what defines a "valley" permutation.

Valley permutations are significant in various applications such as sorting algorithms, data analysis, and combinatorial problems. A common pitfall is to generate all permutations and then filter out the non-valley ones, which is inefficient.

Approach

To solve this problem, we need to generate permutations that inherently follow the valley pattern. Here's a step-by-step approach:

Generate all possible permutations of the sequence [1, 2, ..., n].
Filter out the permutations that do not follow the valley pattern.

While generating all permutations and then filtering is a straightforward approach, it is not optimal. Instead, we can directly generate valley permutations by ensuring the pattern during the generation process.

Naive Solution

The naive solution involves generating all permutations and then checking each one for the valley pattern. This approach is not optimal because it involves generating a large number of permutations and then filtering them.

Optimized Solution

An optimized solution involves generating permutations that follow the valley pattern directly. This can be done using a recursive approach:

Start with an empty permutation.
Recursively add elements to the permutation, ensuring the valley pattern is maintained.
Backtrack if the pattern is violated.

Algorithm

Here is a step-by-step breakdown of the optimized algorithm:

Initialize an empty list to store the valley permutations.
Define a recursive function that takes the current permutation and the remaining elements as arguments.
In the recursive function, if the current permutation is of length n, add it to the list of valley permutations.
Otherwise, for each remaining element, add it to the current permutation and recursively call the function with the updated permutation and remaining elements.
Ensure that the valley pattern is maintained during the recursive calls.

Code Implementation

def generate_valley_permutations(n):
    def is_valley(perm):
        # Check if the permutation follows the valley pattern
        i = 1
        while i < len(perm) and perm[i] > perm[i - 1]:
            i += 1
        while i < len(perm) and perm[i] < perm[i - 1]:
            i += 1
        return i == len(perm)
    
    def backtrack(current_perm, remaining_elements):
        if len(current_perm) == n:
            if is_valley(current_perm):
                result.append(current_perm[:])
            return
        
        for i in range(len(remaining_elements)):
            current_perm.append(remaining_elements[i])
            backtrack(current_perm, remaining_elements[:i] + remaining_elements[i+1:])
            current_perm.pop()
    
    result = []
    backtrack([], list(range(1, n + 1)))
    return result

# Example usage
n = 4
print(generate_valley_permutations(n))

Complexity Analysis

The time complexity of the optimized solution is O(n!), where n is the length of the permutation. This is because we are generating all permutations of length n. The space complexity is O(n) due to the recursion stack and the storage of permutations.

Edge Cases

Potential edge cases include:

n = 0: The output should be an empty list.
n = 1: The output should be [[1]] as there is only one permutation of length 1.

These edge cases are handled by the recursive function and the base cases.

Testing

To test the solution comprehensively, we can use a variety of test cases:

Simple cases: n = 1, n = 2
Medium cases: n = 3, n = 4
Complex cases: n = 5, n = 6

We can use Python's built-in unittest framework to write and run these test cases.

Thinking and Problem-Solving Tips

When approaching such problems, it is important to:

Understand the problem requirements and constraints.
Break down the problem into smaller, manageable parts.
Consider both naive and optimized solutions.
Write clean, readable, and well-commented code.

Practicing similar problems and studying algorithms can help improve problem-solving skills.

Conclusion

In this blog post, we discussed how to generate valley permutations of a given length. We explored both naive and optimized solutions, provided a detailed algorithm, and implemented the solution in Python. Understanding and solving such problems is crucial for developing strong problem-solving skills.

Additional Resources

For further reading and practice, consider the following resources:

Generate Valley Permutations in Python (Time Complexity: O(n!))