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
The core challenge of this problem is to generate all permutations of length n that follow the valley pattern. This pattern is characterized by a sequence that first decreases and then increases. Such permutations are useful in various applications, including sorting algorithms and combinatorial problems.
Potential pitfalls include generating permutations that do not follow the valley pattern or missing some valid permutations.
To solve this problem, we can use a backtracking approach to generate all permutations and then filter out those that do not follow the valley pattern.
1. **Naive Solution**: Generate all permutations of length n and then check each permutation to see if it follows the valley pattern. This approach is not optimal because it generates many invalid permutations.
2. **Optimized Solution**: Use backtracking to generate only those permutations that follow the valley pattern. This approach is more efficient as it avoids generating invalid permutations.
1. Generate all permutations of length n using backtracking.
2. For each permutation, check if it follows the valley pattern.
3. If it does, add it to the result list.
// Function to generate all valley permutations of length n
function generateValleyPermutations(n) {
const results = [];
const current = [];
const used = Array(n + 1).fill(false);
// Helper function to check if the current permutation is a valley
function isValley(arr) {
let i = 1;
while (i < arr.length && arr[i] > arr[i - 1]) i++;
while (i < arr.length && arr[i] < arr[i - 1]) i++;
return i === arr.length;
}
// Backtracking function to generate permutations
function backtrack() {
if (current.length === n) {
if (isValley(current)) {
results.push([...current]);
}
return;
}
for (let i = 1; i <= n; i++) {
if (!used[i]) {
used[i] = true;
current.push(i);
backtrack();
current.pop();
used[i] = false;
}
}
}
backtrack();
return results;
}
// Example usage
const n = 4;
console.log(generateValleyPermutations(n));
The time complexity of the backtracking approach is O(n!), as we generate all permutations of length n. The space complexity is O(n), as we use an array to store the current permutation and a boolean array to track used elements.
1. n = 0: The output should be an empty list.
2. n = 1: The output should be [[1]].
3. n = 2: The output should be [[1, 2], [2, 1]].
To test the solution comprehensively, we can use a variety of test cases, from simple to complex:
console.log(generateValleyPermutations(0)); // []
console.log(generateValleyPermutations(1)); // [[1]]
console.log(generateValleyPermutations(2)); // [[1, 2], [2, 1]]
console.log(generateValleyPermutations(3)); // [[1, 2, 3], [2, 1, 3], [3, 1, 2], [3, 2, 1]]
console.log(generateValleyPermutations(4)); // [[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]]
1. Break down the problem into smaller parts and solve each part step by step.
2. Use backtracking to generate permutations and prune invalid permutations early.
3. Practice solving similar problems to improve problem-solving skills.
Understanding and solving valley permutation problems is important for developing problem-solving skills and understanding combinatorial algorithms. Practice and exploration of similar problems can help improve these skills.
1. LeetCode - Practice coding problems.
2. GeeksforGeeks - Tutorials and explanations of algorithms.
3. MDN Web Docs - JavaScript documentation and tutorials.