Beautiful Arrangement II

class Solution {
public:
    vector<int> generateArray(int n, int k) {
        vector<int> result(n, 0);
        int index = 0;
        for (int value = 1; value < n - k; value++) {
            result[index++] = value;
        }
        for (int j = 0; j <= k; j++) {
            result[index++] = (j % 2 == 0) ? (n - k + j / 2) : (n - j / 2);
        }
        return result;
    }
};

The problem "Beautiful Arrangement II" challenges you to generate an array of integers from 1 to n that consists of exactly k different absolute differences between consecutive elements.

Here's a breakdown of the C++ solution, which utilizes a methodical strategy:

1. The solution defines a function generateArray that takes two parameters, n and k.
2. It initializes a result vector result of size n, with all values set to zero.
3. Utilizes a loop to fill up the first part of the result vector linearly from 1 to n-k, ensuring that the differences between these consecutive elements are precisely 1.
4. The second part of the vector is filled in a pattern designed to alternate and create a larger difference between elements, thus ensuring that k distinct differences are achieved. This section uses more complex arithmetic to determine element placement based on even and odd indices, focusing on decreasing from n towards n-k and inversely.
5. Returns the vector result as an output which will satisfy the condition of having exactly k unique differences between consecutive values.

The solution cleverly adjusts the arrangement of the later elements to achieve the required k distinct differences, balancing simpler initial sequences with a more complex subsequent pattern. This method ensures an efficient creation of the desired array using mathematical patterns.

class Solution {

    public int[] generateSequence(int length, int diff) {
        int[] result = new int[length];
        int index = 0;
        for (int value = 1; value < length - diff; value++) {
            result[index++] = value;
        }
        for (int j = 0; j <= diff; j++) {
            result[index++] = (j % 2 == 0) ? (length - diff + j / 2) : (length - j / 2);
        }
        return result;
    }
}

The Java solution provided generates a sequence of integers based on two parameters: the length of the array and the difference between consecutive elements. The method generateSequence(int length, int diff) is employed to achieve this in several key steps:

1. Initialize an array result of integers with a size equal to length.
2. Iterate from 1 up to length - diff, and store each value in result.
3. In a second loop, compute values based on the modulo operation to decide if the value should be incremented from the middle point of the sequence up or decremented down, then stored in the result array.

The result is a permutation of the numbers from 1 to length such that the absolute difference between any two consecutive integers is close to the specified diff. This algorithm efficiently creates an array with the desired properties by splitting the filling process into two parts: a linear increase and a conditional filling based on the difference requirement.

class Solution:
    def createSequence(self, size: int, diff: int) -> List[int]:
        sequence = [0] * size
        index = 0
        for value in range(1, size - diff):
            sequence[index] = value
            index += 1
        for j in range(diff + 1):
            sequence[index] = size - diff + j // 2 if j % 2 == 0 else size - j // 2
            index += 1
        return sequence

The mentioned Python function createSequence constructs a sequence of integers of a specified length that maintains a provided difference between elements, intended to meet certain conditions for "arrangement beauty."

Here's a summary of the function logic:

• Initialize sequence as a list of zeros with size equal to the input size.
• Populate the sequence with increasing natural numbers until the value is less than size - diff.
• For the remaining part of the sequence, structure the elements to maintain the specified difference. This involves alternating calculations based on whether the index is even or odd.
• The elements added are either increasing by smaller jumps or directly decreasing depending on their position to align with the difference given.

This function benefits the most:

• When creating patterns within a numeric sequence.
• In scenarios where control over specific differences between consecutive numbers in a sequence is required, such as generating test cases for algorithms focusing on sequence properties or constraints.

Always test with various size and diff values to confirm that the sequence meets the desired properties and that it adheres to the intended "beauty" conditions in all cases.