Beautiful Arrangement II

Problem Statement

Given two integers, n and k, the objective is to construct a sequence called answer consisting of n distinct positive integers, each integer within the range from 1 to n. The sequence must fulfill a specific condition related to its consecutive differences. Specifically, when you compute the absolute differences between every two consecutive integers in the sequence (answer), the resulting sequence of differences must contain exactly k distinct integers. Your goal is to return any valid sequence that meets these criteria. If multiple valid sequences exist, you may return any one of them.

Examples

Example 1

Input:

n = 3, k = 1

Output:

[1,2,3]
Explanation: The [1,2,3] has three different positive integers ranging from 1 to 3, and the [1,1] has exactly 1 distinct integer: 1

Example 2

Input:

n = 3, k = 2

Output:

[1,3,2]
Explanation: The [1,3,2] has three different positive integers ranging from 1 to 3, and the [2,1] has exactly 2 distinct integers: 1 and 2.

Constraints

• 1 <= k < n <= 104

Approach and Intuition

The task is to generate a permutation of numbers from 1 to n such that the number of distinct absolute differences between consecutive numbers is exactly k. To achieve this, one can use strategic positioning of numbers to control the differences generated. Here’s a general approach:

1. Understand that the largest difference we can create in an array [1,2,3,...,n] is between 1 and n, which is n-1. Hence, the first point is ensuring our sequence can generate k distinct differences ranging likely between 1 and k.

2. Start by visualizing how the permutations and their differences work. A direct sequence like [1, 2, 3, ..., n] only gives one distinct difference (1). On the other hand, a sequence with a bit more variation such as [1, n, 2, n-1, 3, n-2,...] starts to generate larger and varying differences.

3. Generating the sequence:

   • If k is 1, simply return the sequence [1, 2, 3, ..., n] since all differences are 1.
   • If k is greater than 1, start by arranging the first k+1 elements to cover the k differences. For example, placing numbers in the format [1, k+1, 2, k, 3, k-1,...] ensures that we start the list with the maximum difference of k and introduce decrementing differences as we continue to complete the series with rest of the left numbers.

4. Example elaboration:

   • For n = 3 and k = 1, the sequence [1, 2, 3] fulfills the criteria because all differences are 1 which is the only distinct integer needed.
   • For n = 3 and k = 2, a sequence like [1, 3, 2] will have differences of [2, 1] fulfilling the need for two distinct differences: 1 and 2.

By strategically placing your integers and understanding the concept of differences, a valid permutation can be constructed efficiently to meet any given k. The highest value of k (just below n) suggests a need for maximized manipulation in arrangement to achieve varied and distinct differences, which is achievable within the given constraints.

Solutions

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;
    }
};