Formatted question description: https://leetcode.ca/all/932.html

# 932. Beautiful Array (Medium)

For some fixed N, an array A is beautiful if it is a permutation of the integers 1, 2, ..., N, such that:

For every i < j, there is no k with i < k < j such that A[k] * 2 = A[i] + A[j].

Given N, return any beautiful array A.  (It is guaranteed that one exists.)

Example 1:

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


Example 2:

Input: 5
Output: [3,1,2,5,4]

Note:

• 1 <= N <= 1000

Related Topics:
Divide and Conquer

## Solution 1. Divide and Conquer

Observations:

• Given 2 * A[k] != A[i] + A[j], since 2 * A[k] is even, we can make A[i] an odd number and A[j] an even number. So we can split the array into two parts, all the odd numbers into left and all the even numbers into right.
• We can reuse the solution to 1 2 3 4 5 to construct the solution for 1 3 5 7 9.
// OJ: https://leetcode.com/problems/beautiful-array/

// Time: O(NlogN)
// Space: O(NlogN)
class Solution {
unordered_map<int, vector<int>> m;
vector<int> dfs(int N) {
if (m.count(N)) return m[N];
vector<int> ans(N);
if (N == 1) ans[0] = 1;
else {
int t = 0;
for (int x : dfs((N + 1) / 2)) ans[t++] = 2 * x - 1; // odd
for (int x : dfs(N / 2)) ans[t++] = 2 * x; // even
}
return m[N] = ans;
}
public:
vector<int> beautifulArray(int N) {
return dfs(N);
}
};


Java

class Solution {
public int[] beautifulArray(int N) {
int log = (int) Math.ceil(Math.log(N) / Math.log(2));
int power2 = (int) Math.pow(2, log);
int[] beautifulArrayPower2 = new int[power2];
beautifulArrayPower2[0] = 1;
for (int length = 1; length < power2; length *= 2) {
for (int i = 0; i < length; i++)
beautifulArrayPower2[i] *= 2;
for (int i = 0; i < length; i++)
beautifulArrayPower2[length * 2 - i - 1] = beautifulArrayPower2[i] - 1;
}
int[] beautifulArray = new int[N];
int index = 0;
for (int i = 0; i < power2; i++) {
int num = beautifulArrayPower2[i];
if (num <= N) {
beautifulArray[index] = num;
index++;
}
}
return beautifulArray;
}
}