# 1681. Minimum Incompatibility

## Description

You are given an integer array nums​​​ and an integer k. You are asked to distribute this array into k subsets of equal size such that there are no two equal elements in the same subset.

A subset's incompatibility is the difference between the maximum and minimum elements in that array.

Return the minimum possible sum of incompatibilities of the k subsets after distributing the array optimally, or return -1 if it is not possible.

A subset is a group integers that appear in the array with no particular order.

Example 1:

Input: nums = [1,2,1,4], k = 2
Output: 4
Explanation: The optimal distribution of subsets is [1,2] and [1,4].
The incompatibility is (2-1) + (4-1) = 4.
Note that [1,1] and [2,4] would result in a smaller sum, but the first subset contains 2 equal elements.

Example 2:

Input: nums = [6,3,8,1,3,1,2,2], k = 4
Output: 6
Explanation: The optimal distribution of subsets is [1,2], [2,3], [6,8], and [1,3].
The incompatibility is (2-1) + (3-2) + (8-6) + (3-1) = 6.


Example 3:

Input: nums = [5,3,3,6,3,3], k = 3
Output: -1
Explanation: It is impossible to distribute nums into 3 subsets where no two elements are equal in the same subset.


Constraints:

• 1 <= k <= nums.length <= 16
• nums.length is divisible by k
• 1 <= nums[i] <= nums.length

## Solutions

Solution 1: Preprocessing + State Compression + Dynamic Programming

Let’s assume that the size of each subset after partitioning is $m$, so $m=\frac{n}{k}$, where $n$ is the length of the array.

We can enumerate all subsets $i$, where $i \in [0, 2^n)$, if the binary representation of subset $i$ has $m$ ones, and the elements in subset $i$ are not repeated, then we can calculate the incompatibility of subset $i$, denoted as $g[i]$, i.e., $g[i]=\max_{j \in i} {nums[j]} - \min_{j \in i} {nums[j]}$.

Next, we can use dynamic programming to solve.

We define $f[i]$ as the minimum sum of incompatibilities when the current partitioned subset state is $i$. Initially, $f[0]=0$, which means no elements are partitioned into the subset, and the rest $f[i]=+\infty$.

For state $i$, we find all undivided and non-repeated elements, represented by a state $mask$. If the number of elements in state $mask$ is greater than or equal to $m$, then we enumerate all subsets $j$ of $mask$, and satisfy $j \subset mask$, then $f[i \cup j]=\min {f[i \cup j], f[i]+g[j]}$.

Finally, if $f[2^n-1]=+\infty$, it means that it cannot be partitioned into $k$ subsets, return $-1$, otherwise return $f[2^n-1]$.

The time complexity is $O(3^n)$, and the space complexity is $O(2^n)$. Here, $n$ is the length of the array.

• class Solution {
public int minimumIncompatibility(int[] nums, int k) {
int n = nums.length;
int m = n / k;
int[] g = new int[1 << n];
Arrays.fill(g, -1);
for (int i = 1; i < 1 << n; ++i) {
if (Integer.bitCount(i) != m) {
continue;
}
Set<Integer> s = new HashSet<>();
int mi = 20, mx = 0;
for (int j = 0; j < n; ++j) {
if ((i >> j & 1) == 1) {
break;
}
mi = Math.min(mi, nums[j]);
mx = Math.max(mx, nums[j]);
}
}
if (s.size() == m) {
g[i] = mx - mi;
}
}
int[] f = new int[1 << n];
final int inf = 1 << 30;
Arrays.fill(f, inf);
f[0] = 0;
for (int i = 0; i < 1 << n; ++i) {
if (f[i] == inf) {
continue;
}
Set<Integer> s = new HashSet<>();
for (int j = 0; j < n; ++j) {
if ((i >> j & 1) == 0 && !s.contains(nums[j])) {
}
}
if (s.size() < m) {
continue;
}
for (int j = mask; j > 0; j = (j - 1) & mask) {
if (g[j] != -1) {
f[i | j] = Math.min(f[i | j], f[i] + g[j]);
}
}
}
return f[(1 << n) - 1] == inf ? -1 : f[(1 << n) - 1];
}
}

• class Solution {
public:
int minimumIncompatibility(vector<int>& nums, int k) {
int n = nums.size();
int m = n / k;
int g[1 << n];
memset(g, -1, sizeof(g));
for (int i = 1; i < 1 << n; ++i) {
if (__builtin_popcount(i) != m) {
continue;
}
unordered_set<int> s;
int mi = 20, mx = 0;
for (int j = 0; j < n; ++j) {
if (i >> j & 1) {
if (s.count(nums[j])) {
break;
}
s.insert(nums[j]);
mi = min(mi, nums[j]);
mx = max(mx, nums[j]);
}
}
if (s.size() == m) {
g[i] = mx - mi;
}
}
int f[1 << n];
memset(f, 0x3f, sizeof(f));
f[0] = 0;
for (int i = 0; i < 1 << n; ++i) {
if (f[i] == 0x3f3f3f3f) {
continue;
}
unordered_set<int> s;
for (int j = 0; j < n; ++j) {
if ((i >> j & 1) == 0 && !s.count(nums[j])) {
s.insert(nums[j]);
}
}
if (s.size() < m) {
continue;
}
for (int j = mask; j; j = (j - 1) & mask) {
if (g[j] != -1) {
f[i | j] = min(f[i | j], f[i] + g[j]);
}
}
}
return f[(1 << n) - 1] == 0x3f3f3f3f ? -1 : f[(1 << n) - 1];
}
};

• class Solution:
def minimumIncompatibility(self, nums: List[int], k: int) -> int:
n = len(nums)
m = n // k
g = [-1] * (1 << n)
for i in range(1, 1 << n):
if i.bit_count() != m:
continue
s = set()
mi, mx = 20, 0
for j, x in enumerate(nums):
if i >> j & 1:
if x in s:
break
mi = min(mi, x)
mx = max(mx, x)
if len(s) == m:
g[i] = mx - mi
f = [inf] * (1 << n)
f[0] = 0
for i in range(1 << n):
if f[i] == inf:
continue
s = set()
for j, x in enumerate(nums):
if (i >> j & 1) == 0 and x not in s:
if len(s) < m:
continue
while j:
if g[j] != -1:
f[i | j] = min(f[i | j], f[i] + g[j])
j = (j - 1) & mask
return f[-1] if f[-1] != inf else -1


• func minimumIncompatibility(nums []int, k int) int {
n := len(nums)
m := n / k
const inf = 1 << 30
f := make([]int, 1<<n)
g := make([]int, 1<<n)
for i := range g {
f[i] = inf
g[i] = -1
}
for i := 1; i < 1<<n; i++ {
if bits.OnesCount(uint(i)) != m {
continue
}
s := map[int]struct{}{}
mi, mx := 20, 0
for j, x := range nums {
if i>>j&1 == 1 {
if _, ok := s[x]; ok {
break
}
s[x] = struct{}{}
mi = min(mi, x)
mx = max(mx, x)
}
}
if len(s) == m {
g[i] = mx - mi
}
}
f[0] = 0
for i := 0; i < 1<<n; i++ {
if f[i] == inf {
continue
}
s := map[int]struct{}{}
for j, x := range nums {
if _, ok := s[x]; !ok && i>>j&1 == 0 {
s[x] = struct{}{}
}
}
if len(s) < m {
continue
}
for j := mask; j > 0; j = (j - 1) & mask {
if g[j] != -1 {
f[i|j] = min(f[i|j], f[i]+g[j])
}
}
}
if f[1<<n-1] == inf {
return -1
}
return f[1<<n-1]
}

• function minimumIncompatibility(nums: number[], k: number): number {
const n = nums.length;
const m = Math.floor(n / k);
const g: number[] = Array(1 << n).fill(-1);
for (let i = 1; i < 1 << n; ++i) {
if (bitCount(i) !== m) {
continue;
}
const s: Set<number> = new Set();
let [mi, mx] = [20, 0];
for (let j = 0; j < n; ++j) {
if ((i >> j) & 1) {
if (s.has(nums[j])) {
break;
}
mi = Math.min(mi, nums[j]);
mx = Math.max(mx, nums[j]);
}
}
if (s.size === m) {
g[i] = mx - mi;
}
}
const inf = 1e9;
const f: number[] = Array(1 << n).fill(inf);
f[0] = 0;
for (let i = 0; i < 1 << n; ++i) {
if (f[i] === inf) {
continue;
}
const s: Set<number> = new Set();
for (let j = 0; j < n; ++j) {
if (((i >> j) & 1) === 0 && !s.has(nums[j])) {
}
}
if (s.size < m) {
continue;
}
for (let j = mask; j; j = (j - 1) & mask) {
if (g[j] !== -1) {
f[i | j] = Math.min(f[i | j], f[i] + g[j]);
}
}
}
return f[(1 << n) - 1] === inf ? -1 : f[(1 << n) - 1];
}

function bitCount(i: number): number {
i = i - ((i >>> 1) & 0x55555555);
i = (i & 0x33333333) + ((i >>> 2) & 0x33333333);
i = (i + (i >>> 4)) & 0x0f0f0f0f;
i = i + (i >>> 8);
i = i + (i >>> 16);
return i & 0x3f;
}


• public class Solution {
public int MinimumIncompatibility(int[] nums, int k) {
int n = nums.Length;
int m = n / k;
int[] g = new int[1 << n];
Array.Fill(g, -1);
for (int i = 1; i < 1 << n; ++i) {
if (bitCount(i) != m) {
continue;
}
HashSet<int> s = new();
int mi = 20, mx = 0;
for (int j = 0; j < n; ++j) {
if ((i >> j & 1) == 1) {
if (s.Contains(nums[j])) {
break;
}
mi = Math.Min(mi, nums[j]);
mx = Math.Max(mx, nums[j]);
}
}
if (s.Count == m) {
g[i] = mx - mi;
}
}
int[] f = new int[1 << n];
int inf = 1 << 30;
Array.Fill(f, inf);
f[0] = 0;
for (int i = 0; i < 1 << n; ++i) {
if (f[i] == inf) {
continue;
}
HashSet<int> s = new();
for (int j = 0; j < n; ++j) {
if ((i >> j & 1) == 0 && !s.Contains(nums[j])) {
}
}
if (s.Count < m) {
continue;
}
for (int j = mask; j > 0; j = (j - 1) & mask) {
if (g[j] != -1) {
f[i | j] = Math.Min(f[i | j], f[i] + g[j]);
}
}
}
return f[(1 << n) - 1] == inf ? -1 : f[(1 << n) - 1];
}

private int bitCount(int x) {
int cnt = 0;
while (x > 0) {
x &= x - 1;
++cnt;
}
return cnt;
}
}