1681. Minimum Incompatibility

Problem Link

Description

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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

Solution

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Python3

class Solution:
    def minimumIncompatibility(self, nums: List[int], k: int) -> int:
        scores = collections.defaultdict(int)
        n = len(nums)
        per = n // k
        candidates = []
        
        for mask in range(1 << n):
            count = 0
            
            for j in range(n):
                if mask & (1 << j) > 0:
                    count += 1
            
            if count == per:
                s = set()
                
                for j in range(n):
                    if mask & (1 << j) > 0: 
                        s.add(nums[j])
                
                if len(s) != per: continue
                
                candidates.append(mask) 
                scores[mask] = max(s) - min(s)
        
        @cache
        def go(mask):
            if mask == 0: return 0
            
            res = float('inf')
            
            for candidate in candidates:
                if (mask & candidate) == candidate:
                    res = min(res, go(mask ^ candidate) + scores[candidate])
            
            return res
        
        ans = go((1 << n) - 1)
        
        return -1 if ans == float('inf') else ans