Ezvor

3801. Minimum Cost to Merge Sorted Lists

HardArray Preparing judge… LeetCode

You are given a 2D integer array lists, where each lists[i] is a non-empty array of integers sorted in non-decreasing order.

You may repeatedly choose two lists a = lists[i] and b = lists[j], where i != j, and merge them. The cost to merge a and b is:

len(a) + len(b) + abs(median(a) - median(b)), where len and median denote the list length and median, respectively.

After merging a and b, remove both a and b from lists and insert the new merged sorted list in any position. Repeat merges until only one list remains.

Return an integer denoting the minimum total cost required to merge all lists into one single sorted list.

The median of an array is the middle element after sorting it in non-decreasing order. If the array has an even number of elements, the median is the left middle element.

 

Example 1:

Input: lists = [[1,3,5],[2,4],[6,7,8]]

Output: 18

Explanation:

Merge a = [1, 3, 5] and b = [2, 4]:

  • len(a) = 3 and len(b) = 2
  • median(a) = 3 and median(b) = 2
  • cost = len(a) + len(b) + abs(median(a) - median(b)) = 3 + 2 + abs(3 - 2) = 6

So lists becomes [[1, 2, 3, 4, 5], [6, 7, 8]].

Merge a = [1, 2, 3, 4, 5] and b = [6, 7, 8]:

  • len(a) = 5 and len(b) = 3
  • median(a) = 3 and median(b) = 7
  • cost = len(a) + len(b) + abs(median(a) - median(b)) = 5 + 3 + abs(3 - 7) = 12

So lists becomes [[1, 2, 3, 4, 5, 6, 7, 8]], and total cost is 6 + 12 = 18.

Example 2:

Input: lists = [[1,1,5],[1,4,7,8]]

Output: 10

Explanation:

Merge a = [1, 1, 5] and b = [1, 4, 7, 8]:

  • len(a) = 3 and len(b) = 4
  • median(a) = 1 and median(b) = 4
  • cost = len(a) + len(b) + abs(median(a) - median(b)) = 3 + 4 + abs(1 - 4) = 10

So lists becomes [[1, 1, 1, 4, 5, 7, 8]], and total cost is 10.

Example 3:

Input: lists = [[1],[3]]

Output: 4

Explanation:

Merge a = [1] and b = [3]:

  • len(a) = 1 and len(b) = 1
  • median(a) = 1 and median(b) = 3
  • cost = len(a) + len(b) + abs(median(a) - median(b)) = 1 + 1 + abs(1 - 3) = 4

So lists becomes [[1, 3]], and total cost is 4.

Example 4:

Input: lists = [[1],[1]]

Output: 2

Explanation:

The total cost is len(a) + len(b) + abs(median(a) - median(b)) = 1 + 1 + abs(1 - 1) = 2.

 

Constraints:

  • 2 <= lists.length <= 12
  • 1 <= lists[i].length <= 500
  • -109 <= lists[i][j] <= 109
  • lists[i] is sorted in non-decreasing order.
  • The sum of lists[i].length will not exceed 2000.
ArrayTwo PointersBinary SearchDynamic ProgrammingBit Manipulation

Hints

Hint 1
Use dynamic programming and bitmasks
Hint 2
Precompute the medians for every mask
Hint 3
Let dp[mask] represent the minimum cost to merge all lists in mask
Hint 4
dp[mask] = min(dp[s] + dp[mask ^ s] + cost(s, mask ^ s)), where s and mask ^ s are nonempty disjoint submasks whose union is mask
Hint 5
Use the precomputed medians to compute costs efficiently