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) = 2median(a) = 3 and median(b) = 2cost = len(a) + len(b) + abs(median(a) - median(b)) = 3 + 2 + abs(3 - 2) = 6So 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) = 3median(a) = 3 and median(b) = 7cost = len(a) + len(b) + abs(median(a) - median(b)) = 5 + 3 + abs(3 - 7) = 12So 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) = 4median(a) = 1 and median(b) = 4cost = len(a) + len(b) + abs(median(a) - median(b)) = 3 + 4 + abs(1 - 4) = 10So 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) = 1median(a) = 1 and median(b) = 3cost = len(a) + len(b) + abs(median(a) - median(b)) = 1 + 1 + abs(1 - 3) = 4So 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 <= 121 <= lists[i].length <= 500-109 <= lists[i][j] <= 109lists[i] is sorted in non-decreasing order.lists[i].length will not exceed 2000.