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This approach leverages binary search to efficiently minimize the penalty. The idea is to use binary search to determine the minimum possible 'penalty', where penalty is defined as the maximum number of balls allowed in any bag.
We set our binary search range from 1 to the maximum number of balls in any bag initially. For each mid value (potential penalty), we check if it's possible to achieve this penalty with the given number of operations. We do this by calculating how many operations we'd need to make every bag have at most 'mid' number of balls. If we can do that within the allowed number of operations, we adjust our search range to potentially lower penalties; otherwise, we increase our search range.
Time Complexity: O(n log(max(nums)))
Space Complexity: O(1)
1def can_achieve_penalty(nums, max_operations, penalty):
2 operations = 0
3 for num in nums:
4 if num > penalty:
5 operations +=
The Python version utilizes a binary search within the interval [1, maximum number of balls], trying to minimize the penalty. The auxiliary function can_achieve_penalty checks if it is feasible to redistribute the balls to satisfy the mid penalty limit.
Another approach is to continuously divide the largest bags using a priority queue to keep track of the largest bag sizes. By always splitting the largest bag first, we aim to reduce the penalty faster. This works similarly to the binary search approach in terms of dividing the problem space but uses a different method to handle priorities.
Time Complexity: O(n log n + n log(max(nums)))
Space Complexity: O(1)
1using System.Collections.Generic;
public class Solution {
public int MinimumPenalty(int[] nums, int maxOperations) {
var pq = new SortedSet<(int value, int index)>(Comparer<(int, int)>.Create((a, b) =>
a.value == b.value ? a.index.CompareTo(b.index) : b.value.CompareTo(a.value)));
int index = 0;
foreach (var num in nums) {
pq.Add((num, index++));
}
while (maxOperations > 0 && pq.Max.value > 1) {
var largest = pq.Max;
pq.Remove(largest);
int split = (largest.value + 1) / 2;
pq.Add((split, index++));
pq.Add((largest.value - split, index++));
--maxOperations;
}
return pq.Max.value;
}
public static void Main() {
var sol = new Solution();
Console.WriteLine(sol.MinimumPenalty(new int[]{9}, 2)); // Output: 3
}
}In C#, a SortedSet is utilized to simulate a max-heap behavior by complementing set ordering to prioritize the largest element. It allows efficient removal of the largest ball bag for operations.