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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)
1using System;
2
3public class Solution {
4 private bool CanAchievePenalty(int[] nums, int maxOperations, int penalty) {
5 int operations = 0;
6 foreach (int num in nums) {
7 if (num > penalty) {
8 operations += (num - 1) / penalty;
9 }
10 if (operations > maxOperations) {
11 return false;
12 }
13 }
14 return true;
15 }
16
17 public int MinimumPenalty(int[] nums, int maxOperations) {
18 int left = 1, right = 1000000000;
19 while (left < right) {
20 int mid = left + (right - left) / 2;
21 if (CanAchievePenalty(nums, maxOperations, mid)) {
22 right = mid;
23 } else {
24 left = mid + 1;
25 }
26 }
27 return left;
28 }
29
30 public static void Main() {
31 var sol = new Solution();
32 Console.WriteLine(sol.MinimumPenalty(new int[]{9}, 2)); // Output: 3
33 }
34}In C#, the solution involves binary searching for the smallest possible penalty, using the CanAchievePenalty method to check potential solutions. This solution is efficient and handles large input sizes effectively by leveraging binary search.
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)
1
The greedy approach employs priority queue logic indirectly by sorting the array in descending order initially. It repeatedly considers the largest current value, using a strategy similar to the binary search solution.