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The problem can be solved efficiently using a binary search on the minimum magnetic force. Start by sorting the basket positions, then perform a binary search to find the maximum valid minimum force. The basic idea is to check if it is possible to place all the balls with at least a certain minimum distance (force) between any two of them. If it is possible, try for a larger minimum distance, otherwise try smaller.
Time Complexity: O(n log(n) + log(maxDistance) * n), where n is the number of baskets.
Space Complexity: O(1)
1def can_place_balls(position, m, min_dist):
2 count = 1
3 last_position = position[0]
4 for i in range(1, len(position)):
5 if position[i] - last_position >= min_dist:
6 count += 1
7 last_position = position[i]
8 if count >= m:
9 return True
10 return False
11
12def max_distance(position, m):
13 position.sort()
14 left, right = 1, position[-1] - position[0]
15 answer = 0
16 while left <= right:
17 mid = (left + right) // 2
18 if can_place_balls(position, m, mid):
19 answer = mid
20 left = mid + 1
21 else:
22 right = mid - 1
23 return answer
24
25position = [1, 2, 3, 4, 7]
26m = 3
27print(max_distance(position, m))In the Python solution, the positions are sorted first. Then a binary search determines the maximum minimum distance possible. The `can_place_balls` function checks if it’s feasible to place the balls such that all of them are at least `min_dist` apart, supporting the binary search process.
This approach involves using a binary search for determining the optimal force, aided by a greedy strategy to verify if a particular minimum force is attainable. By iterating over the sorted positions, balls are placed as far apart as possible greedily.
Time Complexity: O(n log(n) + n log(maxDistance)), where n is the number of baskets.
Space Complexity: O(1)
1
class Solution {
public bool CanPlaceBalls(int[] position, int m, int dist) {
int count = 1, last_position = position[0];
for (int i = 1; i < position.Length; ++i) {
if (position[i] - last_position >= dist) {
count++;
last_position = position[i];
if (count == m) {
return true;
}
}
}
return false;
}
public int MaxDistance(int[] position, int m) {
Array.Sort(position);
int left = 1, right = position[position.Length - 1] - position[0];
while (left < right) {
int mid = left + (right - left + 1) / 2;
if (CanPlaceBalls(position, m, mid)) {
left = mid;
} else {
right = mid - 1;
}
}
return left;
}
static void Main() {
Solution sol = new Solution();
int[] position = {5, 4, 3, 2, 1, 1000000000};
int m = 2;
Console.WriteLine(sol.MaxDistance(position, m));
}
}The C# program integrates a greedy and binary search method to establish the maximum value of the minimum magnetic force. The `CanPlaceBalls` method secures the placements iteratively and regulates the search bounds based on feasibility.