DSAPROB19 - Editorial

editorial
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Problem Link - Ball Position

Problem Statement:
You are given K electromagnetic balls that need to be placed on N coordinates on a number line. These balls attract each other, so the goal is to place them as far apart as possible.

Your task is to find the maximum possible value of the minimum distance between any two balls after placing all K balls on the given coordinates optimally.

Approach:
The key idea is to maximize the minimum distance between K electromagnetic balls placed on N sorted coordinates. The solution leverages binary search on possible distances combined with a greedy placement strategy to determine if the placement of balls is feasible for a given minimum distance.

  1. Sorting: First, sort the coordinates so that we can attempt placing balls sequentially in increasing order.

  2. Binary Search on Minimum Distance: Perform binary search on the possible minimum distances between balls, ranging from 0 to the maximum distance (difference between the smallest and largest coordinates).

  3. Greedy Placement Check: For each midpoint mid (in the binary search), use a greedy method to check if we can place all K balls such that the distance between any two consecutive balls is at least mid. Start placing the first ball at the first position and place subsequent balls at the next available position that maintains at least mid distance from the last placed ball.

  4. Adjust Search: If the placement is successful for mid, try for a larger distance by moving the left boundary of the search to mid + 1.

    If unsuccessful, reduce the right boundary to mid - 1 to check for smaller distances.

  5. Final Result: The largest mid for which the placement is feasible gives the maximum possible minimum distance.

Time Complexity:
The overall time complexity is O(Nlog⁡(N) + Nlog(⁡maxDistance)), where sorting positions takes O(Nlog⁡(N)), the binary search runs in O(log⁡(maxDistance)), and the helper function check runs in O(N) for each binary search step.

Space Complexity:
The space complexity is O(N), since we are using an additional array to store the positions