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#!/bin/python3
import math
import os
// ... rest of solution available after purchaseExplanation
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To tackle the problem of selecting k points from n distinct points on a line such that the minimum distance between any two chosen points is maximized, you need to think strategically about how to approach the problem efficiently.
First, recognize the nature of the problem: you want to maximize the minimum distance between any pair in the selected set of points. Intuitively, this means you want to spread out the chosen points as evenly as possible along the x-axis, preventing any two points from being too close.
Step 1: Sort the points
Since the points lie on a one-dimensional axis, start by sorting the array of x-coordinates. Sorting places the points in ascending order and helps you evaluate distances between points in a systematic and ordered manner.
Step 2: Understand the search space
The minimum distance between any two chosen points lies between 0 and the maximum possible distance, which is the distance between the smallest and largest points in the sorted list. So, your candidate answer for the minimum distance will lie within this range.
Step 3: Use binary search on the minimum distance
Binary search is a powerful technique here because you are searching for a value (the minimum distance) within a known range, and you want to find the maximum feasible value.
- Define low as 0 (or 1 if you want to consider a positive minimum distance) and high as the maximum distance between the furthest points.
- For each mid value between low and high, you will check if it’s possible to pick k points so that the minimum distance between any two chosen points is at least mid.
Step 4: Feasibility check (greedy approach)
For a given candidate minimum distance mid:
- Start from the first point (lowest coordinate) and select it.
- Move through the sorted array and select the next point only if the distance between it and the last selected point is at least mid.
- Continue this until you have selected k points or run out of points.
- If you manage to select k points with this condition, mid is feasible. Otherwise, it’s not.
This greedy approach works because you always try to place points as close to the left as possible while maintaining the required distance, which ensures you don't miss any feasible placements.
Step 5: Adjust the binary search bounds
- If mid is feasible, it means you can try to increase the minimum distance to find a better (larger) answer. Update low to mid + 1 to search for a larger feasible distance.
- If mid is not feasible, decrease high to mid - 1 to look for a smaller distance.
Step 6: Final result
When the binary search ends, high will hold the maximum feasible minimum distance. This value maximizes the smallest gap between any two chosen points.
Step 7: Complexity considerations
Sorting the points takes O(n log n). Each feasibility check is O(n) because you iterate through the sorted points once. The binary search on distance values runs in O(log D), where D is the range of distances (max - min). Hence, the overall complexity is O(n log n + n log D), efficient enough for large inputs.
Step 8: Edge cases to consider
- When k = 2, the maximum minimum distance is simply the distance between the two farthest points.
- When k = n, the minimum distance is the smallest gap between consecutive points after sorting.
- Points clustered very closely might limit the minimum distance, so the binary search ensures you find the optimal spacing.
Step 9: Why this approach works
Brute forcing all subsets would be computationally expensive. The combination of sorting, binary search on the minimum distance, and a greedy feasibility check exploits the problem's structure, allowing you to efficiently zero in on the optimal solution without exhaustively enumerating point selections.
Summary:
1. Sort points to enable systematic distance calculations.
2. Use binary search on possible minimum distances between points.
3. For each candidate distance, check feasibility with a greedy selection method.
4. Narrow down the search space based on feasibility until the maximum minimum distance is found.
This structured approach ensures an optimal and scalable solution to the problem.
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