Maximize the Distance Between Points on a Square

Maximize the Distance Between Points on a Square - Problem

Imagine you're a city planner tasked with placing k emergency stations on the perimeter of a square city block to maximize safety coverage. You want to ensure that the closest distance between any two stations is as large as possible.

Given a square with side length side and corners at (0, 0), (0, side), (side, 0), and (side, side), you have a list of candidate locations points on the boundary where stations can be placed. Each point is represented as [x, y].

Your goal is to select exactly k points from the available locations such that the minimum Manhattan distance between any two selected points is maximized.

The Manhattan distance between two points (x₁, y₁) and (x₂, y₂) is |x₁ - x₂| + |y₁ - y₂|.

Return the maximum possible minimum Manhattan distance between the selected k points.

Input & Output

example_1.py — Small Square

$ Input: side = 2, k = 2, points = [[0,0], [1,0], [2,0], [2,1], [2,2], [1,2], [0,2], [0,1]]

› Output: 4

💡 Note: We can select points [0,0] and [2,2] (opposite corners). The Manhattan distance between them is |0-2| + |0-2| = 4, which is the maximum possible minimum distance for k=2 points.

example_2.py — More Points

$ Input: side = 3, k = 3, points = [[0,0], [0,1], [0,2], [0,3], [1,3], [2,3], [3,3], [3,2], [3,1], [3,0], [2,0], [1,0]]

› Output: 4

💡 Note: We can select points [0,0], [3,1], and [1,3]. The minimum distance between any pair is 4, achieved by multiple pairs. This gives us optimal spacing for 3 points.

example_3.py — Edge Case

$ Input: side = 1, k = 4, points = [[0,0], [0,1], [1,1], [1,0]]

› Output: 2

💡 Note: With only 4 corner points available and needing to select all 4, the minimum distance between adjacent corners is 2 (like [0,0] and [0,1]).

Constraints

1 ≤ side ≤ 10⁹
2 ≤ k ≤ points.length ≤ 10⁵
points[i] is on the boundary of the square
All points are distinct and lie exactly on the square's perimeter

Visualization

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Asked in

G Google 35 ⊞ Microsoft 28 a Amazon 22 f Meta 18

The optimal solution uses binary search on the answer combined with a greedy placement strategy. First, convert all boundary points to a linear coordinate system (0 to 4×side), then binary search for the maximum feasible minimum distance. For each candidate distance, greedily check if k points can be placed with at least that distance apart. Time complexity: O(n log n + n log D) where D is the maximum possible distance.

Common Approaches

✓ Sliding Window

⏱️ Time: N/A Space: N/A

Binary Search + Greedy (Optimal)

⏱️ Time: O(n log n + n log D) Space: O(n)

This optimal approach uses binary search on the answer. We first convert all boundary points to a linear coordinate system, then binary search on the minimum distance. For each candidate distance, we greedily check if we can place k points with at least that distance apart.

Brute Force (Try All Combinations)

⏱️ Time: O(C(n,k) × k²) Space: O(k)

This approach generates all possible combinations of k points from the given points array, calculates the minimum Manhattan distance for each combination, and returns the maximum of these minimum distances.

Algorithm Steps — Algorithm Steps

Code -

solution.c — C

Time & Space Complexity

Time Complexity

⏱️

✓ Linear Growth

Space Complexity

✓ Linear Space

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