How do you solve Minimize Manhattan Distances on LeetCode?

Minimize Manhattan Distances (LeetCode #3102) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n) time complexity and O(n) space complexity. Key insight: Understanding the transformation of coordinates can significantly reduce the complexity of the problem.

What is the brute force solution for Minimize Manhattan Distances?

The brute force approach for Minimize Manhattan Distances is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute-force approach involves removing each point one by one and calculating the maximum Manhattan distance between the remaining points. This is straightforward but inefficient, especially for larger datasets. The optimal Optimal Solution approach improves this to O(n).

What algorithm or data structure is used to solve Minimize Manhattan Distances?

Minimize Manhattan Distances uses the following concepts: Array, Math, Geometry, Sorting, Ordered Set. The recommended patterns to study are: Coordinate Transformation, Sorting, Array Manipulation.

What are the interview tips for Minimize Manhattan Distances?

Always start with a brute-force approach to understand the problem better before optimizing. Think about how to leverage mathematical properties of the problem to simplify calculations. Practice transforming coordinates or data structures to find patterns in distance problems.

What are common mistakes when solving Minimize Manhattan Distances?

Failing to consider edge cases where all points are the same, leading to incorrect distance calculations. Not optimizing the distance calculations by using coordinate transformations.

#3102

Minimize Manhattan Distances

Hard

ArrayMathGeometrySortingOrdered SetCoordinate TransformationSortingArray Manipulation

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(n)
Space	O(1)	O(n)

💡

Intuition

Time O(n)Space O(n)

Instead of checking all pairs of points, we can leverage the properties of Manhattan distance. By transforming the coordinates, we can efficiently calculate the maximum distances without needing to remove points one by one.

⚙️

Algorithm

4 steps

1Step 1: Transform each point to two new coordinates: x - y and x + y.
2Step 2: Find the maximum and minimum values of these transformed coordinates.
3Step 3: The maximum distance after removing one point can be computed using the differences of these max and min values.
4Step 4: Return the minimum of the maximum distances calculated for each transformed coordinate.

solution.py10 lines

1def min_max_distance(points):
2    x_minus_y = [x - y for x, y in points]
3    x_plus_y = [x + y for x, y in points]
4
5    max_x_minus_y = max(x_minus_y)
6    min_x_minus_y = min(x_minus_y)
7    max_x_plus_y = max(x_plus_y)
8    min_x_plus_y = min(x_plus_y)
9
10    return min(max_x_minus_y - min_x_minus_y, max_x_plus_y - min_x_plus_y)

ℹ

Complexity note: This complexity arises because we traverse the list of points a constant number of times (twice) to compute the transformed coordinates and their max/min values.

1Understanding the transformation of coordinates can significantly reduce the complexity of the problem.
2Manhattan distance properties allow us to compute maximum distances efficiently.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.