963. Minimum Area Rectangle II

Problem Description

In this problem, we are given a set of points in the X-Y plane, with each point represented as a pair of coordinates [x_i, y_i]. The task is to find the minimum area of any rectangle that can be formed using these points. Importantly, the sides of these rectangles do not have to be parallel to the X and Y axes, which means we could be dealing with rectangles at any orientation in the plane.

The output should be the area of the smallest such rectangle. If no rectangle can be formed from the given points, we should return 0. In the answer, a small tolerance is allowed - the returned result must be within 10^-5 (0.00001) of the actual minimum area.

The challenge is to find a way to efficiently check all combinations of points to determine which ones can form the corners of a valid rectangle and calculate the minimum area among all possible rectangles.

Intuition

The solution makes use of a brute-force approach, iterating through each combination of three points (x1, y1), (x2, y2), (x3, y3) to determine whether a fourth point exists that would complete a rectangle. For any given three points, the idea is to calculate the fourth point (x4, y4) which would form two vectors representing two adjacent sides of the rectangle.

To ensure that these sides are perpendicular (which is a requirement for a rectangle), the dot product of the vectors is calculated. If the dot product is zero, we have a right angle, and thus a candidate for rectangle corners.

Here's the breakdown of the solution:

1. Store all given points in a set for constant time look-up.
2. Iterate over each pair of points (x1, y1) and (x2, y2), treating them as adjacent corners of a prospective rectangle.
3. For each pair, explore another point (x3, y3) to act as the third corner.
4. Calculate the potential fourth corner's coordinates (x4, y4) using vector addition.
5. Check if the calculated fourth point actually exists in the provided points set.
6. Confirm that the angle between the vectors representing potential rectangle sides is a right angle by checking if their dot product is zero.
7. If we have a valid rectangle, calculate its area using the lengths of the vectors, which are the rectangle's sides.
8. Keep track of the minimum area found so far.
9. After all points have been considered, return the minimum area or 0 if no rectangle is found.

By iterating over all possible combinations and considering only those sets of points that pass the perpendicularity test, the solution finds the rectangle with the minimum area. This approach guarantees finding the correct answer, but it may be slow for a large set of points because of its cubic time complexity.

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Solution Approach

The Reference Solution Approach relies on algorithmic geometry and set data structures to solve the problem efficiently. Here's a step-by-step breakdown of the solution implementation:

1. Using a Set for Lookup: A set s is created to store all the points. A set is used here because it allows for O(1) complexity for look-up operations, which is crucial since we want to check the existence of the potential fourth point quickly in our algorithm.

2. Iterating Over Combinations of Points: The solution uses three nested loops to iterate over all possible combinations of three different points(x1, y1), (x2, y2), and (x3, y3). The outermost loop picks the first point, the second loop picks the second point, and the third loop picks the third point.

3. Calculating the Fourth Point: With (x1, y1), (x2, y2), and (x3, y3) determined, the coordinates for the potential fourth point (x4, y4) are calculated. This is done by vector addition:

   1x4 = x2 - x1 + x3
   2y4 = y2 - y1 + y3

   This represents the diagonal translation from the first point to the second and then from the first to the third, arriving at the fourth point's location.

4. Verifying Perpendicular Sides: To ensure the points can form a rectangle, the algorithm verifies that the vectors representing the sides are perpendicular. This is done by computing the dot product of the two vectors:

   1v21 = (x2 - x1, y2 - y1)  # Vector from point 1 to point 2
   2v31 = (x3 - x1, y3 - y1)  # Vector from point 1 to point 3

   The dot product of v21 and v31 should be zero for the sides to be perpendicular:

   1v21[0] * v31[0] + v21[1] * v31[1] == 0

5. Calculating Area of Rectangle: If a valid rectangle is found, its area is calculated using the lengths of the sides:

   1w = sqrt(v21[0] ** 2 + v21[1] ** 2)  # Width of the rectangle
   2h = sqrt(v31[0] ** 2 + v31[1] ** 2)  # Height of the rectangle

   The area w * h is computed and compared with the running minimum.

6. Tracking the Minimum Area: A variable ans is initialized to inf (infinity) to keep track of the minimum area seen so far. Whenever a new rectangle area is computed, we update ans with the lesser of the current minimum or the new area.

7. Returning the Result: After all possible point combinations have been considered, the solution checks if ans is still inf. If it is, it means no rectangle was found, so we return 0. If a minimum area was found, we return that minimum area.

This approach systematically explores all combinations of points that could possibly form rectangles, ensuring no potential rectangle is missed. Although the solution's complexity is high (O(n^3)), it is acceptable given that the problem size (the number of points) is not excessively large.

Example Walkthrough

Let's consider a small example to illustrate the solution approach.

Suppose we are given the following set of points: [(1, 0), (0, 1), (2, 1), (1, 2)], and we need to find the minimum area of a rectangle that can be formed with these points.

Following the solution approach:

1. Using a Set for Lookup: The first step is to store all points in a set for quick lookup. Let's define our set s:

   1s = {(1, 0), (0, 1), (2, 1), (1, 2)}

2. Iterating Over Combinations of Points: We start iterating over combinations of three different points. Let's take (x1, y1) = (1, 0), (x2, y2) = (0, 1), and (x3, y3) = (2, 1).

3. Calculating the Fourth Point: We calculate the potential fourth point (x4, y4) using vector addition:

   1x4 = x2 - x1 + x3 = 0 - 1 + 2 = 1
   2y4 = y2 - y1 + y3 = 1 - 0 + 1 = 2

   So the fourth point's coordinates are (x4, y4) = (1, 2).

4. Verifying Perpendicular Sides: We have to check if the sides are perpendicular, which is a requirement for a rectangle. We calculate the vectors from point 1 to point 2 v21, and from point 1 to point 3 v31:

   1v21 = (x2 - x1, y2 - y1) = (0 - 1, 1 - 0) = (-1, 1)
   2v31 = (x3 - x1, y3 - y1) = (2 - 1, 1 - 0) = (1, 1)

   The dot product of v21 and v31 is (-1*1 + 1*1) = 0. Since the dot product is zero, v21 and v31 are perpendicular.

5. Calculating Area of Rectangle: Now that we confirmed we have a rectangle, we can calculate its area:

   1w = sqrt(v21[0] ** 2 + v21[1] ** 2) = sqrt((-1) ** 2 + 1 ** 2) = sqrt(2)
   2h = sqrt(v31[0] ** 2 + v31[1] ** 2) = sqrt(1 ** 2 + 1 ** 2) = sqrt(2)

   The area of the rectangle is w * h = sqrt(2) * sqrt(2) = 2.

6. Tracking the Minimum Area: Since this is our first rectangle, we set the minimum area ans = 2.

7. Returning the Result: After completing the iterations through all possible point combinations (since our example is small, we only had a single combination to check), we find that the minimum area ans is 2. Therefore, the minimum area of a rectangle that can be formed with our given set of points is 2.

This simple example demonstrates the steps outlined in the solution approach, and how it can be used to find the minimum area rectangle given a set of points in the X-Y plane.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code can be determined by analyzing the nested loops and the operations inside them:

• The first loop runs over all the points, which gives us O(n).
• Nested within the first loop, the second loop runs over all other points except the one chosen by the first loop, contributing O(n-1) complexity.
• Inside the second loop, there's another loop that starts from the current index of the second loop and goes over the remaining points, which, in the worst case, contributes O(n/2) (this is an approximation since it depends on the current index of the second loop).

Putting these together, the complexity of the loops is approximately O(n) * O(n-1) * O(n/2), which simplifies to O(n^3) for the worst case.

Additionally, within the innermost loop, there are constant-time operations such as checking if a point exists in the set, and basic arithmetic operations. The dot product and the calculation of rectangle area also take constant time, which does not affect the overall O(n^3) time complexity.

Space Complexity

The space complexity can be considered by looking at the data structures used:

• A set s that holds all the points. In the worst case, every point is unique, so the space complexity for the set is O(n).
• Constant amount of extra space is used for variables such as x1, y1, x2, y2, x3, y3, x4, y4, v21, v31, w, h, and ans.

Therefore, the overall space complexity of the algorithm is O(n) for storing the set of points.

Learn more about how to find time and space complexity quickly using problem constraints.