Problem Description

You are given an integer n representing the dimensions of an n x n grid, with the origin located at the bottom-left corner. Additionally, a 2D array rectangles is provided, where each element is in the form [start_x, start_y, end_x, end_y]. This array details the coordinates of non-overlapping rectangles on the grid:

• (start_x, start_y): Represents the bottom-left corner of the rectangle.
• (end_x, end_y): Represents the top-right corner.

The challenge is to determine if it is possible to make either two horizontal cuts or two vertical cuts on the grid such that:

• Each of the three sections resulting from these cuts contains at least one rectangle.
• Every rectangle belongs to exactly one section.

Your task is to return true if such cuts can be made, else return false.

Intuition

To solve this problem, the approach involves detecting potential cutting lines that can divide the grid into three sections, each containing at least one rectangle. The idea is to explore both horizontal and vertical potential cutting lines using the coordinates of the rectangles provided.

Here's the reasoning:

1. Representation of Coordinates:

   • For each rectangle, define two intersections: one when the rectangle begins and one when it ends. This will help track the number of overlapping rectangles as we traverse through the grid.

2. Sorting and Traversal:

   • Sort these events based on their coordinates, and for ties, prioritize 'end' events over 'start' events. This ensures that when counting overlaps, ending a rectangle does not prematurely affect the count of intersections.

3. Counting Line Intersections:

   • As the sorted list of events is traversed, an overlap counter is maintained: incrementing on a 'start' and decrementing on an 'end'.
   • A potential cutting line is determined when the overlap count reaches zero, meaning the grid is clear at that line. We need at least three such clear lines to form three distinct sections.

By applying these steps both horizontally and vertically, if either direction provides partitioning with three valid sections, it means the grid can be cut accordingly. Therefore, the solution checks these conditions and returns whether such a partition exists.

Learn more about Sorting patterns.

Solution Approach

The solution involves an algorithm that efficiently checks if the provided rectangles can be divided with vertical or horizontal cuts such that each section contains at least one rectangle.

1. Coordinate Representation:

   • For each rectangle in the list, create two events for both x-coordinates and y-coordinates:
     • A 'start' event when a rectangle begins at (start_x, start_y).
     • An 'end' event when a rectangle ends at (end_x, end_y).
   • These events are represented as tuples (coordinate, type) where type is 1 for 'start' and 0 for 'end'.

2. Data Structures:

   • Use lists to store these coordinate events for both x and y directions separately.

3. Sorting:

   • Sort both the x and y coordinate events. The sorting key ensures that on a tie in coordinate value, 'end' events come before 'start' events, using the sort key (coordinate, event type).

4. Counting Line Intersections:

   • Implement the countLineIntersections method, which traverses the sorted coordinate list. It maintains an overlap counter:
     • Increment the counter for a 'start' event.
     • Decrement the counter for an 'end' event.
   • A clear line occurs when the overlap counter reaches zero.
   • Count the number of these clear lines where the overlap is zero.

5. Validation:

   • The method proceeds by checking both the vertical (x_coordinates) and horizontal (y_coordinates) possibilities:
     • If either dimension allows for three clear lines, then it returns True.
   • Otherwise, return False.

The algorithm effectively leverages sorting and a linear traversal to achieve the desired solution, ensuring that it checks for partitionability in both vertical and horizontal directions.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach:

Suppose we have a grid of size 4 x 4 (n = 4) and the following list of rectangles:

rectangles = [
    [1, 0, 2, 2],  # Rectangle A
    [2, 0, 4, 1],  # Rectangle B
    [0, 2, 1, 4]   # Rectangle C
]

Step 1: Coordinate Representation

• For each rectangle, create coordinate events for both x-coordinates and y-coordinates:
  • Rectangle A has x-events (1, 1) and (2, 0); y-events (0, 1) and (2, 0)
  • Rectangle B has x-events (2, 1) and (4, 0); y-events (0, 1) and (1, 0)
  • Rectangle C has x-events (0, 1) and (1, 0); y-events (2, 1) and (4, 0)

Step 2: Data Structures

• x-events: [(1, 1), (2, 1), (4, 0), (0, 1), (2, 0), (1, 0)]
• y-events: [(0, 1), (2, 0), (1, 0), (2, 1), (4, 0), (0, 1)]

Step 3: Sorting

• Sort the coordinate events, prioritizing 'end' events before 'start' events when there is a tie:
  • Sorted x-events: [(0, 1), (1, 1), (1, 0), (2, 1), (2, 0), (4, 0)]
  • Sorted y-events: [(0, 1), (0, 1), (1, 0), (2, 0), (2, 1), (4, 0)]

Step 4: Counting Line Intersections

• Count clear lines in x and y:

  • Traverse x-events, maintain an overlap counter:

    • Start overlapping at x=0: overlap = 1
    • x=1: overlap increases to 2, then decreases to 1
    • x=2: overlap increases to 2, then decreases to 1
    • x=4: overlap decreases to 0 (clear line found)

  • Three clear distinct sections are not evident in x direction.

  • Traverse y-events:

    • Start overlapping at y=0: overlap = 2
    • y=1: overlap decreases to 1
    • y=2: overlap decreases to 0 (clear line found), and overlap increases to 1
    • y=4: overlap decreases to 0 (another clear line)

  • Three distinct sections are also not evident in y direction.

Step 5: Validation

• Since neither x nor y events form three distinct sections with clear lines, the grid cannot be properly partitioned with the given rectangles.
• Therefore, the function would return False.

Solution Implementation

Time and Space Complexity

The code iterates through the list of rectangles and performs operations on each of them. Let's analyze the complexities:

Time Complexity

1. Iterating Over Rectangles: The loop at for rect in rectangles: iterates over n rectangles, resulting in an O(n) complexity for this part.
2. Appending Coordinates: Appending coordinates of start and end points for each rectangle involves adding two elements for y-coordinates and two for x-coordinates, maintaining the O(n) complexity.
3. Sorting Operations: Both y_coordinates and x_coordinates lists are sorted, each containing 2n points. Sorting these lists has a complexity of O(n log n).
4. Count Line Intersections: The countLineIntersections method processes each sorted list once, iterating over 2n elements per list. This yields a linear complexity of O(2n) which is simplified to O(n) as constants are negligible.

Thus, the overall time complexity of the solution is O(n log n) due to the sorting step.

Space Complexity

1. Coordinate Storage: Two lists y_coordinates and x_coordinates are used, each storing 2n elements, leading to a space complexity of O(n) for each list.
2. Additional Variables: Other variables occupy constant space, which does not affect overall complexity significantly.

Hence, the overall space complexity is O(n), dominated by the lists storing the coordinates.

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