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In this article, we will see how Sweep Line Technique can be used to compute union of rectangles.



Let's suppose a list of (axis-aligned) rectangles and each rectangle is represented as [x1, y1, x2, y2] , where (x1, y1) are the coordinates of the bottom-left corner, and (x2, y2) are the coordinates of the top-right corner of the ith rectangle.
Our objective is to find the total area covered by all rectangles in the plane.



This problem can be solved very easily using Sweep Line technique, using the concepts of events and active events. However, to solve this problem we would need Dual Sweep Line Technique, where we will have to sweep line along both x-axis and y-axis. The algorithm and code below explains the technique very well.

Algorithm:



Here we are calculating union of all given rectangles. So any area in the 2D plane should be calculated only once, which means we should be careful about any area that is overlapped by two or more rectangles, because they need to be included only once.
A union B = A + B - (A intersection B).

So what we can do is:
  1. We can sweep a horizontal line from bottom to top as we see the horizontal lines of the rectangles (marking start/bottom and end/top-end of a rectangle) we calculate the area till that y-coordinate level. So at the end, by the time we reach the top most horizontal line(s) and are done processing the top most y-coordinate level, we will be done calculating the whole area.

    For every level, we calculate the area till that level only once even when we have more than one horizontal line at the same level (y-coordinate).
  2. For every level we sweep a vertical line from left to right and calculate the width of the rectangles present at that level.

    So from above, we see we need to do line sweep in both vertical and horizontal direction.For every line sweep operation along y-axis, whenever we find at least one new event present at that level (y-coordinate) we do a horizontal line sweep along x-axis to measure the total widths of all the rectangles present at that level (y-coordinate).

    To achieve this, we need to keep track of the WIDTH and HEIGHT of every rectangle. Since we will be sweeping line from bottom to top, we need to have the base (y-coordinate) and top-end (y-coordinate) of the rectangles as event along with their width (start x-coordinate and end x-coordinate for that rectangle).


        
     18 |          H ___________ G
        |           |           |
     16 |           |           |
        |           |           |
     14 |           | I____ L   |
        |           | |    |    |
     12 |           | |____|    |
        |  P______ O| J     K   |
     10 |   |     | |           |
        |  M|_____| N           |
      8 |           |           |
        |  D________|______ C   |
      6 |   |       |      |    |
        |   |       |      |    |
      4 |   |      E|______|____|F
        |   |              |
      2 |  A|______________|B
        |   
        |_____________________________________
       0    2    4    6    8    10    11    12




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Related Must-Read Topics:

  1. 1d Range Search
  2. Closest Pair
  3. Convex Hull
  4. 2D Intervals Union
    & Skyline Problem
  5. Overlapping 1D Intervals
  6. Merging Overlapping 1D Intervals
  7. Separating Overlapping 1D Intervals


The above content is written by:

Abhishek Dey

Abhishek Dey

A Visionary Software Engineer With A Mission To Empower Every Person & Every Organization On The Planet To Achieve More

Microsoft | University of Florida

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