Intersection of N rectangles

Question

I\'m looking for an algorithm to solve this problem:

Given N rectangles on the Cartesian coordinate, find out if the intersection of those rectangles is empty or no

Answer 1

Abstract

Either use a sorting algorithm according to smallest X value of the rectangle, or store your rectangles in an R-tree and search it.

Straight-forward approach (with sorting)

Let us denote low_x() - the smallest (leftmost) X value of a rectangle, and high_x() - the highest (rightmost) X value of a rectangle.

Algorithm:

Sort the rectangles according to low_x().                   # O(n log n)

For each rectangle in sorted array:                         # O(n)
    Finds its highest X point.                              # O(1)
    Compare it with all rectangles whose low_x() is smaller # O(log n)
        than this.high(x)

Complexity analysis

This should work on O(n log n) on uniformly distributed rectangles.

The worst case would be O(n^2), for example when the rectangles don't overlap but are one above another. In this case, generalize the algorithm to have low_y() and high_y() too.

Data-structure approach: R-Trees

R-trees (a spatial generalization of B-trees) are one of the best ways to store geospatial data, and can be useful in this problem. Simply store your rectangles in an R-tree, and you can spot intersections with a straightforward O(n log n) complexity. (n searches, log n time for each).