如何在给定的矩形列表中找到与特定矩形相邻的所有矩形？

Question

Given a rectangle R1 and a list of of rectangles R2,R3,.... How can I find all rectangles that are connected with the main rectangle R1.

I don't just need the rectangle that are directly connected to R1 but also all that are indirectly connected to R1. For Example if R2 is connected to R1 and R3 is connected to R2. R3 is considered connected to R1.

Rectangles are given in the form (xmin, ymin, xmax, ymax). All rectangles are parallel to the axis. Rectangles are considered connected when they are either overlapping or touching. When they just touch in the corner they are not considered connected.

Example:

____________
_111________
_11122______
____22______
____22______
____333333__ 
____22______
__55___4444_
__55___4444_ 

In this example R1,R2,R3 are connected with each other. So I need to return R1,R2,R3.

R4, and R5 are not connected.

An obvious solution would be to compare each each rectangle with eachother O(n^2). But I think there should be faster solutions. I have tried to use Implement a sweep line algorithm with an Interval Tree. But it is to slow. I need a solution in O(n log n)

Answer 1

Make 2 lists, one containing X-coordinate objects and one containing Y-containing objects.

The object will consist of (the coordinate, the rectangle number and whether the coordinate corresponds to min or max).

Sort both lists O(nlogn) and traverse any one. On encountering a min-coordinate add the rectangle in a stack, on encountering a max-coordinate remove the rectangle. If multiple rectangles are added together in a stack, add them in a group, while removing a rectangle the group will correspond to all the overlapping rectangles. Store the information of such groups. Do the same procedure using the other list. The common rectangles in groups obtained from both X and Y lists will be the overlapping rectangles.

The solution is O(nlogn).