Axis Aligned? Rectangles

Describe? Algorithm? takes? unsorted. Array? axis‐aligned? rectangles? Returns? rectangles?? Overlaps, huh? What if? There? Is such? A? pair.? ? Axis‐aligned? Means? That? All? Rectangle? Sides? is either? parallel? Or? Perpendicular? to? The? X‐? and? Y‐axis. ?? You? Can? Assume? each? Rectangle? Object? Has? Variables?in? It: the. x‐y? Coordinates? Upper‐left Corner and the bottom‐right? corner.

good? Answer: ? Create? A. Sorted? array? The? X? Coordinates? Left? and? Right? Edges?? The rectangles. ?? Then, the use? A? "Scanline"? Move? From? Left, right? through? The? Rectangles. ?? Keep? A? Binary? Search? The tree? containing? Y. Coordinates? Top? and? Bottom? Edges? Rectangles? That? Overlap Scanline. ?? For? Element? The?? Array,? Check? Whether? Is it? A? Left? or? Edge.? ? If? Is it? A? right? Edge,? remove? The? Corresponding? Top? and? Bottom? Edges? Bst.?? What If? It is A?left edge,? Search? The? Bst? For? rectangles? That? Overlap? Current? Rectangle, what if? There? Is One,?return? Overlap.?? Then,? Add? the?y? Coordinates?? The? Top? and? Bottom?edges? The. Rectangle? To the? Bst.?? The? Search? Takes? O (n?log?n)? time,? Since? It? takes? O (n?log?n)? To? Sort? The? Rectangles? and? each? The?? 2n? Iteration S?takes? O (log?n)? Time.

C + + Sample Code (Find all pairs of rectangles-overlap):

1#include <iostream>2#include <vector>3#include <map>4#include <algorithm>5 using namespacestd;6 7 structRect {8     int_x1, _x2, _y1, _y2;9RectintX1,intX2,intY1,inty2):Ten _x1 (x1), _x2 (x2), _y1 (y1), _y2 (y2) {} One }; A  - structXedge { -     int_x; the     int_is_left; -     int_i_rect; -Xedge (intXintIs_left,inti_rect): - _x (x), _is_left (Is_left), _i_rect (I_rect) {} + }; -  + structYedge { A     int_y; at     int_is_top; -     int_i_rect; -Yedge (intYintIs_top,inti_rect): - _y (y), _is_top (Is_top), _i_rect (I_rect) {} - }; -  in intMain () { -Vector<rect> RCTs;//vector that stores rectangles toVector<xedge> xes;//Left and right edges of the rectangles +     intx1, x2, y1, y2; -     inti =0; the      while(CIN >> x1 >> x2 >> y1 >>y2) { * rcts.push_back (Rect (x1, x2, y1, y2)); $Xes.push_back (Xedge (x1,1, i));Panax NotoginsengXes.push_back (Xedge (x2,0, i)); -i++; the     } +  A     //Sort left and right rectangles according to their x-coordinates the sort (Xes.begin (), Xes.end (), +[](Constxedge& E1,Constxedge&E2) { -             if(e1._x! = e2._x)returnE1._x <e2._x; $             Else if(E1._is_left! =e2._is_left) $                 returnE1._is_left >E2._is_left; -             Else returnE1._i_rect <E2._i_rect; -         }); the  -     //A binary search tree for top and bottom edges that overlap the "scanline"Wuyimultimap<int, yedge>Edge_map; the      for(auto& xe:xes) {//XE: "Scanline" -         if(!xe._is_left) { WuAuto Range =Edge_map.equal_range (rcts[xe._i_rect]._y1); -              for(Auto iter = Range.first; iter! = Range.second; + +ITER) { About                 if(Iter->second._i_rect = =xe._i_rect) { $ Edge_map.erase (ITER); -                      Break; -                 } -             } ARange =Edge_map.equal_range (rcts[xe._i_rect]._y2); +              for(Auto iter = Range.first; iter! = Range.second; + +ITER) { the                 if(Iter->second._i_rect = =xe._i_rect) { - Edge_map.erase (ITER); $                      Break; the                 } the             } the}Else { the             intY1 =rcts[xe._i_rect]._y1; -             inty2 =Rcts[xe._i_rect]._y2; inAuto Iter1 =edge_map.lower_bound (y1); theAuto Iter2 =edge_map.upper_bound (y2); the              for(Auto iter = iter1; iter! = Iter2; + +ITER) { Aboutcout << Xe._i_rect <<" "<< Iter->second._i_rect <<Endl; the             } theEdge_map.insert (Make_pair (Y1, Yedge (Y1,0, Xe._i_rect))); theEdge_map.insert (Make_pair (y2, Yedge (y2,1, Xe._i_rect))); +         } -     } the Bayi     return 0; the}

Axis Aligned? Rectangles