Quadtrees are trees used to efficiently store data of points on a two-dimensional space. In this tree, each node has at most four children.
We can construct a quadtree from a two-dimensional area using the following steps:
- Divide the current two dimensional space into four boxes.
- If a box contains one or more points in it, create a child object, storing in it the two dimensional space of the box
- If a box does not contain any points, do not create a child for it
- Recurse for each of the children.
Quadtrees are used in image compression, where each node contains the average colour of each of its children. The deeper you traverse in the tree, the more the detail of the image.
Quadtrees are also used in searching for nodes in a two-dimensional area. For instance, if you wanted to find the closest point to given coordinates, you can do it using quadtrees.
Insert Function
The insert functions is used to insert a node into an existing Quad Tree. This function first checks whether the given node is within the boundaries of the current quad. If it is not, then we immediately cease the insertion. If it is within the boundaries, we select the appropriate child to contain this node based on its location.
This function is O(Log N) where N is the size of distance.
Search Function
The search function is used to locate a node in the given quad. It can also be modified to return the closest node to the given point. This function is implemented by taking the given point, comparing with the boundaries of the child quads and recursing.
This function is O(Log N) where N is size of distance.
The program given below demonstrates storage of nodes in a quadtree.
// C++ Implementation of Quad Tree #include <iostream> #include <cmath> using namespace std; // Used to hold details of a point struct Point { int x; int y; Point(int _x, int _y) { x = _x; y = _y; } Point() { x = 0; y = 0; } }; // The objects that we want stored in the quadtree struct Node { Point pos; int data; Node(Point _pos, int _data) { pos = _pos; data = _data; } Node() { data = 0; } }; // The main quadtree class class Quad { // Hold details of the boundary of this node Point topLeft; Point botRight; // Contains details of node Node *n; // Children of this tree Quad *topLeftTree; Quad *topRightTree; Quad *botLeftTree; Quad *botRightTree; public: Quad() { topLeft = Point(0, 0); botRight = Point(0, 0); n = NULL; topLeftTree = NULL; topRightTree = NULL; botLeftTree = NULL; botRightTree = NULL; } Quad(Point topL, Point botR) { n = NULL; topLeftTree = NULL; topRightTree = NULL; botLeftTree = NULL; botRightTree = NULL; topLeft = topL; botRight = botR; } void insert(Node*); Node* search(Point); bool inBoundary(Point); }; // Insert a node into the quadtree void Quad::insert(Node *node) { if (node == NULL) return; // Current quad cannot contain it if (!inBoundary(node->pos)) return; // We are at a quad of unit area // We cannot subdivide this quad further if (abs(topLeft.x - botRight.x) <= 1 && abs(topLeft.y - botRight.y) <= 1) { if (n == NULL) n = node; return; } if ((topLeft.x + botRight.x) / 2 >= node->pos.x) { // Indicates topLeftTree if ((topLeft.y + botRight.y) / 2 >= node->pos.y) { if (topLeftTree == NULL) topLeftTree = new Quad( Point(topLeft.x, topLeft.y), Point((topLeft.x + botRight.x) / 2, (topLeft.y + botRight.y) / 2)); topLeftTree->insert(node); } // Indicates botLeftTree else { if (botLeftTree == NULL) botLeftTree = new Quad( Point(topLeft.x, (topLeft.y + botRight.y) / 2), Point((topLeft.x + botRight.x) / 2, botRight.y)); botLeftTree->insert(node); } } else { // Indicates topRightTree if ((topLeft.y + botRight.y) / 2 >= node->pos.y) { if (topRightTree == NULL) topRightTree = new Quad( Point((topLeft.x + botRight.x) / 2, topLeft.y), Point(botRight.x, (topLeft.y + botRight.y) / 2)); topRightTree->insert(node); } // Indicates botRightTree else { if (botRightTree == NULL) botRightTree = new Quad( Point((topLeft.x + botRight.x) / 2, (topLeft.y + botRight.y) / 2), Point(botRight.x, botRight.y)); botRightTree->insert(node); } } } // Find a node in a quadtree Node* Quad::search(Point p) { // Current quad cannot contain it if (!inBoundary(p)) return NULL; // We are at a quad of unit length // We cannot subdivide this quad further if (n != NULL) return n; if ((topLeft.x + botRight.x) / 2 >= p.x) { // Indicates topLeftTree if ((topLeft.y + botRight.y) / 2 >= p.y) { if (topLeftTree == NULL) return NULL; return topLeftTree->search(p); } // Indicates botLeftTree else { if (botLeftTree == NULL) return NULL; return botLeftTree->search(p); } } else { // Indicates topRightTree if ((topLeft.y + botRight.y) / 2 >= p.y) { if (topRightTree == NULL) return NULL; return topRightTree->search(p); } // Indicates botRightTree else { if (botRightTree == NULL) return NULL; return botRightTree->search(p); } } }; // Check if current quadtree contains the point bool Quad::inBoundary(Point p) { return (p.x >= topLeft.x && p.x <= botRight.x && p.y >= topLeft.y && p.y <= botRight.y); } // Driver program int main() { Quad center(Point(0, 0), Point(8, 8)); Node a(Point(1, 1), 1); Node b(Point(2, 5), 2); Node c(Point(7, 6), 3); center.insert(&a); center.insert(&b); center.insert(&c); cout << "Node a: " << center.search(Point(1, 1))->data << "\n"; cout << "Node b: " << center.search(Point(2, 5))->data << "\n"; cout << "Node c: " << center.search(Point(7, 6))->data << "\n"; cout << "Non-existing node: " << center.search(Point(5, 5)); return 0; } |
Output:
Node a: 1 Node b: 2 Node c: 3 Non-existing node: 0
Exercise:
Implement a Quad Tree which returns 4 closest nodes to a given point.
Further References:
http://jimkang.com/quadtreevis/
https://en.wikipedia.org/wiki/Quadtree
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