Octree | Insertion and Searching

Octree is a tree data structure in which each internal node can have at most 8 children. Like Binary tree which divides the space into two segments, Octree divides the space into at most eight-part which is called as octanes. It is used to store the 3-D point which takes a large amount of space. If all the internal node of the Octree contains exactly 8 children, then it is called full Octree. It is also useful for high-resolution graphics like 3D computer graphics.

The Octree can be formed from 3D volume by doing the following steps: 

1. Divide the current 3D volume into eight boxes
2. If any box has more than one point then divide it further into boxes
3. Do not divide the box which has one or zero points in it
4. Do this process repeatedly until all the box contains one or zero point in it

The above steps are shown in figure. 

If S is the number of points in each dimension then the number of nodes that are formed in Octree is given by this formula . 

Insertion in Octree:

• To insert a node in Octree, first of all, we check if a node exists or not if a node exists then return otherwise we go recursively
• First, we start with the root node and mark it as current
• Then we find the child node in which we can store the point
• If the node is empty then it is replaced with the node we want to insert and make it a leaf node
• If the node is the leaf node then make it an internal node and if it is an internal node then go to the child node, This process is performed recursively until an empty node is not found
• The time complexity of this function is where, N is the number of nodes

Search in Octree:

• This function is used to search the point exist is the tree or not
• Start with the root node and search recursively if the node with given point found then return true, if an empty node or boundary point or empty point is encountered then return false
• If an internal node is found go that node. The time complexity of this function is also O(Log N) where, N is the number of nodes

Below is the implementation of the above approach

CPP14

// Implementation of Octree in c++
#include <iostream>
#include <vector>
using namespace std;

#define TopLeftFront 0
#define TopRightFront 1
#define BottomRightFront 2
#define BottomLeftFront 3
#define TopLeftBottom 4
#define TopRightBottom 5
#define BottomRightBack 6
#define BottomLeftBack 7

// Structure of a point
struct Point {
    int x;
    int y;
    int z;
    Point()
        : x(-1), y(-1), z(-1)
    {
    }

    Point(int a, int b, int c)
        : x(a), y(b), z(c)
    {
    }
};

// Octree class
class Octree {

    // if point == NULL, node is internal node.
    // if point == (-1, -1, -1), node is empty.
    Point* point;

    // Represent the boundary of the cube
    Point *topLeftFront, *bottomRightBack;
    vector<Octree*> children;

public:
    // Constructor
    Octree()
    {
        // To declare empty node
        point = new Point();
    }

    // Constructor with three arguments
    Octree(int x, int y, int z)
    {
        // To declare point node
        point = new Point(x, y, z);
    }

    // Constructor with six arguments
    Octree(int x1, int y1, int z1, int x2, int y2, int z2)
    {
        // This use to construct Octree
        // with boundaries defined
        if (x2 < x1
            || y2 < y1
            || z2 < z1) {
            cout << "boundary points are not valid" << endl;
            return;
        }

        point = nullptr;
        topLeftFront
            = new Point(x1, y1, z1);
        bottomRightBack
            = new Point(x2, y2, z2);

        // Assigning null to the children
        children.assign(8, nullptr);
        for (int i = TopLeftFront;
             i <= BottomLeftBack;
             ++i)
            children[i] = new Octree();
    }

    // Function to insert a point in the octree
    void insert(int x,
                int y,
                int z)
    {

        // If the point already exists in the octree
        if (find(x, y, z)) {
            cout << "Point already exist in the tree" << endl;
            return;
        }

        // If the point is out of bounds
        if (x < topLeftFront->x
            || x > bottomRightBack->x
            || y < topLeftFront->y
            || y > bottomRightBack->y
            || z < topLeftFront->z
            || z > bottomRightBack->z) {
            cout << "Point is out of bound" << endl;
            return;
        }

        // Binary search to insert the point
        int midx = (topLeftFront->x
                    + bottomRightBack->x)
                   / 2;
        int midy = (topLeftFront->y
                    + bottomRightBack->y)
                   / 2;
        int midz = (topLeftFront->z
                    + bottomRightBack->z)
                   / 2;

        int pos = -1;

        // Checking the octant of
        // the point
        if (x <= midx) {
            if (y <= midy) {
                if (z <= midz)
                    pos = TopLeftFront;
                else
                    pos = TopLeftBottom;
            }
            else {
                if (z <= midz)
                    pos = BottomLeftFront;
                else
                    pos = BottomLeftBack;
            }
        }
        else {
            if (y <= midy) {
                if (z <= midz)
                    pos = TopRightFront;
                else
                    pos = TopRightBottom;
            }
            else {
                if (z <= midz)
                    pos = BottomRightFront;
                else
                    pos = BottomRightBack;
            }
        }

        // If an internal node is encountered
        if (children[pos]->point == nullptr) {
            children[pos]->insert(x, y, z);
            return;
        }

        // If an empty node is encountered
        else if (children[pos]->point->x == -1) {
            delete children[pos];
            children[pos] = new Octree(x, y, z);
            return;
        }
        else {
            int x_ = children[pos]->point->x,
                y_ = children[pos]->point->y,
                z_ = children[pos]->point->z;
            delete children[pos];
            children[pos] = nullptr;
            if (pos == TopLeftFront) {
                children[pos] = new Octree(topLeftFront->x,
                                           topLeftFront->y,
                                           topLeftFront->z,
                                           midx,
                                           midy,
                                           midz);
            }

            else if (pos == TopRightFront) {
                children[pos] = new Octree(midx + 1,
                                           topLeftFront->y,
                                           topLeftFront->z,
                                           bottomRightBack->x,
                                           midy,
                                           midz);
            }
            else if (pos == BottomRightFront) {
                children[pos] = new Octree(midx + 1,
                                           midy + 1,
                                           topLeftFront->z,
                                           bottomRightBack->x,
                                           bottomRightBack->y,
                                           midz);
            }
            else if (pos == BottomLeftFront) {
                children[pos] = new Octree(topLeftFront->x,
                                           midy + 1,
                                           topLeftFront->z,
                                           midx,
                                           bottomRightBack->y,
                                           midz);
            }
            else if (pos == TopLeftBottom) {
                children[pos] = new Octree(topLeftFront->x,
                                           topLeftFront->y,
                                           midz + 1,
                                           midx,
                                           midy,
                                           bottomRightBack->z);
            }
            else if (pos == TopRightBottom) {
                children[pos] = new Octree(midx + 1,
                                           topLeftFront->y,
                                           midz + 1,
                                           bottomRightBack->x,
                                           midy,
                                           bottomRightBack->z);
            }
            else if (pos == BottomRightBack) {
                children[pos] = new Octree(midx + 1,
                                           midy + 1,
                                           midz + 1,
                                           bottomRightBack->x,
                                           bottomRightBack->y,
                                           bottomRightBack->z);
            }
            else if (pos == BottomLeftBack) {
                children[pos] = new Octree(topLeftFront->x,
                                           midy + 1,
                                           midz + 1,
                                           midx,
                                           bottomRightBack->y,
                                           bottomRightBack->z);
            }
            children[pos]->insert(x_, y_, z_);
            children[pos]->insert(x, y, z);
        }
    }

    // Function that returns true if the point
    // (x, y, z) exists in the octree
    bool find(int x, int y, int z)
    {
        // If point is out of bound
        if (x < topLeftFront->x
            || x > bottomRightBack->x
            || y < topLeftFront->y
            || y > bottomRightBack->y
            || z < topLeftFront->z
            || z > bottomRightBack->z)
            return 0;

        // Otherwise perform binary search
        // for each ordinate
        int midx = (topLeftFront->x
                    + bottomRightBack->x)
                   / 2;
        int midy = (topLeftFront->y
                    + bottomRightBack->y)
                   / 2;
        int midz = (topLeftFront->z
                    + bottomRightBack->z)
                   / 2;

        int pos = -1;

        // Deciding the position
        // where to move
        if (x <= midx) {
            if (y <= midy) {
                if (z <= midz)
                    pos = TopLeftFront;
                else
                    pos = TopLeftBottom;
            }
            else {
                if (z <= midz)
                    pos = BottomLeftFront;
                else
                    pos = BottomLeftBack;
            }
        }
        else {
            if (y <= midy) {
                if (z <= midz)
                    pos = TopRightFront;
                else
                    pos = TopRightBottom;
            }
            else {
                if (z <= midz)
                    pos = BottomRightFront;
                else
                    pos = BottomRightBack;
            }
        }

        // If an internal node is encountered
        if (children[pos]->point == nullptr) {
            return children[pos]->find(x, y, z);
        }

        // If an empty node is encountered
        else if (children[pos]->point->x == -1) {
            return 0;
        }
        else {

            // If node is found with
            // the given value
            if (x == children[pos]->point->x
                && y == children[pos]->point->y
                && z == children[pos]->point->z)
                return 1;
        }
        return 0;
    }
};

// Driver code
int main()
{
    Octree tree(1, 1, 1, 5, 5, 5);

    tree.insert(1, 2, 3);
    tree.insert(1, 2, 3);
    tree.insert(6, 5, 5);

    cout << (tree.find(1, 2, 3)
                 ? "Found\n"
                 : "Not Found\n");

    cout << (tree.find(3, 4, 4)
                 ? "Found\n"
                 : "Not Found\n");
    tree.insert(3, 4, 4);

    cout << (tree.find(3, 4, 4)
                 ? "Found\n"
                 : "Not Found\n");

    return 0;
}

Output:

Point already exist in the tree
Point is out of bound
found
not found
found

Applications:

1. It is used in 3D computer graphics games
2. It is also used to find nearest neighboring objects in 3D space
3. It is also used for color quantization

Further References:
https://en.wikipedia.org/wiki/Octree