What is K dimension tree?
A K-D Tree(also called as K-Dimensional Tree) is a binary search tree where data in each node is a K-Dimensional point in space. In short, it is a space partitioning(details below) data structure for organizing points in a K-Dimensional space. A non-leaf node in K-D tree divides the space into two parts, called as half-spaces. Points to the left of this space are represented by the left subtree of that node and points to the right of the space are represented by the right subtree. We will soon be explaining the concept on how the space is divided and tree is formed. For the sake of simplicity, let us understand a 2-D Tree with an example. The root would have an x-aligned plane, the root’s children would both have y-aligned planes, the root’s grandchildren would all have x-aligned planes, and the root’s great-grandchildren would all have y-aligned planes and so on.
Generalization: Let us number the planes as 0, 1, 2, …(K – 1). From the above example, it is quite clear that a point (node) at depth D will have A aligned plane where A is calculated as: A = D mod K
How to determine if a point will lie in the left subtree or in right subtree? If the root node is aligned in planeA, then the left subtree will contain all points whose coordinates in that plane are smaller than that of root node. Similarly, the right subtree will contain all points whose coordinates in that plane are greater-equal to that of root node.
Creation of a 2-D Tree: Consider following points in a 2-D plane: (3, 6), (17, 15), (13, 15), (6, 12), (9, 1), (2, 7), (10, 19)
- Insert (3, 6): Since tree is empty, make it the root node.
- Insert (17, 15): Compare it with root node point. Since root node is X-aligned, the X-coordinate value will be compared to determine if it lies in the right subtree or in the left subtree. This point will be Y-aligned.
- Insert (13, 15): X-value of this point is greater than X-value of point in root node. So, this will lie in the right subtree of (3, 6). Again Compare Y-value of this point with the Y-value of point (17, 15) (Why?). Since, they are equal, this point will lie in the right subtree of (17, 15). This point will be X-aligned.
- Insert (6, 12): X-value of this point is greater than X-value of point in root node. So, this will lie in the right subtree of (3, 6). Again Compare Y-value of this point with the Y-value of point (17, 15) (Why?). Since, 12 < 15, this point will lie in the left subtree of (17, 15). This point will be X-aligned.
- Insert (9, 1):Similarly, this point will lie in the right of (6, 12).
- Insert (2, 7):Similarly, this point will lie in the left of (3, 6).
- Insert (10, 19): Similarly, this point will lie in the left of (13, 15).
How is space partitioned?
All 7 points will be plotted in the X-Y plane as follows:
- Point (3, 6) will divide the space into two parts: Draw line X = 3.
- Point (2, 7) will divide the space to the left of line X = 3 into two parts horizontally. Draw line Y = 7 to the left of line X = 3.
- Point (17, 15) will divide the space to the right of line X = 3 into two parts horizontally. Draw line Y = 15 to the right of line X = 3.
- Point (6, 12) will divide the space below line Y = 15 and to the right of line X = 3 into two parts. Draw line X = 6 to the right of line X = 3 and below line Y = 15.
- Point (13, 15) will divide the space below line Y = 15 and to the right of line X = 6 into two parts. Draw line X = 13 to the right of line X = 6 and below line Y = 15.
- Point (9, 1) will divide the space between lines X = 3, X = 6 and Y = 15 into two parts. Draw line Y = 1 between lines X = 3 and X = 13.
- Point (10, 19) will divide the space to the right of line X = 3 and above line Y = 15 into two parts. Draw line Y = 19 to the right of line X = 3 and above line Y = 15.
Following is C++ implementation of KD Tree basic operations like search, insert and delete.
C++
// A C++ program to demonstrate operations of KD tree#include<bits/stdc++.h>using namespace std;const int k = 2;// A structure to represent node of kd treestruct Node{ int point[k]; // To store k dimensional point Node *left, *right;};// A method to create a node of K D treestruct Node* newNode(int arr[]){ struct Node* temp = new Node; for (int i=0; i<k; i++) temp->point[i] = arr[i]; temp->left = temp->right = NULL; return temp;}// Inserts a new node and returns root of modified tree// The parameter depth is used to decide axis of comparisonNode *insertRec(Node *root, int point[], unsigned depth){ // Tree is empty? if (root == NULL) return newNode(point); // Calculate current dimension (cd) of comparison unsigned cd = depth % k; // Compare the new point with root on current dimension 'cd' // and decide the left or right subtree if (point[cd] < (root->point[cd])) root->left = insertRec(root->left, point, depth + 1); else root->right = insertRec(root->right, point, depth + 1); return root;}// Function to insert a new point with given point in// KD Tree and return new root. It mainly uses above recursive// function "insertRec()"Node* insert(Node *root, int point[]){ return insertRec(root, point, 0);}// A utility method to determine if two Points are same// in K Dimensional spacebool arePointsSame(int point1[], int point2[]){ // Compare individual pointinate values for (int i = 0; i < k; ++i) if (point1[i] != point2[i]) return false; return true;}// Searches a Point represented by "point[]" in the K D tree.// The parameter depth is used to determine current axis.bool searchRec(Node* root, int point[], unsigned depth){ // Base cases if (root == NULL) return false; if (arePointsSame(root->point, point)) return true; // Current dimension is computed using current depth and total // dimensions (k) unsigned cd = depth % k; // Compare point with root with respect to cd (Current dimension) if (point[cd] < root->point[cd]) return searchRec(root->left, point, depth + 1); return searchRec(root->right, point, depth + 1);}// Searches a Point in the K D tree. It mainly uses// searchRec()bool search(Node* root, int point[]){ // Pass current depth as 0 return searchRec(root, point, 0);}// Driver program to test above functionsint main(){ struct Node *root = NULL; int points[][k] = {{3, 6}, {17, 15}, {13, 15}, {6, 12}, {9, 1}, {2, 7}, {10, 19}}; int n = sizeof(points)/sizeof(points[0]); for (int i=0; i<n; i++) root = insert(root, points[i]); int point1[] = {10, 19}; (search(root, point1))? cout << "Found\n": cout << "Not Found\n"; int point2[] = {12, 19}; (search(root, point2))? cout << "Found\n": cout << "Not Found\n"; return 0;} |
Java
import java.util.*;class Node { int[] point; Node left, right; public Node(int[] arr) { this.point = arr; this.left = this.right = null; }}class KDtree { int k = 2; public Node newNode(int[] arr) { return new Node(arr); } public Node insertRec(Node root, int[] point, int depth) { if (root == null) { return newNode(point); } int cd = depth % k; if (point[cd] < root.point[cd]) { root.left = insertRec(root.left, point, depth + 1); } else { root.right = insertRec(root.right, point, depth + 1); } return root; } public Node insert(Node root, int[] point) { return insertRec(root, point, 0); } public boolean arePointsSame(int[] point1, int[] point2) { for (int i = 0; i < k; ++i) { if (point1[i] != point2[i]) { return false; } } return true; } public boolean searchRec(Node root, int[] point, int depth) { if (root == null) { return false; } if (arePointsSame(root.point, point)) { return true; } int cd = depth % k; if (point[cd] < root.point[cd]) { return searchRec(root.left, point, depth + 1); } return searchRec(root.right, point, depth + 1); } public boolean search(Node root, int[] point) { return searchRec(root, point, 0); } public static void main(String[] args) { KDtree kdTree = new KDtree(); Node root = null; int[][] points = { { 3, 6 }, { 17, 15 }, { 13, 15 }, { 6, 12 }, { 9, 1 }, { 2, 7 }, { 10, 19 } }; int n = points.length; for (int i = 0; i < n; i++) { root = kdTree.insert(root, points[i]); } int[] point1 = { 10, 19 }; System.out.println(kdTree.search(root, point1) ? "Found" : "Not Found"); int[] point2 = { 12, 19 }; System.out.println(kdTree.search(root, point2) ? "Found" : "Not Found"); }} |
Python3
# A Python program to demonstrate operations of KD treeimport sys# Number of dimensionsk = 2# A structure to represent node of kd treeclass Node: def __init__(self, point): self.point = point self.left = None self.right = None# A method to create a node of K D treedef newNode(point): return Node(point)# Inserts a new node and returns root of modified tree# The parameter depth is used to decide axis of comparisondef insertRec(root, point, depth): # Tree is empty? if not root: return newNode(point) # Calculate current dimension (cd) of comparison cd = depth % k # Compare the new point with root on current dimension 'cd' # and decide the left or right subtree if point[cd] < root.point[cd]: root.left = insertRec(root.left, point, depth + 1) else: root.right = insertRec(root.right, point, depth + 1) return root# Function to insert a new point with given point in# KD Tree and return new root. It mainly uses above recursive# function "insertRec()"def insert(root, point): return insertRec(root, point, 0)# A utility method to determine if two Points are same# in K Dimensional spacedef arePointsSame(point1, point2): # Compare individual coordinate values for i in range(k): if point1[i] != point2[i]: return False return True# Searches a Point represented by "point[]" in the K D tree.# The parameter depth is used to determine current axis.def searchRec(root, point, depth): # Base cases if not root: return False if arePointsSame(root.point, point): return True # Current dimension is computed using current depth and total # dimensions (k) cd = depth % k # Compare point with root with respect to cd (Current dimension) if point[cd] < root.point[cd]: return searchRec(root.left, point, depth + 1) return searchRec(root.right, point, depth + 1)# Searches a Point in the K D tree. It mainly uses# searchRec()def search(root, point): # Pass current depth as 0 return searchRec(root, point, 0)# Driver program to test above functionsif __name__ == '__main__': root = None points = [[3, 6], [17, 15], [13, 15], [6, 12], [9, 1], [2, 7], [10, 19]] n = len(points) for i in range(n): root = insert(root, points[i]) point1 = [10, 19] if search(root, point1): print("Found") else: print("Not Found") point2 = [12, 19] if search(root, point2): print("Found") else: print("Not Found") # This code is contributed by Prajwal Kandekar |
C#
// C# implementation for the above approach// Importing the System namespace which contains classes used in the programusing System;// Defining the Node class with an integer array to hold the point and references to left and right nodesclass Node { public int[] point; public Node left; public Node right; // Defining a constructor to initialize the Node object public Node(int[] point) { this.point = point; left = null; right = null; }}// Defining the KdTree class to represent a K-d tree with a specified value of kclass KdTree { private int k; // Defining a constructor to initialize the KdTree object public KdTree(int k) { this.k = k; } // Defining a method to create a new Node object public Node newNode(int[] point) { return new Node(point); } // Defining a recursive method to insert a point into the K-d tree private Node insertRec(Node root, int[] point, int depth) { if (root == null) { return newNode(point); } int cd = depth % k; if (point[cd] < root.point[cd]) { root.left = insertRec(root.left, point, depth + 1); } else { root.right = insertRec(root.right, point, depth + 1); } return root; } // Defining a method to insert a point into the K-d tree public Node insert(Node root, int[] point) { return insertRec(root, point, 0); } // Defining a method to check if two points are the same private bool arePointsSame(int[] point1, int[] point2) { for (int i = 0; i < k; i++) { if (point1[i] != point2[i]) { return false; } } return true; } // Defining a recursive method to search for a point in the K-d tree private bool searchRec(Node root, int[] point, int depth) { if (root == null) { return false; } if (arePointsSame(root.point, point)) { return true; } int cd = depth % k; if (point[cd] < root.point[cd]) { return searchRec(root.left, point, depth + 1); } return searchRec(root.right, point, depth + 1); } // Defining a method to search for a point in the K-d tree public bool search(Node root, int[] point) { return searchRec(root, point, 0); }}// Program class to demonstrate the KdTree data structure and its functionalityclass Program { static void Main(string[] args) { // Initialize root node as null Node root = null; // Define an array of points to be inserted into the tree int[][] points = { new int[] { 3, 6 }, new int[] { 17, 15 }, new int[] { 13, 15 }, new int[] { 6, 12 }, new int[] { 9, 1 }, new int[] { 2, 7 }, new int[] { 10, 19 } }; // Get the length of the points array int n = points.Length; // Create a new KdTree object with k=2 KdTree tree = new KdTree(2); // Insert all the points into the tree and update the root node for (int i = 0; i < n; i++) { root = tree.insert(root, points[i]); } // Define a point to search for in the tree int[] point1 = { 10, 19 }; // Search for the point in the tree and print the result if (tree.search(root, point1)) { Console.WriteLine("Found"); } else { Console.WriteLine("Not Found"); } // Define another point to search for in the tree int[] point2 = { 12, 19 }; // Search for the point in the tree and print the result if (tree.search(root, point2)) { Console.WriteLine("Found"); } else { Console.WriteLine("Not Found"); } }}// This code is contributed by Amit Mangal. |
Javascript
// JavaScript code for the above approach// Number of dimensionsconst k = 2;// A structure to represent node of kd treeclass Node { constructor(point) { this.point = point; this.left = null; this.right = null; }}// A method to create a node of K D treefunction newNode(point) { return new Node(point);}// Inserts a new node and returns root of modified tree// The parameter depth is used to decide axis of comparisonfunction insertRec(root, point, depth) { // Tree is empty? if (!root) { return newNode(point); } // Calculate current dimension (cd) of comparison const cd = depth % k; // Compare the new point with root on current dimension 'cd' // and decide the left or right subtree if (point[cd] < root.point[cd]) { root.left = insertRec(root.left, point, depth + 1); } else { root.right = insertRec(root.right, point, depth + 1); } return root;}// Function to insert a new point with given point in// KD Tree and return new root. It mainly uses above recursive// function "insertRec()"function insert(root, point) { return insertRec(root, point, 0);}// A utility method to determine if two Points are same// in K Dimensional spacefunction arePointsSame(point1, point2) { // Compare individual coordinate values for (let i = 0; i < k; i++) { if (point1[i] !== point2[i]) { return false; } } return true;}// Searches a Point represented by "point[]" in the K D tree.// The parameter depth is used to determine current axis.function searchRec(root, point, depth) { // Base cases if (!root) { return false; } if (arePointsSame(root.point, point)) { return true; } // Current dimension is computed using current depth and total // dimensions (k) const cd = depth % k; // Compare point with root with respect to cd (Current dimension) if (point[cd] < root.point[cd]) { return searchRec(root.left, point, depth + 1); } return searchRec(root.right, point, depth + 1);}// Searches a Point in the K D tree. It mainly uses// searchRec()function search(root, point) {// Pass current depth as 0 return searchRec(root, point, 0);}// Driver program to test above functionslet root = null;const points = [ [3, 6], [17, 15], [13, 15], [6, 12], [9, 1], [2, 7], [10, 19],];const n = points.length;for (let i = 0; i < n; i++) { root = insert(root, points[i]);}const point1 = [10, 19];if (search(root, point1)) { console.log("Found");} else { console.log("Not Found");}const point2 = [12, 19];if (search(root, point2)) { console.log("Found");} else { console.log("Not Found");}// This code is contributed by Amit Mangal |
Output:
Found Not Found
Time Complexity: O(n)
Auxiliary Space: O(n)
Refer below articles for find minimum and delete operations.
Advantages of K Dimension tree –
K-d trees have several advantages as a data structure:
- Efficient search: K-d trees are effective in searching for points in a k-dimensional space, such as in nearest neighbor search or range search.
- Dimensionality reduction: K-d trees can be used to reduce the dimensionality of the problem, allowing for faster search times and reducing the memory requirements of the data structure.
- Versatility: K-d trees can be used for a wide range of applications, such as in data mining, computer graphics, and scientific computing.
- Balance: K-d trees are self-balancing, which ensures that the tree remains efficient even when data is inserted or removed.
- Incremental construction: K-d trees can be incrementally constructed, which means that data can be added or removed from the structure without having to rebuild the entire tree.
- Easy to implement: K-d trees are relatively easy to implement and can be implemented in a variety of programming languages.
This article is compiled by Aashish Barnwal. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
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