Organization of B-Tree

Organization of B-Tree:

A B-tree is a balanced tree data structure where each node have contains multiple keys and multiple child pointers. The organization of the B-tree is designed to optimize the search, insertion and deletion operation while maintaining balance....

Implementation of B-Tree:

Java // Java Program for Implementaion B-Tree class BTreeNode { // Variables Declared int[] keys; // Array to store keys int t; // Minimum degree (defines the range for number of keys) BTreeNode[] children; // Array to store child pointers int n; // Current number of keys boolean leaf; // True when node is leaf, else False public BTreeNode(int t, boolean leaf) { this.t = t; this.leaf = leaf; keys = new int[2 * t - 1]; children = new BTreeNode[2 * t]; n = 0; } // Function to search given key in subtree // rooted with this node public BTreeNode search(int key) { int i = 0; while (i < n && key > keys[i]) i++; if (keys[i] == key) return this; if (leaf) return null; return children[i].search(key); } // Function to insert a new key // in subtree rooted with this node public void insertNonFull(int key) { int i = n - 1; if (leaf) { while (i >= 0 && key < keys[i]) { keys[i + 1] = keys[i]; i--; } keys[i + 1] = key; n++; } else { while (i >= 0 && key < keys[i]) i--; i++; if (children[i].n == 2 * t - 1) { splitChild(i, children[i]); if (key > keys[i]) i++; } children[i].insertNonFull(key); } } // Function to split the child node public void splitChild(int i, BTreeNode y) { BTreeNode z = new BTreeNode(y.t, y.leaf); z.n = t - 1; for (int j = 0; j < t - 1; j++) z.keys[j] = y.keys[j + t]; if (!y.leaf) { for (int j = 0; j < t; j++) z.children[j] = y.children[j + t]; } y.n = t - 1; for (int j = n; j >= i + 1; j--) children[j + 1] = children[j]; children[i + 1] = z; for (int j = n - 1; j >= i; j--) keys[j + 1] = keys[j]; keys[i] = y.keys[t - 1]; n++; } // Function to print all keys in the // subtree rooted with this node public void printInOrder() { int i; for (i = 0; i < n; i++) { if (!leaf) children[i].printInOrder(); System.out.print(keys[i] + " "); } if (!leaf) children[i].printInOrder(); } } public class BTree { // Pointer to root node private BTreeNode root; // Minimum degree private int t; public BTree(int t) { this.t = t; root = null; } // Function to search a key in this tree public BTreeNode search(int key) { return (root == null) ? null : root.search(key); } // Function to insert a key into the B-tree public void insert(int key) { if (root == null) { root = new BTreeNode(t, true); root.keys[0] = key; root.n = 1; } else { if (root.n == 2 * t - 1) { BTreeNode newRoot = new BTreeNode(t, false); newRoot.children[0] = root; newRoot.splitChild(0, root); int i = 0; if (newRoot.keys[0] < key) i++; newRoot.children[i].insertNonFull(key); root = newRoot; } else { root.insertNonFull(key); } } } // Function to print the entire B-tree public void printBTree() { if (root != null) root.printInOrder(); System.out.println(); } public static void main(String[] args) { // Create a B-tree with minimum degree 3 BTree bTree = new BTree(3); bTree.insert(10); bTree.insert(20); bTree.insert(5); bTree.insert(6); bTree.insert(12); bTree.insert(30); System.out.print("B-tree : "); bTree.printBTree(); int searchKey = 6; BTreeNode foundNode = bTree.search(searchKey); if (foundNode != null) System.out.println("Key " + searchKey + " found in the B-tree."); else System.out.println("Key " + searchKey + " not found in the B-tree."); } }...

Complexity of the Above Method:

Operation Explanation Complexity Search Operation Searching for the key in a B-tree involves traversing the tree from the root node to the leaf node, following the child pointers based on a comparison of the keys. The time complexity of the searching operation is O(log n).The space complexity of the searching operation is O(1) for iterative, and O(log n) for recursive search.Insertion Operation Inserting a key into the B-tree involves finding an appropriate leaf node for the insertion and maintaining B-tree properties by splitting nodes if necessary. The time complexity of insertion operation is O(log n).The space complexity of the insertion operation is O(1og n).Deletion Operation Deleting a key from the B-tree involves the key, removing it from the appropriate leaf node and ensuring the B-tree remains balanced by merging or redistributing the nodes if necessary. The time complexity of the deletion operation is O(log n).The space complexity of the deletion operation is O(log n).Traversal Operation Traversing the B-tree involves visiting all the keys in the tree in a specific order like in-order, pre-order or post-order traversals. The time complexity of traversal operations is O(n).The space complexity of the traversal operation is O(t).Range Queries Range queries involve searching for the keys within the given range in the B-tree. The time complexity of range queries is O(k + log n).The space complexity of range queries is O(log n)....