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5. If k i = k, return false (indicating that the insertion was unsuccessful because a record with key k already exists, and keys should be unique). 6. Let x be the root of subtree S i . 7. Add the record to disk. 8. Insert k (with the record s disk address) into x between k i 1 and k i . 9. Add a dummy leaf node to x. 10. If degree(x) = m, repeat steps 11 13 until degree(x) < m. 11. Let k j be the middle key in node x. 12. Let u and v be the left and right halves of x after removing k j from x. 13. If x is the root, create a new root node containing k j with subtrees u and v. 14. Otherwise, insert k j in x s parent node and attach subtrees u and v. 15. Return true. This insertion process is illustrated in Figure 12.6.
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Figure 12.6 Inserting into a B-tree
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The deletion algorithm for B-trees is similar to the insertion algorithm. All three algorithms run in time proportional to the height of the tree. From Corollary 10.1 on page 188 it follows that that height is proportional to log m n. From Theorem A.2 on page 320, it follows that that is proportional to lgn. Thus we have: Theorem 12.1 In a B-tree, searching, inserting, and deleting all run in O(lgn) time. BINARY SEARCH TREES A binary search tree is a binary tree whose elements include a key field of some ordinal type and which has this property: If k is the key value at any node, then k x for every key x in the node s left subtree and k y for
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Figure 12.7 A binary search tree
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every key y in the node s right subtree. This property, called the BST property, guarantees that an inorder traversal of the binary search tree will produce the elements in increasing order. The BST property is applied for each insertion into the tree: Algorithm 12.3 Inserting into a binary search Tree To insert an element with key value k into a binary search tree: 1. If the tree is empty, insert the new element at the root. Then return. 2. Let p locate the root. 3. If k is less than the key stored at p and if the node at p has no left child, insert the new element as the left child of p. Then return. 4. If k is less than the key stored at p and if the node at p has a left child, let p locate that left child of p. Then go back to step 3. 5. If the node at p has no right child, insert the new element as the right child of p. Then return. 6. Let p locate the right child of p. Then go back to step 3. EXAMPLE 12.4 Inserting into a Binary Search Tree
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Apply Algorithm 12.3 to insert an element with key M into the binary search tree shown in Figure 12.7. Step 1 starts the iterator p at the root K. Since M is greater than K (i.e., it follows it lexicographically) and node K has a right child, the algorithm proceeds to step 6, resetting the iterator p to node P, and then goes back to step 3. Next, since M is less than P (i.e., it precedes it lexicographically) and node P has a left child, the algorithm proceeds to step 4, resetting the iterator p to node N, and then goes back to step 3. Next, since M is also less than N but node N has no left child, the Figure 12.8 A binary search tree algorithm proceeds to step 5, inserts the new element as the left child of node N, and then returns. This is illustrated in Figure 12.8.
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