Foundation of Computing Systems

Foundation of Computing Systems Lecture 11 Trees: Part VI-C IT 60101: Lecture #11

… Continued IT 60101: Lecture #11

Deletion in a Red-Black Tree • Deletion of a node from a red-black tree follows two major steps • Delete the node using the usual deletion operation as in a binary search tree. • This deletion first finds the location of the node to be deleted and then it is replaced by its inorder successor. • If the inorder successor has its child, then the child is moved up to the place of the inorder successor. 2. The above deletion operation may disturb the properties of a red-black tree. To restore the red-black tree, we require a fix-up operation. IT 60101: Lecture #11

Deletion in a Red-Black Tree • Suppose the node that has to be deleted is x and its inorder successor is y. • The following situation and steps to be followed accordingly as discussed below. Case 1:y is NULL, that is, node x does not have any inorder successor. Case 2: y is RED and y has no children as internal nodes Case 3: y is BLACK and it has no children as an internal node Case 4:y is BLACK and it has an internal RED node IT 60101: Lecture #11

Deletion Operation: Case 1 • Case 1:y is NULL, that is, node x does not have any inorder successor. IT 60101: Lecture #11

Deletion Operation: Case 2 • Case 2: y is RED and y has no children as internal nodes IT 60101: Lecture #11

Deletion Operation: Case 3 • Case 3: y is BLACK and it has no children as an internal node IT 60101: Lecture #11

Deletion Operation: Case 4 • Case 4:y is BLACK and it has an internal RED node IT 60101: Lecture #11

Deletion: Fix-up Operation • Suppose, x is the node to be deleted. y is its inorder successor and z be its another child other than the leaf child. • The following three problems may arise while x is deleted from the red-black tree. • y is red and it becomes the new root. This violates the “root property”. • Both z and the parent of y (now after deletion of y the parent of y becomes the parent of z) are red. This violates the “internal property” and leads to a “double-red problem”. • y’s removal causes any path that contained one fewer black node than the path before the removal of y. This violates the “black-depth property”. IT 60101: Lecture #11

Deletion: Fix-up Operation • Assume that z is the left sub-tree with respect to the parent y. The following four cases can occur while fixing up the above-mentioned problem. • Case 1: z’s sibling w is RED • Case 2: z’s sibling w is BLACK and both the children of w are BLACK • Case 3:z’s sibling w is BLACK and w’s left child is RED and w’s right child is BLACK. • Case 4: z’s sibling w is BLACK and w’s RIGHT child is RED. IT 60101: Lecture #11

Fix-up Operation: Case 1 • Case 1: z’s sibling w is RED • Steps: • Change the colour of w to BLACK 2. Switch (flip) the colour of parent of z. 3. Perform a left rotation of w 4. Sibling of z becomes a new sibling • This may lead to any one of the following problems of fix-up Case 2, Case 3 and Case 4 (when the node w is black). • These three cases are distinguished by the colours of children of w. IT 60101: Lecture #11

Fix-up Operation: Case 1 IT 60101: Lecture #11

Fix-up Operation: Case 2 • Case 2: z’s sibling w is BLACK and both the children of w are BLACK • Steps: • Make the colour w of RED • Add an extra black to the parent of z (which was earlier either red or black). • Repeat the fix-up procedure but now changing the parent of z becomes a new z. IT 60101: Lecture #11

Fix-up Operation: Case 3 • Case 3:z’s sibling w is BLACK and w’s left child is RED and w’s right child is BLACK. • Steps: • Exchange the colours of w and its left child. • Perform a right rotation of w’s left child. This rotation results in a new sibling w of z which is a black node with a red child. IT 60101: Lecture #11

Fix-up Operation: Case 4 • Case 4: z’s sibling w is BLACK and w’s RIGHT child is RED. • Steps: • Exchange the colour of w with the colour of parent of z. • Perform a left-rotation of parent of z (or w) • Repeat the fix-up procedure until z reaches the root or z becomes BLACK. IT 60101: Lecture #11

For detail implementation of deletion operation on red-black binary trees see the book • Classic Data Structures • Chapter 7 • PHI, 2nd Edn., 17th Reprint IT 60101: Lecture #11

HB (AVL) Tree vs. RB Tree • Height • A red-black tree with n internal nodes has height at most 2log2(n+1) • An AVL tree with n internal nodes has height at most 1.44log2n (see Lemma 7.11). • Specialization: • An AVL tree is a kind of a red-black tree (in other words, all AVL trees satisfy the properties of a red-black tree but all red-black trees are not AVL trees). • Generalization: • A red-black tree is a kind of B-tree. This is why a red-black tree is also alternatively termed Symmetric B-tree. (see Section 7.5.9: How a red-black tree is a kind of B-tree is discussed in the next section). IT 60101: Lecture #11

Application of RB Tree • A red-black tree is used to improve the speed of searching in a B-tree • Restructuring with RB tree IT 60101: Lecture #11

Restructuring with RB Tree IT 60101: Lecture #11

Foundation of Computing Systems