Foundation of Computing Systems

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Foundation of Computing Systems

• Lecture 11
• Trees Part VI-C

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Continued
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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.

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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

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Deletion Operation Case 1

• Case 1  y is NULL, that is, node x does not have
  any inorder successor.

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Deletion Operation Case 1

• Case 1  y is NULL, that is, node x does not have
  any inorder successor.

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Deletion Operation Case 1

• Case 1  y is NULL, that is, node x does not have
  any inorder successor.

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Deletion Operation Case 2

• Case 2  y is RED and y has no children as
  internal nodes

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Deletion Operation Case 2

• Case 2  y is RED and y has no children as
  internal nodes

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Deletion Operation Case 3

• Case 3  y is BLACK and it has no children as an
  internal node

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Deletion Operation Case 4

• Case 4  y is BLACK and it has an internal RED
  node

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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.
• ys removal causes any path that contained one
  fewer black node than the path before the removal
  of y. This violates the black-depth property.

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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  zs sibling w is RED
• Case 2  zs sibling w is BLACK and both the
  children of w are BLACK
• Case 3  zs sibling w is BLACK and ws left
  child is RED and ws right child is BLACK.
• Case 4  zs sibling w is BLACK and ws RIGHT
  child is RED.

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Fix-up Operation Case 1

• Case 1  zs 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.

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Fix-up Operation Case 1
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Fix-up Operation Case 2

• Case 2  zs 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.

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Fix-up Operation Case 2
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Fix-up Operation Case 3

• Case 3  zs sibling w is BLACK and ws left
  child is RED and ws right child is BLACK.
• Steps
• Exchange the colours of w and its left child.
• Perform a right rotation of ws left child. This
  rotation results in a new sibling w of z which is
  a black node with a red child.

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Fix-up Operation Case 3
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Fix-up Operation Case 4

• Case 4  zs sibling w is BLACK and ws 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.

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Fix-up Operation Case 4
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For detail implementation of deletion operation
on red-black binary trees see the book Classic
Data Structures Chapter 7 PHI, 2nd Edn., 17th
Reprint
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HB (AVL) Tree vs. RB Tree

• Height
• A red-black tree with n internal nodes has height
  at most 2log2(n1)
• 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).

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Application of RB Tree

• A red-black tree is used to improve the speed of
  searching in a B-tree
• Restructuring with RB tree

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Restructuring with RB Tree
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Restructuring with RB Tree
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Restructuring with RB Tree
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Restructuring with RB Tree
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Restructuring with RB Tree
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Restructuring with RB Tree