10-Balanced-Search-Trees

# 10-Balanced-Search-Trees - Algorithms LECTURE 10 Balanced...

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Algorithms L7.1 Professor Ashok Subramanian L ECTURE 10 Balanced Search Trees Red-black trees Height of a red-black tree Rotations Insertion Algorithms

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Algorithms L7.2 Balanced search trees Balanced search tree: A search-tree data structure for which a height of O (lg n ) is guaranteed when implementing a dynamic set of n items. Examples: AVL trees 2-3 trees 2-3-4 trees B-trees Red-black trees
Algorithms L7.3 Red-black trees This data structure requires an extra one- bit color field in each node. Red-black properties: 1. Every node is either red or black. 2. The root and leaves ( NIL ’s) are black. 3. If a node is red, then its parent is black. 4. All simple paths from any node x to a descendant leaf have the same number of black nodes = black-height( x ) .

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Algorithms L7.4 Example of a red-black tree h = 4 8 11 10 18 26 22 3 7 NIL NIL NIL NIL NIL NIL NIL NIL NIL
Algorithms L7.5 Example of a red-black tree 8 11 10 18 26 22 3 7 NIL NIL NIL NIL NIL NIL NIL NIL NIL 1. Every node is either red or black.

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Algorithms L7.6 Example of a red-black tree 8 11 10 18 26 22 3 7 NIL NIL NIL NIL NIL NIL NIL NIL NIL 1. The root and leaves ( NIL ’s) are black.
Algorithms L7.7 Example of a red-black tree 8 11 10 18 26 22 3 7 NIL NIL NIL NIL NIL NIL NIL NIL NIL 1. If a node is red, then its parent is black.

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Algorithms L7.8 Example of a red-black tree 1. All simple paths from any node x to a descendant leaf have the same number of black nodes = black-height ( x ) . 8 11 10 18 26 22 3 7 NIL NIL NIL NIL NIL NIL NIL NIL NIL bh = 2 bh = 1 bh = 1 bh = 2 bh = 0
Algorithms L7.9 Height of a red-black tree Theorem. A red-black tree with n keys has height h 2 lg( n + 1) .

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