Balanced Binary Search Tree

(English slides)

신찬수

Complexities on Binary Search Tree • Search, insertion, and deletion on BST need O( h )

– h = height of BST

– h can be increased to n – 1

– Thus all three operations take O( n ) time

• Solution: maintain the height as less as possible under operations

– BST is called balanced if its height is O( log n ) under operations.

– AVL trees, red-black trees, 2-3 trees, splay trees

AVL trees

• Guarantee the O(log n)-height by controlling the tree height directly.

– maintain the height of left and right subtrees of every node differ by at most

one. height rule

– all leaves are on the last two levels

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When the height rule violates?

• 25 is inserted:

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When the height rule violates?

• 25 is inserted:

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right rotation

Rotations for balancing

• Single rotations

– Left rotation (figure) and right rotation

Rotations for balancing

• Double rotations

– Left-left, left-right, right-left (figure), right-right rotations

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Deletion 1. Delete the node of the key by copying with its predecessor.

2. Balance the height if necessary by rotations while moving to root.

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Animation

http://www.seanet.com/users/arsen/avltree.html

http://www.seanet.com/users/arsen/avltree.html

Red-black tree • A BST is called a red-black tree if the following

four properties are satisfied:

1. Every node is either red or black.

2. Every null is assumed to be black.

3. If a node is red, then both its children are black.

4. Every simple path from a node to a descendant

leaf contains the same number of black nodes.

• Red-balck tree maintains O(log n)-height by

controlling the path length to leaf.

Red-black tree • The height of a red-black tree of n nodes is at most (2log n + 1).

– It means the longest path length is at most (2log n + 1).

• This means that it takes O( log n ) time to search a node with a

specific key.

• Insertion and deletion will violate Red-Black property, so we need

to restore the property (by rotations).

– Insertion performs at most two rotations.

– Deletion performs at most three rotations.

http://www.seanet.com/users/arsen/avltree.html

http://www.seanet.com/users/arsen/avltree.html

Splay tree • AVL tree and red-black tree enforces the height of the tree to be

O( log n ) whenever insertion and deletion endanger the tree balance.

• But it is not the only way to support efficient search, insert, delete operations.

• Splay tree uses the locality of the access frequency:

– The more frequently are key value accessed, the closer is it to the root.

– Moving up the key to the root by rotation whenever it is accessed by operations.

• With m operations, the amortized cost is O( m log n ) time, so the average time per operation is O( log n ). ( n is the maximum number of key values of the tree from start to end. )

Splaying to move a node to the root

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Splaying to move a node to the root

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Splaying to move a node to the root

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Search/insertion/deletion • Search

1. Find the node containing the key value.

2. Splay the node to the root.

• Insertion

1. Find the location to be inserted, and create a node there.

2. Splay the node to the root.

• Deletion

1. Find the node to be deleted, and splay the node to the root.

2. Delete the node (then you have left and right subtrees L and R)

3. Splay the predecessor in L of the node to the root.

4. Join L and R where R becomes right subtree of the root of L.