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Balanced Binary Search Trees
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
• B-trees
• Red-black trees
• Treaps
and Randomized Binary Search TreesRed-black trees
• A binary search tree with an extra bit of storage per node: it’s color
• Color can be red or black
• Tree is approximately balanced such that no path is more than twice as long as any other
• Properties:
• Every node is red or black
• The root is black
• Leaves are nil nodes that are black
• If a node is red then both its children are black
• For each node, all simple paths from the node to descendant leaves contain the same number of black nodes
Example of a red-black tree
• Every node is red or black
• The root is black
• Leaves are nil nodes that are black
• If a node is red then both its children are black
• For each node, all simple paths from the node to descendant leaves contain the same number of black
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Height of a red-black tree
• Theorem
• A red-black tree with n internal nodes (non-nil, nodes with keys) has height
h <= 2lg(n+1)
• Intuitive argument: Merge red nodes into their black parents
Height of a red-black tree
• Resulting tree is a 2-3-4 tree where each node has 2, 3, or 4 children
• The 2-3-4 tree has uniform depth h’ of leaves
Height of a red-black tree
• Since at most half of the leaves on any path are red we must have
• h’ >= h/2
• Note that the number of leaves in each tree is n+1
• n+1 >= 2
h’(from the black height of each node)
• lg(n+1) >= h’ >= h/2
• h < 2lg(n+1)
Corollary
• All the query operations take O(lg n) time on a red-black tree with n nodes
• Search
• Min
• Max
• Successor
• Predecessor
• We can also insert and delete in O(lg n) time but it requires some trickier operations to maintain the red-black tree properties
• Must change colors and perform rotations
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Rotations
• A rotation takes O(1) time while maintaining the ordering of the keys and the BST property
Insertion Idea
• Insert new node as normal to a BST. Color it red and then recolor/rotate up the tree if there are any violations.
• Example: Insert x=15
Insertion Example
• Recolor 11 to black, 10 to red, the right-rotate 10 with 18
Insertion Example
• Recolor, left-rotate 10 with 7 and recolor
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Final Tree
Analysis
• Code is a bit messy with special cases – see online for details
• Can perform O(1) rotations moving up the tree for O(lg n) insertions
• Overall O(lg n) runtime
• Deletion also O(lg n) time
• Similar special cases to consider
Randomized Data Structures
• If we have random input data then we get expected good performance on binary search trees
• BUT on particular inputs we can get bad behavior
• Idea:
• When you can’t randomize the input, randomize the data structure
• Gives expected random behavior, which is O(lg n) for BST, on any input!
Enter... the TREAP
• Tree + Heap, hence TREAP
• Both a BST and Heap are merged together. No array like we did previously with the heap, instead it is part of the node of the BST
• Invented in 1989 by Aragon and Seidel
• Why?
• High probability of O(lg n) performance for any input
• Code is much simpler than red-black trees
• Perhaps the simplest of all balanced BST’s
• Can implement without recursion for extra speed
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Treap
• Treaps maintain a key value and a priority value for each node
• binary search tree property maintained for the keys
• heap property maintained for the priority value
• The heap values are randomly assigned
Key: 10 Priority: 0.97
Key: 5 Priority: 0.31
Key: 2 Priority: 0.24
Key: 8 Priority: 0.15
Key: 15 Priority: 0.85
Key: 12 Priority: 0.12
Key: 85 Priority: 0.76
Key: 97 Priority: 0.11
Treap Insert
• Choose a random priority (in our example between 0-1)
• Insert like normal into the BST
• Rotate up until heap order is restored
• Note that rotations preserve the heap property AND the BST property!
• Example, insert key=20, priority=0.5
Key: 5 Priority: 0.31
Key: 2 Priority: 0.24
Key: 8 Priority: 0.15 Key: 4
Priority: 0.13
Key: 5 Priority: 0.31
Key: 2 Priority: 0.24
Key: 8 Priority: 0.15
Key: 4 Priority: 0.13
Key: 20 Priority: 0.5
Insert Example
• Rotate
Key: 5 Priority: 0.31
Key: 2 Priority: 0.24
Key: 8 Priority: 0.15 Key: 4
Priority: 0.13
Key: 20 Priority: 0.5
Key: 20 Priority: 0.5
Key: 5 Priority: 0.31
Key: 2 Priority: 0.24
Key: 4 Priority: 0.13
Key: 8 Priority: 0.15
Shape of the tree fully specified by random priorities No bad inputs, only bad random numbers!
Delete
• To delete a node:
• Find the node X to delete via BST search
• If X is a leaf, just delete it
• If X has only one child, rotate the child up and delete X
• If X has two children, rotate with child that has the largest priority, maintaining the BST property with the key, and then recursively delete X
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Delete Example
• Delete Key 15
Key: 10 Priority: 0.97
Key: 5 Priority: 0.31
Key: 2 Priority: 0.24
Key: 8 Priority: 0.15
Key: 15 Priority: 0.85
Key: 12 Priority: 0.10
Key: 85 Priority: 0.76
Key: 97 Priority: 0.11
Rotate with Key 85 Priority 0.76
Delete Example
Key: 10 Priority: 0.97
Key: 5 Priority: 0.31
Key: 2 Priority: 0.24
Key: 8 Priority: 0.15
Key: 15 Priority: 0.85
Key: 12 Priority: 0.10
Key: 85 Priority: 0.76
Key: 97 Priority: 0.11
Rotate with Key 12 Priority 0.10
Delete Example
Key: 10 Priority: 0.97
Key: 5 Priority: 0.31
Key: 2 Priority: 0.24
Key: 8 Priority: 0.15
Key: 12 Priority: 0.10
Key: 15 Priority: 0.85
Key: 85 Priority: 0.76
Key: 97 Priority: 0.11
No children, just delete Key 15
Delete Example
Key: 10 Priority: 0.97
Key: 5 Priority: 0.31
Key: 2 Priority: 0.24
Key: 8 Priority: 0.15
Key: 12 Priority: 0.10
Key: 85 Priority: 0.76
Key: 97 Priority: 0.11
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Treap Applet
• https://www.ibr.cs.tu-bs.de/courses/ss98/audii/applets/BST/Treap- Example.html
• Java applet so kind of hard to run nowadays (could still run in IE after editing Java Console permissions)
Treap Summary
• Implements Binary Search Tree
• At a more abstract level, a Dictionary – fast to insert, retrieve, delete
