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Red-Black Trees

Oct 21, 2014

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Data Structures. Red-Black Trees. Design and Analysis of Algorithms I. Binary Search Tree Structure. The Height of a BST. Balanced Search Trees. Red-Black Invariants. Example #1. Example #2. Height Guarantee. Height Guarantee ( con’d ).

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Red-Black Trees

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Data Structures Red-Black Trees Design and Analysis of Algorithms I
Binary Search Tree Structure
The Height of a BST
Balanced Search Trees
Red-Black Invariants
Example #1
Example #2
Height Guarantee
Height Guarantee (con’d)
Which of the search tree operations have to be re-implemented so that the Red-Black invariants are maintained? Search Delete Insert and Delete None of the above

Red-Black Trees

Red-Black Trees. Red-black trees: Overview. Red-black trees are a variation of binary search trees to ensure that the tree is balanced . Height is O (lg n ), where n is the number of nodes. Operations take O (lg n ) time in the worst case .

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Red Black Trees

Red Black Trees. Colored Nodes Definition Binary search tree. Each node is colored red or black. Root and all external nodes are black. No root-to-external-node path has two consecutive red nodes. All root-to-external-node paths have the same number of black nodes. Red Black Trees.

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Red-Black Trees. Bottom-Up Deletion. Recall “ordinary” BST Delete. If node to be deleted is a leaf, just delete it. If node to be deleted has just one child, replace it with that child (splice)

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Red-black trees. Outline. In this topic, we will cover: The idea behind a red-black tree Defining balance Insertions and deletions The benefits of red-black trees over AVL trees. Red-Black Trees. A red black tree “colours” each node within a tree either red or black

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Red-Black Trees. Definitions and Bottom-Up Insertion. Red-Black Trees. Definition: A red-black tree is a binary search tree in which: Every node is colored either Red or Black. Each NULL pointer is considered to be a Black “node”. If a node is Red, then both of its children are Black.

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Red Black Trees. Lecture 6. Balanced search trees. Red-black trees. Example of a red-black tree. Example of a red-black tree. Example of a red-black tree. Example of a red-black tree. Example of a red-black tree. Height of a red-black tree. Height of a red-black tree.

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Red-Black Trees. A Red-Black Tree with NULLs shown Black-Height of the tree = 4. Red-Black Trees. Definition: A red-black tree is a binary search tree where: Every node is either red or black. Each NULL pointer is considered to be a black node

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Red-Black Trees

Red-Black Trees. Red-Black Trees. Definition: A red-black tree is a binary search tree where: Every node is either red or black. Each NULL pointer is considered to be a black “node” If a node is red, then both of its children are black.

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Red Black Trees. Top-Down Deletion. Recall the rules for BST deletion. If vertex to be deleted is a leaf, just delete it. If vertex to be deleted has just one child, replace it with that child

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Red-black Trees . Consider a b-tree of order 4. A node must have at least 2 children and as many as 4. A node must have at least 1 key value and as many as 3. We have always represented the key values as an array, but what if we did it as a tree?. Red-black Trees Example.

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Red-Black Trees. CS302 Data Structures Dr. George Bebis. Red-Black Trees. Binary search tree with an additional attribute for its nodes: color which can be red or black Constrains the way nodes can be colored on any path from the root to a leaf.

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Red Black Trees. Top-Down Insertion. Review of Bottom-Up Insertion. In B-Up insertion, “ordinary” BST insertion was used, followed by correction of the tree on the way back up to the root This is most easily done recursively

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Red- Black Trees. CSIT 402 Data Structures II. Binary Search Trees. Average case and worst case Big O for insertion deletion access Balance is important. Unbalanced trees give worse than log N times for the basic tree operations Can balance be guaranteed?. Red Black Trees.

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Red-Black Trees. Bottom-Up Deletion. Recall “ordinary” BST Delete. If vertex to be deleted is a leaf, just delete it. If vertex to be deleted has just one child, replace it with that child

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Red-Black Trees. CS 583 Analysis of Algorithms. Outline. Red-Black Trees Definitions Rotations Augmenting Data Structures Definitions Dynamic order statistics Determining the rank of an element Maintaining subtree sizes. Definitions.

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Red-Black Trees. 6. v. 8. 3. z. 4. A red-black tree is a representation of a (2,4) tree by means of a binary tree whose nodes are colored red or black In comparison with its associated (2,4) tree, a red-black tree has same logarithmic time performance

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