Left-leaning red–black tree

Left-leaning red–black tree
Type	tree
Invented	2008
Invented by	Robert Sedgewick
Time complexity in big O notation Operation Average Worst case Search O(log n) O(log n) Insert O(log n) O(log n) Delete O(log n) O(log n) Space complexity Space O(n) O(n)

A left-leaning red–black (LLRB) tree is a type of self-balancing binary search tree. It is a variant of the red–black tree and guarantees the same asymptotic complexity for operations, but is designed to be easier to implement.

Properties

A left-leaning red-black tree satisfies all the properties of a red-black tree:

1. Every node is either red or black.
2. A NIL node is considered black.
3. A red node does not have a red child.
4. Every path from a given node to any of its descendant NIL nodes goes through the same number of black nodes.
5. The root is black (by convention).

Additionally, the left-leaning property states that:

6. A node must not have a red right child.

Relation to 2–3 trees

In the same way conventional red-black trees are related to 2–3–4 trees, left-leaning red-black trees are isomorphic to 2–3 trees, which are a subtype of 2–3–4 trees. This means that for every left-leaning red-black tree, there is a unique corresponding 2–3 tree, and vice versa. Precisely, each (red left child, black parent) pair corresponds to a degree 3 node in a 2–3 tree, and all other black nodes correspond to degree 2 nodes.

Analysis

All of the red-black tree algorithms that have been proposed are characterized by a worst-case search time bounded by a small constant multiple of log N in a tree of N keys, and the behavior observed in practice is typically that same multiple faster than the worst-case bound, close to the optimal log N nodes examined that would be observed in a perfectly balanced tree.

Specifically, in a left-leaning red-black 2–3 tree built from N random keys:

• A random successful search examines log₂ N − 0.5 nodes.
• The average tree height is about 2 log₂ N
• The average size of left subtree exhibits log-oscillating behavior.

Bibliography

• Robert Sedgewick's Java implementation of LLRB from his 2008's paper
• Robert Sedgewick. 20 Apr 2008. Animations of LLRB operations
• Open Data Structures - Section 9.2.2 - Left-Leaning Red–Black Trees, Pat Morin

External links

• Robert Sedgewick. Left-leaning Red–Black Trees. Direct link to PDF.
• Robert Sedgewick. Left-Leaning Red–Black Trees slides from April 2008.
• Linus Ek, Ola Holmström and Stevan Andjelkovic. May 19, 2009. Formalizing Arne Andersson trees and Left-leaning Red–Black trees in Agda
• Julien Oster. March 22, 2011. An Agda implementation of deletion in Left-leaning Red–Black trees
• Kazu Yamamoto. 2011.10.19. Purely Functional Left-Leaning Red–Black Trees
• Left-Leaning Red-Black Trees Considered Harmful

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This page was last edited on 12 April 2024, at 19:16