Left-leaning red–black tree

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.

Left-leaning red–black tree
Typetree
Invented2008
Invented byRobert 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)

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:

  1. 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


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