Lec10 平衡搜索树

MIT算法导论课程：Lec10 平衡搜索树，对应书上的章节：Chapter 13

1. 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
2-3-4 trees
B-trees
Red-black trees

2. Red-black trees

This data structure requires an extra one bit color field in each node.

Red-black properties:

Every node is either red or black.
The root and leaves (NIL’s) are black.
If a node is red, then its parent is black.
All simple paths from any node x to a descendant leaf have the same number of black nodes = black-height(x ).

3. Height of a red-black tree

Theorem. A red-black tree with n keys has height \(h\leq 2\lg(n+1)\).

Proof：（书上采用的是induction的方法证明）

INTUITION :

Merge red nodes into their black parents.
This process produces a tree in which each node has 2, 3, or 4 children.
The 2-3-4 tree has uniform depth h′ of leaves.
We have h′ ≥ h/2, since at most half the leaves on any path are red.
The number of leaves in each tree is

\[ \begin{aligned} n+1 & \geq2^{h'}\\ \lg (n+1) &\geq h'\geq h/2\\ h&\leq 2\lg(n+1) \end{aligned} \]

Query operations

The queries SEARCH, MIN, MAX, SUCCESSOR, and PREDECESSOR all run in \(O(\lg n)\) time on a red-black tree with n nodes.

Modifying operations

The operations INSERT and DELETE cause modifications to the red-black tree:

the operation itself,
color changes,
restructuring the links of the tree via “rotations”.

4. Rotations

5. Insertion

IDEA: Insert x in tree. Color x red. Only red-black property 3 might be violated. Move the violation up the tree by recoloring until it can be fixed with rotations and recoloring.

Pseudocode: