mirror of
https://github.com/jackyzha0/quartz.git
synced 2025-12-24 05:14:06 -06:00
48 lines
1.7 KiB
Markdown
48 lines
1.7 KiB
Markdown
---
|
|
title: "balancing-binary-search-trees"
|
|
aliases: Balancing BST, balancing
|
|
tags:
|
|
- cosc201
|
|
---
|
|
|
|
the height of a [BST](notes/binary-search-tree.md) is the length of its longest chain. Most operations are $O(n)$ where n is the height of the tree. In an Ideal situation each layer of the tree is full. The height of the tree is logarithmic to the number of nodes.
|
|
|
|
When a tree is being used only occainsonally, we can afford to simply rebalance it periodically. However when it is in constant use we cannot afford this cost
|
|
|
|
long branches are a problem
|
|
the performance bounds for all BST operations are linear of the length of the longest branch of the tree
|
|
|
|
ideal shape is a similar to a [heap](notes/heap.md) (wide and shallow).
|
|
|
|
we want the longest branch to be $\theta(lg\ n)$
|
|
|
|
one way is to do an [In order](notes/tree-traversal.md#In%20order) and save to a sorted array. then construct a new tree by repeatedly bisecting the array. and recursively building the left and right subtrees
|
|
|
|
need some local operation that helps to restore balance
|
|
|
|
# Rotation
|
|
## How
|
|
suppose that in this bst there is a longer chain in e than else where
|
|
|
|
|
|
|
|
imagine twisting d to be the root
|
|
|
|
|
|
|
|
changes are
|
|
- b's right child is now c
|
|
- d's left child is not b
|
|
- b parent now points to d
|
|
|
|
## When
|
|
|
|
|
|
sometimes two rotations are needed
|
|
|
|
basic idea is to modify the add and delete operations fo the BST to be somewhat self-balancing. This does not need to be perfect
|
|
|
|
We need a rule to decide when the tree is "balanced enough" and also strategies for fixing problems when the rule is violated.
|
|
|
|
We only need to fix the area local to the add or delete operations
|