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Splay Trees: A Self-Balancing Binary Search Tree, Study notes of Computer Science
An overview of splay trees, a self-balancing binary search tree. It covers the motivation for using splay trees, their basic operations, and the splaying process. Splay trees are easy to implement and perform well in practice, making them a popular choice for data structures.
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CSE 2100
Splay Trees
Overview
Motivation
Basic Operations
Splaying
Implementation
Motivation
Objective...
A simple solution to unbalanced tree
Avoid pathological cases
Get balanced over time
What we will do
Understand the data structure
Understand the implementation
What we will not do
The analysis
Balanced Trees
Why we care....
Balanced implies cheap to search
Height=4! h
Nodes=31! 2
h+
-
Unbalanced Tree
What is the problem?
Height =
Nodes =
Question
Can we
Have Another binary search tree with
The same content
and
Better balanced?
Answer...
Sure!
Idea
Choose a better root!
Answer...
Sure!
Idea
Choose a better root!
“Re-wire” the tree
Can we do any better in this case?
The big Picture
Splay Trees are
Binary search tree
Self-rebalancing
They are not guaranteed to be balanced
They tend towards a balance state
Why do we study them?
Splay trees are easy to implement
Splay trees do extremely well in practice
Splay tree carry a very low overhead
Compared to other balanced trees
How It All Works
One simple key idea
Good news
No additional information to store or update
What must be defined
Exactly how an element floats up the tree!
Each time the tree is accessed (insertion, lookup
or deletion), the deepest node reached is splayed
(it floats) to the root of the tree.
The binary search tree property is maintained
during the splaying.
Splaying
A splaying considers
Up to 3 nodes
X! The node being splayed
Y! The parent of X
Z! The grand-parent of X (if it exist)
Up to 4 sub-trees
The two subtrees of X
The sibling of X
The aunt/uncle of X
Up to 3 cases
Based on how X,Y,Z are hooked up together
Case 1: Zig-Zig
Actually, this covers two cases
X
Y=parent(X)
Z=grand-parent(X)
The node we splay
Objectives:
X moves closer to the root
The partition is maintained
Case 1: Zig-Zig
Let's do it!
X
Y
Z
Z
Y
X
Case 1: Zig-Zig
Let's do it!
X
Y
Z
Z
Y
X
Case 1: Zig-Zig
Let's do it!
X
Y
Z
Z
Y
X
Case 1: Zig-Zig
Let's do it!
X
Y
Z
Z
Y
X
Case 2: Zig-Zag
Actually, this also covers two cases
X
Y=parent(X)
Z=grand-parent(X)
The node we splay
Objectives:
X moves closer to the root
The partition is maintained
The height gets reduced!
Case 2: Zig-Zag
Let's do it!
X
Y
Z
X
Z
Y
Case 2: Zig-Zag
Let's do it!
X
Y
Z
X
Z
Y
Case 2: Zig-Zag
Let's do it!
X
Y
Z
X
Z
Y
Case 2: Zig-Zag
Let's do It!
X
Y
Z
X
Z
Y
Case 2: Zig-Zag
Let's do it!
X
Y
Z
X
Z
Y
We squished the tree!
Case 3: plain Zig
Again, this also covers two cases
W
X
Y=parent(X)
The node we splay
Objectives:
X moves to the root
The partition is maintained
The height gets reduced!