Splay Trees CSE 326 Data Structures

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Splay TreesCSE 326 Data Structures

• Richard Anderson
• July 11, 2005

Textbook reference Section 4.5
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Classroom Presenter and Student Submissions
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AVL Trees

• What is the most important aspect of AVL trees?

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Lecture Outline

• Amortized Complexity
• Splaying Binary Search Trees
• Double rotations
• Search Tree operations using splay trees
• Why it works

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Amortized cost

• Cost per operation vs. cost of a sequence of
  operations
• Total cost of ownership
• Examples
• Owning a car
• Small recurring costs gas, parking
• Big expenses insurance, repairs
• Processing email
• Dealing with arriving messages
• Cleaning up inbox once every three months

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Data Structure ExampleBuffering recent items

• Problem
• Allow access to the most recent k items received
• Solution
• Maintain an array of recent items, add items at
  the end, compact the array when full
• Array of size 2k
• If space left just add the item
• If full shift all elements k positions to the left

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Problem What is the amortized cost of inserting
k2 items?

• Number of insertions
• Cost per insertion
• Total cost of insertions

• Number of array compactions
• Cost per compaction
• Total cost of compactions

Total overall cost Amortized cost per item
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Balancing trees

• Basic binary search trees
• No explicit balancing
• Order determined by order of insertion
• AVL Trees
• Explicit operations to keep tree balanced
• Guarantees on height of tree with respect to
  number of nodes
• Self adjusting trees
• Operations applied which help keep the tree
  balanced

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Splay Trees

• Splay trees are tree structures that
• Are not perfectly balanced all the time
• Data most recently accessed is near the root.
• The procedure
• After node X is accessed, perform splaying
  operations to bring X to the root of the tree.
• Do this in a way that leaves the tree more
  balanced as a whole

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The result

• Splay trees support insert, find, and delete at
  O(log n) Amortized cost
• The cost of a sequence of n insert, find, and
  delete operations is O(n log n)
• No guarantee (better than O(n)) on the cost of a
  single operation

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Splaying a node
Elements in increasing order A,B,,G

• Rotate a node to the root of the tree two levels
  at a time

D
G
G
G
G
A
A
A
A
D
F
F
C
D
E
C
B
E
E
E
D
B
B
F
C
C
D
B
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ZIG-ZIG
Z
Y
D
X
C
A
B
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ZIG-ZAG
Show the ZIG-ZAG transformation to bring X to the
root
Z
Y
A
D
X
B
C
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ZAG-ZIG, ZAG-ZAG, ZIG, ZAG

• Other two level transformations similar
• ZIG and ZAG used to handle the case of a path to
  root of odd length

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Find operation

• Find the value as in a binary search tree
• Once the value is found, splay it to the root
• This will rearrange nodes in the tree, generally
  making future operations less expensive

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Splay E
E
Items order A, , I, smallest to largest
H
B
I
A
D
F
G
C
I
H
A
B
G
F
C
D
E
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Splay E (submit your answer)
E
I
A
H
B
A
G
C
I
D
F
B
H
C
G
D
F
E
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Insert and Remove

• Insert x
• Apply standard BST insert
• Splay x to the root
• Remove x
• Splay x to the root
• Delete the root
• Splay largest element of left subtree to the root
• Merge trees

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Remove
X
X
Z
Z
Z
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Describe how you find the largest element in a
tree
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Splaying summary

• Every time we access an element, we bring it to
  the root with a splay
• This ensures that the work done in accessing a
  node also rebalances the tree

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Splay tree performance

• Claim n Insert, Remove, Find operations take
  O(n log n) time
• Intuitive argument long splay operations reduce
  the depth of many nodes
• Rigorous argument accounting based proof to
  show each step costs O(log n)

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Splay Trees

• What is the most important aspect of Splay trees?