Analysis of Data Structures Binary Search Trees

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Analysis of Data StructuresBinary Search Trees
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Goal

• Analyzing data structures
• Example binary search trees
• Overview
• Definition
• Properties
• Operations
• Analyzing properties running times of
  operations

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Storing and modifying data

• Arrays (sorted)
• Linked lists (unsorted)

fast searching, slow insertion
slow searching, fast insertion
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Data structures for maintaining sets
Search
Insert/delete
Unsorted array
T(n)
T(1)
Sorted array
T(log n)
T(n)
Unsorted list
T(n)
T(1)
T(n)
T(n)
Sorted list
Balanced search tree
T(log n)
T(log n)
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Trees

• Each of the n nodes contains
• data (number, object, etc..)
• pointers to its children (also trees!)
• Primitive operations
• Accessing data O(1) time
• Traversing link O(1) time

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Binary trees

• Every node has 2 children (can be dummies)

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Binary search trees

• Binary tree with comparable values
• For a node with value x
• Left subtree x
• Right subtree x

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Tree property height

• Height h of a tree length of the longest path
• 2 h n (why?)

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Height of a binary tree
max

• min

1 1 1 1 1
1 2 4 n/2
n
2log n
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Operation searching for an element

• Searching for 7 start at root
• At every node
• youve found it!
• otherwise, choose left or right child
• Until you find 7, or youre at a leaf node
• What is the running time?

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O(h)
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Insertion

• Search for 7
• Replace leaf node
• If already presentjust go on
• What is the running time?

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O(h)
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Inorder tree traversal

• Visit nodes sequentially

O(n) time
x
LC
RC
1, 4, 5, 6, 6, 8, 9 .
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Removal

• Search for 7
• If node has at least one leaf node as a child
• Delete node, and attach other child to parent

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Removal - 2

• Remove 8
• Left child is a leaf,so attach right child to
  parent
• But what if we want to remove 4?

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Removal - 3

• Find inorder successor
• It always exists! (And its left childis a leaf
  node)
• Replace yourself with this number
• Remove successor in the previously described way
• How (fast) can we find the inorder successor in
  this case?

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Again O(h)
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Overview