AVL Tree and Heap

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AVL Tree and Heap

• Data Structures
• Rutgers University

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Binary search tree

• Binary search trees The run times can be far
  worse than O(log n) for search, insert and delete
• Worst case when completely lopsided, or skewed on
  one side or the other. ? O(n) time for
  search/insert/delete.
• To maintain the O(log n) time, the height of the
  tree would have to be maintained at O(log n) ?
  Need to be in balanced condition
• To achieve this ? rebalance at every insert or
  delete

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Height balanced binary search trees

• AVL tree
• Named after the inventors, (Russina
  mathematicians) G.M. Adelson-Velskii and Em.M.
  Landis (1962)
• Red-black trees
• We will not cover them in this course

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AVL Tree Structure

• AVL Tree is Height balanced binary search tree
• Ideally, for every node, the heights of its left
  and right subtrees are equal.
• AVL definition of height balance
• An AVL tree is a binary search tree in which the
  heights of the left and right subtrees of every
  node differ by at most 1.

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Examples
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Examples
Recursive definition An AVL tree is a binary
search tree in which the left and right subtrees
of the root are AVL trees whose heights differ by
at most 1.
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Height Imbalance During Insertion or Deletion
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Rotation

• Locally rearrange the structure of an AVL tree

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Heaps

• Heaps or based on the notion of a complete tree
• A complete tree is filled from the left
• All the leaver are on the same level or two
  adjacent ones
• All nodes at the lowest level are as far to the
  left as possible
• A binary tree is completely full ?? 2h1-1 nodes
  (h is height)
• A binary tree is complete iff
• It is empty or
• Its left subtree is complete of height h-1 and
  its right subtree is completely full of height
  h-2 or
• Its left subtree is completely full of height h-1
  and its right subtree is complete of height h-1

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Heaps

• A binary tree has the heap property iff
• It is empty or
• The key in the root is larger than that in either
  child and both subtrees have the heap property
• Heap can be used as a priority queue
• Highest priority item is at the root and is
  trivially extracted
• If root is deleted, we are left with two
  sub-trees and we must recreate a single tree with
  heap property
• We can both extract the highest priority item and
  insert a new item in O(log n) time.

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Deletion from a Heap
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5. Tree is now a heap again We need to make at
most h interchanges of a root of a subtree with
one of its children to restore the heap
property. Thus deletion from a heap is O(h) or
O(log n).
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Addition to a Heap
Follow the reverse procedure Place it in the next
leaf position and move it up
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We need O(h) or O(log n) exchanges
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Storage of complete trees

• Use n sequential locations in an array

• Number the nodes from 1 at the root
• and place
• The left child of node k is at 2k
• The right child of node k is at 2k1

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Viewed as an array, we can see that the nth node
is always in index position n
Delete extract the root, swap the last item to
its place, and call MoveDown recursively Insertion
similar and use MoveUp recursively