Chapter 21 The Binary Heap

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Chapter 21The Binary Heap

• Bernard Chen
• Spring 2006

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Whats priority queue
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Problem find MIN and Delete

• Possible methods
• unsorted array
• sorted array
• Binary Search tree

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Whats Binary Heap

• The Binary Heap supports the insertion of new
  items and delete of MIN item in logarithmic
  worst-case time.
• It uses only an array to implement. (Instead of
  linked list)
• It is the classic method used to implement
  priority queues

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Structure Property

• A COMPLETE BINARY TREE is a tree that complete
  filled.

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Complete Binary Tree
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Advantages of Complete Tree

• The height of a complete binary tree is at most
  logN
• Left and right pointers are not needed

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Advantages of Complete Tree

• The parent is in position i/2
• The left child is in position 2i
• The right child is in position 2i1

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Heap-Order property

• In a Heap, for every node X with parent P, the
  key in P is smaller than or equal to the key in X

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21.2 Implementation of the Basic Operations

• Insert operation
• Delete operation

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21.2.1 The INSERT operation

• Insertion is implemented by creating a hole at
  the next available location and then percolating
  it up until the new item can be placed in it
  without violating a heap order.
• Insert takes constant time on average but
  logarithmic time in worst case.

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Insert step 1
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Insert step 2
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Insert step 3
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Insert step 4
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Insert step 5
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21.2.2 The DeleteMIN Operation

• Deletion of the min involves placing the former
  last item in a hole that is created at the root.
• The hole is percolated down the tree through min
  children until the item can be placed without
  violating the heap order property.

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DeleteMIN step 1
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DeleteMIN step 2
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DeleteMIN step 3
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DeleteMIN step 4
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DeleteMIN step 5
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21.3 the buildHeap OperationLinear-Time Heap
construction

• The buildHeap operation can be done in linear
  time by applying a percolate down routine to
  nodes in reverse order

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BuildHeap step 1
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BuildHeap step 2
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BuildHeap step 3
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BuildHeap step 4
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BuildHeap step 5
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BuildHeap step 6
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BuildHeap step 7
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BuildHeap step 8
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BuildHeap step 8