Computer Science II

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Computer Science II

• Heaps

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What is a heap?

• A tree TR,T0,T1,...Tn-1where
• Every subtree is a heap, and
• The root of T is greater than or equal to the
  root of every subtree of T.
• (Note An alternate definition states that R is
  less than or equal to each of its children)

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Perfect Trees

• This is a perfect tree
• Its also a heap!

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Complete Trees

• This is a complete tree, but its not perfect
• Its a complete binary tree!

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Array Representation
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• Number the nodes, level by level

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Array Representation
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• Create an array, where the node values are placed
  in matching index positions

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Array Representation
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• Note how for any node, the index of its left
  child is 2i, and for its right child its 2i1

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Building a Heap

• Q Given the following complete binary tree, how
  would turn it into a heap?

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Building a Heap

• Q How about this tree?

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Building a Heap

• Heapify(A,i)
• 1. l ? 2i
• 2. r ? 2i 1
• 3. If l ? HeapSize and Al gt Ai
• 4. then largest ? l
• 5. else largest ? i
• 6. if r ? HeapSize and Ar gt Alargest
• 7. then largest ? r
• 8. if largest ? i
• 9. then Ai ? Alargest
• 10. Heapify(A, largest)

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Building a Heap

• Build-Heap(A)
• 1. HeapSize ? length(A)
• 2. for i ? ?length(A)/2? downto 1
• 3. do Heapify(A,i)

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Building a Heap
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• First, using an array, turn the array into a heap
• That is, for each node i, make sure that it is
  larger than nodes 2i and 2i1

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Building a Heap
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• Swap first and last elements

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Building a Heap
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• Eliminate last element from process

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Building a Heap
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• You now have an almost-heap
• Percolate the new root down to where it belongs
  to form a new heap

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Building a Heap
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• You now have an almost-heap
• Percolate the new root down to where it belongs
  to form a new heap

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Building a Heap
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• You now have an almost-heap
• Percolate the new root down to where it belongs
  to form a new heap

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Building a Heap
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• Swap first and last elements

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Building a Heap
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• Eliminate last element from process

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Building a Heap
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• And so on

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Heapsort

• Heapsort(A)
• 1. Build-Heap(A)
• 2. for i ? length(A) downto 2
• 3. do A1 ? Ai
• 4. HeapSize ? HeapSize - 1
• 5. Heapify(A,1)

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Questions?

• How many comparisons does it take to build a
  heap?
• Best case?
• Worst case?
• Average case?

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Questions?

• How many comparisons does it take perform a
  heapsort?
• Best case?
• Worst case?
• Average case?

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Questions?

• How would you insert an element into a Heap?
• How would you delete an element?