Heaps,heapsort and priority queue

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Heaps,heapsort and priority queue

• Binhai Zhu
• Computer Science Department, Montana State
  University

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Definition

• A heap is a binary tree with the following
  properties
• (1) The value of each node is not less than
  the values stored in each of its children. //heap
  property
• (2) The tree is perfectly balanced and leaves
  in the last level are all in the leftmost
  positions.

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Example
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Example
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Definition
A---array lengthA---number of elements in the
array heap-sizeA---number of elements in the
heap stored in the array A root of the
tree/heap---A1
Q. Given index i, how do we find its
parent/children?
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Example
Q. Given index i, how do we find its
parent/children?
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Example
Q. Given index i, how do we find its
parent/children?
parent(i)
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Example
Q. Given index i, how do we find its
parent/children?
parent(i) return i/2
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Example
Q. Given index i, how do we find its
parent/children?
parent(i) return i/2
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left(i)
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Example
Q. Given index i, how do we find its
parent/children?
parent(i) return i/2
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left(i) return 2i
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Example
Q. Given index i, how do we find its
parent/children?
parent(i) return i/2
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left(i) return 2i
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right(i) return 2i1
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Heap property
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Heap property Aparent(i)Ai
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The height of a node is the number of edges on
the longest simple downward path from the node to
a leaf.The height of a tree is the height of
its root.Take-home exercise An n-element heap
has height log n . Prove it!
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Now we cover a fundamental procedure regarding
heaps the heapify procedure.Condition binary
trees rooted at left(i) and right(i) are heaps,
but at Ai the heap property is violated i.e.,
Ai maybe smaller than its children.
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Now we cover a fundamental procedure regarding
heaps the heapify procedure.Condition binary
trees rooted at left(i) and right(i) are heaps,
but at Ai the heap property is violated i.e.,
Ai maybe smaller than its children.
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• Heapify(A,i)
• l ? left (i), r ? right(i)
• If l heap-sizeA and Al gt Ai
• then largest ? l
• else largest ? i //find the largest index
  among Ai,Aleft(i),Aright(i)
• If r heap-sizeA and Ar gt Alargest
• then largest ? r
• If largest ? i
• then exchange Ai ? Alargest
• Heapify(A,largest)

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• Heapify(A,i)
• l ? left (i), r ? right(i)
• If l heap-sizeA and Al gt Ai
• then largest ? l
• else largest ? i //find the largest index
  among Ai,Aleft(i),Aright(i)
• If r heap-sizeA and Ar gt Alargest
• then largest ? r
• If largest ? i
• then exchange Ai ? Alargest
• Heapify(A,largest)

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• Heapify(A,i)
• l ? left (i), r ? right(i)
• If l heap-sizeA and Al gt Ai
• then largest ? l
• else largest ? i //find the largest index
  among Ai,Aleft(i),Aright(i)
• If r heap-sizeA and Ar gt Alargest
• then largest ? r
• If largest ? i
• then exchange Ai ? Alargest
• Heapify(A,largest)
• Runnting time O(log n)

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We can use Heapify to convert an unorganized
array into a heap. Build-Heap(A) 1.heap-sizeA ?
lengthA 2.for i ? lengthA/2 downto 1 3
do Heapify(A,i)
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Q Where do we start?
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Q Where do we start? i5
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Old Ai
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i4, after heapify
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Old Ai
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Old Ai
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i3, after Heapify
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Old Ai
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Old Ai
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i2, run Heapify
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Old Ai
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i2, after Heapify
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Old Ai
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i1
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Old Ai
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i1
i1, after Heapify
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Old Ai
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i1
i1, after Heapify
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New Ai
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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

After Heapify(A,1)
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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

After Heapify(A,1)
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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

After Heapify(A,1)
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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

After Heapify(A,1)
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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

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After a few rounds
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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

After Heapify(A,1)
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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

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The array is sorted!
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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

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Running time?
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Heapsort(A)

1. Build-Heap(A)
2. for ilengthA down to 2
3. exchange A1 and Ai
4. heap-sizeAheap-sizeA-1
5. Heapify(A,1)

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Running time? O(n log n)!
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Priority Queue

• A priority queue is a data structure for
  maintaining a set S of elements, each with an
  associated value called key. //think of
  scheduling a set of jobs
• It supports the following operations
• Insert(S,x)
• Max(S)
• Delete-Max(S) //Of course Min is the symmetric
  case, depending on real application

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Heap-Delete-Max(A)

• If heap-sizeA lt 1
• then print heap underflow
• max ? A1
• A1 ? Aheap-sizeA
• heap-sizeA ? heap-sizeA-1
• Heapify(A,1)
• return max

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Heap-Delete-Max(A)

• If heap-sizeA lt 1
• then print heap underflow
• max ? A1
• A1 ? Aheap-sizeA
• heap-sizeA ? heap-sizeA-1
• Heapify(A,1)
• return max

Running time?
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Heap-Insert(A,key)

• heap-sizeA ? heap-sizeA 1
• i ? heap-sizeA
• While i gt 1 and Aparent(i) lt key
• do Ai ? Aparent(i)
• i ? parent(i)
• Ai ? key

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Heap-Insert(A,key)

• heap-sizeA ? heap-sizeA 1
• i ? heap-sizeA
• While i gt 1 and Aparent(i) lt key
• do Ai ? Aparent(i)
• i ? parent(i)
• Ai ? key

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Heap-Insert(A,key)

• heap-sizeA ? heap-sizeA 1
• i ? heap-sizeA
• While i gt 1 and Aparent(i) lt key
• do Ai ? Aparent(i)
• i ? parent(i)
• Ai ? key

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Heap-Insert(A,key)

• heap-sizeA ? heap-sizeA 1
• i ? heap-sizeA
• While i gt 1 and Aparent(i) lt key
• do Ai ? Aparent(i)
• i ? parent(i)
• Ai ? key

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Running time?
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Heap-Insert(A,key)

• heap-sizeA ? heap-sizeA 1
• i ? heap-sizeA
• While i gt 1 and Aparent(i) lt key
• do Ai ? Aparent(i)
• i ? parent(i)
• Ai ? key

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Question What if we have to increase the key 15
to 18?
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