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6.006 Introduction to Algorithms
Spring 2008
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Lecture 9
Sorting II: Heaps
Lecture 9: Sorting II: Heaps
Lecture Overview
• Review: Heaps and MAX HEAPIFY
• Building a Heap
• Heap Sort
• Priority Queues (Recitation)
Readings
CLRS 6.1-6.4
Review
Heaps:
Parent(i) = �i/2�
Left(i) = 2i
Right(i) = 2i + 1
Max heap property:
A[Parent(i)] ≥ A[i]
• MAX HEAPIFY(A, 2)
heap size(A) = 10
A[2] ←→ A[4]
• MAX HEAPIFY(A,4)
A[4] ←→ A[9]
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6.006 Spring 2008
Lecture 9
Sorting II: Heaps
6.006 Spring 2008
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Violation
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etc
O(lg n) time
Figure 1: Review from last lecture
Building a Heap
A[1 · · · n] converted to a max heap Observation: Elements A[�n/2 + 1� · · · n] are all leaves
of the tree and can’t have children.
BUILD MAX HEAP(A):
heap size(A) = length(A)
O(n) times for i ← � length[A]/2� downto 1
O(lg n) time
do MAX HEAPIFY(A, i)
O(n lg n) overall
See Figure 2 for an example.
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Lecture 9
Sorting II: Heaps
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MAX-HEAPIFY (A,3)
Swap A[3] and A[7]
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MAX-HEAPIFY (A,2)
Swap A[2] and A[5]
Swap A[5] and A[10]
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MAX-HEAPIFY (A,5)
no change
MAX-HEAPIFY (A,4)
Swap A[4] and A[8]
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MAX-HEAPIFY (A,1)
Swap A[1] with A[2]
Swap A[2] with A[4]
Swap A[4] with A[9]
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Figure 2: Example: Building Heaps
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Lecture 9
Sorting II: Heaps
6.006 Spring 2008
Sorting Strategy
• Build max heap from unordered array
• Find maximum element (A[1])
• Put it in correct position A[n], A[n] goes to A[1]
New root could violate max heap property but children remain max heaps.
• Discard node n from heap (decrement heapsize)
Heap Sort Algorithm
O(n lg n) BUILD MAX HEAP(A):
n times for i =length[A] downto 2
do exchange A[1] ←→ A[i]
heap size[A] = heap size[A] − 1
O(lg n)
MAX HEAPIFY(A, 1)
O(n lg n) overall
See Figure 3 for an illustration.
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Lecture 9
Sorting II: Heaps
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heap_size = 9
MAX_HEAPIFY (A,1)
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not part of heap
Note: cannot run MAX_HEAPIFY with heapsize of 10
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MAX_HEAPIFY (A,1)
and so on . . .
Figure 3: Illustration: Heap Sort Algorithm
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not part of heap
Lecture 9
Sorting II: Heaps
6.006 Spring 2008
Priority Queues
This is an abstract datatype as it can be implemented in different ways.
INSERT(S, X) :
MAXIMUM(S):
EXTRACT MAX(S):
INCREASE KEY(S, x, k):
inserts X into set S
returns element of S with largest key
removes and returns element with largest key
increases the value of element x’s key to new value k
(assumed to be as large as current value)
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