Heapsort
A minimalist's approach
Jeff Chastine
Heapsort
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Like MERGESORT, it runs in O(n lg n)
Unlike MERGESORT, it sorts in place
Based off of a “heap”, which has several uses
The word “heap” doesn’t refer to memory
management
Jeff Chastine
The Heap
• A binary heap is a nearly complete binary tree
• Implemented as an array A
• Two similar attributes:
– length[A] is the size (number of slots) in A
– heap-size[A] is the number of elements in A
– Thus, heap-size[A]  length[A]
– Also, no element past A[heap-size[A]] is an
element
Jeff Chastine
The Heap
• Can be a min-heap or a max-heap
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Simple Functions
PARENT(i)
return (i/2)
LEFT(i)
return (2i)
RIGHT(i)
return (2i + 1)
Jeff Chastine
Properties
• Max-heap property:
– A[PARENT(i)]  A[i]
• Min-heap property:
– A[PARENT(i)]  A[i]
• Max-heaps are used for sorting
• Min-heaps are used for priority queues (later)
• We define the height of a node to be the longest
path from the node to a leaf.
• The height of the tree is (lg n)
Jeff Chastine
MAX-HEAPIFY
• This is the heart of the algorithm
• Determines if an individual node is smaller
than its children
• Parent swaps with largest child if that child is
larger
• Calls itself recursively
• Runs in O(lg n) or O(h)
Jeff Chastine
HEAPIFY
MAX-HEAPIFY (A, i)
l ← LEFT (i)
r ← RIGHT(i)
if l ≤ heap-size[A] and A[l] > A[i]
then largest ← l
else largest ← i
if r ≤ heap-size[A] and A[r]>A[largest]
then largest ← r
if largest ≠ i
then exchange A[i] with A[largest]
MAX-HEAPIFY (A, largest)
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Of Note
• The children’s subtrees each have size at most
2n/3 – when the last row is exactly ½ full
• Therefore, the running time is:
T (n) = T(2n/3) + (1) = O(lg n)
Jeff Chastine
BUILD-HEAP
• Use MAX-HEAPIFY in bottom up manner
• Why does the loop start at length[A]/2?
• At the start of each loop, each node i is the
root of a max-heap!
BUILD-HEAP (A)
heap-size[A] ← length[A]
for i ← length[A]/2 downto 1
do MAX-HEAPIFY(A, i)
Jeff Chastine
Analysis of Building a Heap
• Since each call to MAX-HEAPIFY costs O(lg n)
and there are O(n) calls, this is O(n lg n)...
• Can derive a tighter bound: do all nodes take
log n time?
• Has at most n/2h+1 nodes at any height (the
more the height, the less nodes there are)
• It takes O(h) time to insert a node of height h.
Jeff Chastine
Sum up the work at each level
h
 n 

 2h1 O(h)  n 2h
h 0
h 0
lg n
lg n
Height h is
logarithmic
The number of
nodes at height h
Multiplied by their height
• Thus, the running time is 2n = O(n)
Jeff Chastine
HEAPSORT
HEAPSORT (A)
BUILD-HEAP(A)
for i ← length[A] downto 2
do exchange A[1] with A[i]
heap-size[A] ← heap-size[A] - 1
MAX-HEAPIFY(A, 1)
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Swap A[1]  A[i]
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heap-size  heap-size – 1
MAX-HEAPIFY(A, 1)
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Swap A[1]  A[i]
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heap-size  heap-size – 1
MAX-HEAPIFY(A, 1)
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Swap A[1]  A[i]
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heap-size  heap-size – 1
MAX-HEAPIFY(A, 1)
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Swap A[1]  A[i]
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heap-size  heap-size – 1
MAX-HEAPIFY(A, 1)
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Swap A[1]  A[i]
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heap-size  heap-size – 1
MAX-HEAPIFY(A, 1)
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Swap A[1]  A[i]
Jeff Chastine
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heap-size  heap-size – 1
MAX-HEAPIFY(A, 1)
Jeff Chastine
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Swap A[1]  A[i]
Jeff Chastine
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heap-size  heap-size – 1
MAX-HEAPIFY(A, 1)
Jeff Chastine
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Swap A[1]  A[i]
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heap-size  heap-size – 1
MAX-HEAPIFY(A, 1)
Jeff Chastine
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Swap A[1]  A[i]
Jeff Chastine
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heap-size  heap-size – 1
MAX-HEAPIFY(A, 1)
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Priority Queues
• A priority queue is a heap that uses a key
• Common in operating systems (processes)
• Supports HEAP-MAXIMUM, EXTRACT-MAX,
INCREASE-KEY, INSERT
HEAP-MAXIMUM (A)
1 return A[1]
Jeff Chastine
HEAP-EXTRACT-MAX (A)
1 if heap-size[A] < 1
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then error “heap underflow”
3 max  A[1]
4 A[1]  A[heap-size[A]]
5 heap-size[A]  heap-size[A] – 1
6 MAX-HEAPIFY (A, 1)
7 return max
Jeff Chastine
HEAP-INCREASE-KEY(A, i, key)
1 if key < A[i]
2
then error “new key smaller than current”
3 A[i]  key
4 while i > 1 and A[PARENT(i)] < A[i]
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do exchange A[i]  A[PARENT(i)]
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i  PARENT(i)
Note: runs in O(lg n)
Jeff Chastine
MAX-HEAP-INSERT (A, key)
1 heap-size[A]  heap-size[A] + 1
2 A[heap-size[A]]  -
3 HEAP-INCREASE-KEY(A, heap-size[A], key)
Jeff Chastine