heap sort in the design anad analysis of algorithms

HEAP
A heap is a data structure that stores a
collection of objects (with keys), and has the
following properties:
 Complete Binary tree
 Heap Order
It is implemented as an array where each
node in the tree corresponds to an element
of the array.

HEAP
 The binary heap data structures is an array that can
be viewed as a complete binary tree. Each node of
the binary tree corresponds to an element of the
array. The array is completely filled on all levels
except possibly lowest.
19
12 16
4
1 7
16
19 1 4
12 7
Array A

HEAP
 The root of the tree A[1] and given index i of a
node, the indices of its parent, left child and right
child can be computed
PARENT (i)
return floor(i/2)
LEFT (i)
return 2i
RIGHT (i)
return 2i + 1

HEAP ORDER PROPERTY
 For every node v, other than the root, the key
stored in v is greater or equal (smaller or equal for
max heap) than the key stored in the parent of v.
 In this case the maximum value is stored in the root

DEFINITION
 Max Heap
 Store data in ascending order
 Has property of
A[Parent(i)] ≥ A[i]
 Min Heap
 Store data in descending order
 Has property of
A[Parent(i)] ≤ A[i]

MAX HEAP EXAMPLE
16
19 1 4
12 7
Array A
19
12 16
4
1 7

MIN HEAP EXAMPLE
12
7 19
16
4
1
Array A
1
4 16
12
7 19

INSERTION
 Algorithm
1. Add the new element to the next available position at the
lowest level
2. Restore the max-heap property if violated
 General strategy is percolate up (or bubble up): if the
parent of the element is smaller than the element, then
interchange the parent and child.
OR
Restore the min-heap property if violated
 General strategy is percolate up (or bubble up): if the
parent of the element is larger than the element, then
interchange the parent and child.

19
12 16
4
1 7
19
12 16
4
1 7 17
19
12 17
4
1 7 16
Insert 17
swap
Percolate up to maintain the heap
property

DELETION
 Delete max
 Copy the last number to the root ( overwrite the
maximum element stored there ).
 Restore the max heap property by percolate down.
 Delete min
 Copy the last number to the root ( overwrite the
minimum element stored there ).
 Restore the min heap property by percolate down.

HEAP SORT
A sorting algorithm that works by first organizing the
data to be sorted into a special type of binary tree called
a heap

PROCEDURES ON HEAP
 Heapify
 Build Heap
 Heap Sort

HEAPIFY
 Heapify picks the largest child key and compare it to the
parent key. If parent key is larger than heapify quits, otherwise
it swaps the parent key with the largest child key. So that the
parent is now becomes larger than its children.
Heapify(A, i)
{
l  left(i)
r  right(i)
if l <= heapsize[A] and A[l] > A[i]
then largest l
else largest  i
if r <= heapsize[A] and A[r] > A[largest]
then largest  r
if largest != i
then swap A[i]  A[largest]
Heapify(A, largest)
}

BUILD HEAP
 We can use the procedure 'Heapify' in a bottom-up fashion to
convert an array A[1 . . n] into a heap. Since the elements in
the subarray A[n/2 +1 . . n] are all leaves, the procedure
BUILD_HEAP goes through the remaining nodes of the tree
and runs 'Heapify' on each one. The bottom-up order of
processing node guarantees that the subtree rooted at
children are heap before 'Heapify' is run at their parent.
Buildheap(A)
{
heapsize[A] length[A]
for i |length[A]/2 //down to 1
do Heapify(A, i)
}

HEAP SORT ALGORITHM
 The heap sort algorithm starts by using procedure BUILD-
HEAP to build a heap on the input array A[1 . . n]. Since the
maximum element of the array stored at the root A[1], it can
be put into its correct final position by exchanging it with A[n]
(the last element in A). If we now discard node n from the
heap than the remaining elements can be made into heap.
Note that the new element at the root may violate the heap
property. All that is needed to restore the heap property.
Heapsort(A)
{
Buildheap(A)
for i  length[A] //down to 2
do swap A[1]  A[i]
heapsize[A]  heapsize[A] - 1
Heapify(A, 1)
}

Example: Convert the following array to a heap
16 4 7 1 12 19
Picture the array as a complete binary tree:
16
4 7
12
1 19

16
4 7
12
1 19
16
4 19
12
1 7
16
12 19
4
1 7
19
12 16
4
1 7
swap
swap
swap

HEAP SORT
 The heapsort algorithm consists of two phases:
- build a heap from an arbitrary array
- use the heap to sort the data
 To sort the elements in the decreasing order, use a min heap
 To sort the elements in the increasing order, use a max heap
19
12 16
4
1 7

EXAMPLE OF HEAP SORT
19
12 16
4
1 7
19
12 16 1 4 7
Array A
Sorted:
Take out biggest
Move the last element
to the root

12 16
4
1
7
19
12 16 1 4
7
Array A
Sorted:
HEAPIFY()
swap

12
16
4
1
7
19
12
16 1 4
7
Array A
Sorted:

12
16
4
1
7
19
12 16
1 4
7
Array A
Sorted:
Take out biggest
Move the last element
to the root

12
4
1
7
19
12 16
1
4 7
Array A
Sorted:

12
4
1
7
19
12 16
1
4 7
Array A
Sorted:
HEAPIFY()
swap

12
4
1
7
19
12 16
1
4 7
Array A
Sorted:
Take out biggest
Move the last
element to the
root

4
1
7
19
12 16
1 4 7
Array A
Sorted:
swap

4 1
7
19
12 16
1
4
7
Array A
Sorted:

4 1
7
19
12 16
1 4 7
Array A
Sorted:
Move the last
element to the
root
Take out biggest

4
1
19
12 16
1
4 7
Array A
Sorted:
HEAPIFY()
swap

4
1
19
12 16
1 4 7
Array A
Sorted:
Move the last
element to the
root
Take out biggest

1
19
12 16
1 4 7
Array A
Sorted:
Take out biggest

TIME ANALYSIS
 Build Heap Algorithm will run in O(n) time
 There are n-1 calls to Heapify each call requires
O(log n) time
 Heap sort program combine Build Heap program
and Heapify, therefore it has the running time of O(n
log n) time
 Total time complexity: O(n log n)

TIME COMPLEXITY OF MAX HEAPIFY
 Maximum number of nodes of Height h is

COMPARISON WITH QUICK SORT AND MERGE
SORT
 Quick sort is typically somewhat faster, due to better cache
behavior and other factors, but the worst-case running time for
quick sort is O (n2), which is unacceptable for large data sets
and can be deliberately triggered given enough knowledge of
the implementation, creating a security risk.
 The quick sort algorithm also requires Ω (log n) extra storage
space, making it not a strictly in-place algorithm. This typically
does not pose a problem except on the smallest embedded
systems, or on systems where memory allocation is highly
restricted. Constant space (in-place) variants of quick sort are
possible to construct, but are rarely used in practice due to
their extra complexity.

COMPARISON WITH QUICK SORT AND MERGE
SORT (CONT)
 Thus, because of the O(n log n) upper bound on heap sort’s
running time and constant upper bound on its auxiliary
storage, embedded systems with real-time constraints or
systems concerned with security often use heap sort.
 Heap sort also competes with merge sort, which has the same
time bounds, but requires Ω(n) auxiliary space, whereas heap
sort requires only a constant amount. Heap sort also typically
runs more quickly in practice. However, merge sort is simpler
to understand than heap sort, is a stable sort, parallelizes
better, and can be easily adapted to operate on linked lists
and very large lists stored on slow-to-access media such as
disk storage or network attached storage. Heap sort shares
none of these benefits; in particular, it relies strongly on
random access.

CONCLUSION
 The primary advantage of the heap sort is its
efficiency. The execution time efficiency of the heap
sort is O(n log n). The memory efficiency of the
heap sort, unlike the other n log n sorts, is constant,
O(1), because the heap sort algorithm is not
recursive.
 The heap sort algorithm has two major steps. The
first major step involves transforming the complete
tree into a heap. The second major step is to
perform the actual sort by extracting the largest
element from the root and transforming the
remaining tree into a heap.

heap sort in the design anad analysis of algorithms

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