Chapter 6
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Chapter 6. Heapsort Lee, Hsiu-Hui Ack: This presentation is based on the lecture slides from Hsu, Lih-Hsing, as well as various materials from the web. Why sorting. 1. Sometimes the need to sort information is inherent in the application.

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Chapter 6

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Chapter 6

Chapter 6

Heapsort

Lee, Hsiu-Hui

Ack: This presentation is based on the lecture slides from Hsu, Lih-Hsing, as well as various materials from the web.


Why sorting

Why sorting

1. Sometimes the need to sort information is inherent in the application.

2. Algorithms often use sorting as a key subroutine.

3. There is a wide variety of sorting algorithms, and they use rich set of techniques.

4. Sorting problem has a nontrivial lower bound

5. Many engineering issues come to fore when implementing sorting algorithms.

Hsiu-Hui Lee


Sorting algorithm

Sorting algorithm

  • Insertion sort :

    • In place: only a constant number of elements of the input array are even sorted outside the array.

  • Merge sort :

    • not in place.

  • Heap sort : (chapter 6)

    • Sorts n numbers in place in O(n lgn)

Hsiu-Hui Lee


Sorting algorithm1

Sorting algorithm

  • Quick sort : (chapter 7)

    • worst time complexity O(n2)

    • Average time complexity O(n lg n)

  • Decision tree model : (chapter 8)

    • Lower bound O (n lg n)

    • Counting sort

    • Radix sort

  • Order statistics

Hsiu-Hui Lee


6 1 heaps binary heap

6.1 Heaps (Binary heap)

  • Thebinary heap data structure is an array object that can be viewed as a complete tree.

Parent(i)

return 

Left(i)

return 2i 

Right(i)

return2i+1

Hsiu-Hui Lee


Heap property

Heap property

  • Max-heap : A[Parent(i)] A[i]

  • Min-heap : A[Parent(i)]≤ A[i]

  • The height of a node in a tree: the number of edges on the longest simple downward path from the node to a leaf.

  • The height of a tree: the height of the root

  • Theheight of a heap: O(lg n).

Hsiu-Hui Lee


Basic procedures on heap

Basic procedures on heap

  • Max-Heapify

  • Build-Max-Heap

  • Heapsort

  • Max-Heap-Insert

  • Heap-Extract-Max

  • Heap-Increase-Key

  • Heap-Maximum

Hsiu-Hui Lee


6 2 maintaining the heap property

6.2 Maintaining the heap property

Max-Heapify(A, i)

  • To maintain the max-heap property.

  • Assume that the binary trees rooted at Left(i) and Right(i) are heaps, but that A[i] may be smaller than its children, thus violating the heap property.

Hsiu-Hui Lee


Chapter 6

Max-Heapify(A,i)

1 l Left(i)

2 r  Right(i)

3 if l ≤ heap-size[A] and A[l] > A[i]

4 then largest  l

5 else largest  i

6 if r ≤ heap-size[A] and A[r] > A[largest]

7then largest  r

8 if largest  i

9 then exchange A[i]  A[largest]

10 Max-Heapify(A, largest)

Alternatively O(h) (h: height)

Hsiu-Hui Lee


Max heapify a 2 heap size a 10

Max-Heapify(A,2)heap-size[A] = 10

Max-Heapify(A,2)

-> Max-Heapify(A,4)

->Max-Heapify(A,9)

Hsiu-Hui Lee


6 3 building a heap

6.3 Building a Heap

Build-Max-Heap(A)

  • To produce a max-heap from an unordered input array.

    Build-Max-Heap(A)

    1 heap-size[A]  length[A]

    2 fori  length[A]/2 downto 1

    3 do Max-Heapify(A, i)

Hsiu-Hui Lee


Build max heap a heap size a 10

Build-Max-Heap(A)heap-size[A] = 10

i=5

Max-Heapify(A, i)

i=4

Max-Heapify(A, i)

Hsiu-Hui Lee


Chapter 6

i=3

Max-Heapify(A, i)

i=2

Max-Heapify(A, i)

Hsiu-Hui Lee


Chapter 6

i=1

Max-Heapify(A, i)

Hsiu-Hui Lee


Chapter 6

Hsiu-Hui Lee


6 4 heapsort algorithm

6.4 Heapsort algorithm

Heapsort(A)

  • To sort an array in place.

    Heapsort(A)

    1 Build-Max-Heap(A)

    2 fori length[A] down to 2

    3 do exchange A[1]A[i]

    4 heap-size[A]  heap-size[A]-1

    5 Max-Heapify(A,1)

Hsiu-Hui Lee


Operation of heapsort

Operation of Heapsort

Hsiu-Hui Lee


Chapter 6

Analysis: O(n lg n)

Hsiu-Hui Lee


6 5 priority queues

6.5 Priority Queues

  • A priority queue is a data structure that maintain a set S of elements, each with an associated value call a key.

  • A max-priority queue support the following operations:

    Insert(S, x) O(lg n)

    inserts the element x into the set S.

    Maximum(S) O(1)

    returns the element of S with the largest key.

Hsiu-Hui Lee


Chapter 6

Extract-Max(S)O(lg n)

removes and returns the element of S with the largest key.

Increase-Key(S,x,k) O(lg n)

increases the value of element x’s key to the new value k, where k x’s current key value

Hsiu-Hui Lee


Heap maximum

Heap-Maximum

Heap-Maximum(A)

1 returnA[1]

Hsiu-Hui Lee


Heap extract max

Heap_Extract-Max

Heap_Extract-Max(A)

1 if heap-size[A] < 1

2 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

Hsiu-Hui Lee


Heap increase key

Heap-Increase-Key

Heap-Increase-Key (A, i, key)

1 if key < A[i]

  • then error “new key is smaller than

    current key”

    3 A[i]  key

    4 whilei > 1 and A[Parent(i)] < A[i]

    5 do exchange A[i]  A[Parent(i)]

    6 i  Parent(i)

Hsiu-Hui Lee


Heap increase key a i 15

Heap-Increase-Key(A,i,15)

Hsiu-Hui Lee


Heap insert

Heap_Insert

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)

Hsiu-Hui Lee


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