Heap sort

1. 12/08/2006 CMSC 131 Fall 2006
Rance Cleaveland
©2006 Univeristy of Maryland
Lecture 41:
Heapsort
Last time:
1. Analyzing selection sort
2. Trees
Today:
1. Project #8 assigned!
2. Heaps
3. Heapsort

2. CMSC 131 Fall 2006
Rance Cleaveland
©2006 University of Maryland
1
Recall: Binary Trees
Data structures consisting of
Nodes (where data it stored)
Edges
Every node has one parent except
root, which has none
Every node has ≤ two children
Nodes with no children = leaves
Nodes with children = internal
Nodes can be divided into levels
based on distance from root
The height of a tree is the longest
path from root
A binary tree is complete if:
Every level except the last is full
In the last level, every leaf is as far to
the left as possible
A binary tree is perfect if every level
(including the last) is full
‘a’
‘y’ ‘J’
‘3’‘c’

3. CMSC 131 Fall 2006
Rance Cleaveland
©2006 University of Maryland
2
The Tree Quiz
1. Let T be a perfect tree of height n
a. How many leaves does T have?
2n
b. How many internal nodes does T have?
2n - 1
c. How many total nodes does T have?
2n+1 - 1
2. Let T be a complete binary tree of
height n
a. How many leaves does T have (give
range)?
2n-1 to 2n, inclusive
b. How many total nodes may T have (give
range)?
2n to 2n+1 - 1, inclusive
3. Let T be a complete binary tree with n
nodes. What is its height?
Floor (rounding down) of log2n
‘a’
‘y’ ‘J’
‘3’‘c’
‘z’ ‘:’

4. CMSC 131 Fall 2006
Rance Cleaveland
©2006 University of Maryland
3
Heaps
Assume data stored in
tree nodes is ordered
by ≤
A heap is:
A complete binary tree
such that:
Every parent’s data is ≥
its child(ren)’s data
Fact: greatest value is
stored at root!
17
14 11
1015

5. CMSC 131 Fall 2006
Rance Cleaveland
©2006 University of Maryland
4
Heapsort
Idea:
Keep unsorted elements in a heap
Keep sorted elements in array, filling in
from right (largest) to left (smallest)
Maintain following property:
Every unsorted element ≤ every sorted
element (cf. selection sort)
To increase sorted part:
Remove root (largest element) from heap
and put into correct position in array
Restore remaining heap elements into a
heap
Questions
How do you “restore” a heap?
How do you create a heap in the first
place?
17
14 11
1015
Unsorted elements
242219
Sorted elements

6. CMSC 131 Fall 2006
Rance Cleaveland
©2006 University of Maryland
5
Restoring a Heap
After removing root of a
heap …
… how do we
rearrange the
remainder into a heap?
One approach:
Move bottom-most, right-
most leaf to root
Repeatedly swap node
with greater of children
until heap property
restored
17
14 11
1015

7. CMSC 131 Fall 2006
Rance Cleaveland
©2006 University of Maryland
6
Example
11
14
1015
14 11
1015
15
14
1011
15
11
1014

8. CMSC 131 Fall 2006
Rance Cleaveland
©2006 University of Maryland
7
What Is Complexity of
Restoring a Heap?
Each swap operation takes O(1)
Swap always results in level of “out-of-place”
node increasing
How many times can level increase, in terms
of n = # of nodes in tree?
log2n!
So complexity of restoration operation is
O(log2n)

9. CMSC 131 Fall 2006
Rance Cleaveland
©2006 University of Maryland
8
Creating a Heap
To create a heap, we repeatedly insert
elements from original array into heap
How to insert a new node into a heap?
Put new node into heap as bottom-most, right-
most leaf
Swap node with parent until heap property
restored

10. CMSC 131 Fall 2006
Rance Cleaveland
©2006 University of Maryland
9
Example: Inserting 13
17
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1015
13
17
11
1315
1014

11. CMSC 131 Fall 2006
Rance Cleaveland
©2006 University of Maryland
10
Complexity of Insertion
Each swap is O(1)
Swap decreases level of new node by one
So complexity of insertion in terms of n = # of
nodes in heap is:
O(log2n)

12. CMSC 131 Fall 2006
Rance Cleaveland
©2006 University of Maryland
11
Complexity of Heap Creation
Recall heap-creation procedure (a is array to
be sorted)
start with empty heap h
for i = 0, …, a.length
insert a[i] into h
Complexity analysis
Insertion called for each element in a (so, n times)
Each insertion costs O(log2n)
So complexity of heap creation is O(n log2n)

13. CMSC 131 Fall 2006
Rance Cleaveland
©2006 University of Maryland
12
Complexity of Heapsort
Here is heapsort pseudo-code for sorting array a
Create heap h from a
for j = a.length-1, …, 0 do
remove root from h and assign to a[j]
restore heap
Complexity analysis
Heap creation: O(n log2n)
Each loop iteration: O(log2n)
Number of iterations: n
Total: O(n log2n)!

14. CMSC 131 Fall 2006
Rance Cleaveland
©2006 University of Maryland
13
0
1,000
2,000
3,000
4,000
5,000
6,000
7,000
0 8 16 24 32 40 48 56 64 72 80
Selection sort
Heapsort
Comparing Sorts
# of elements in array
runningtime

15. CMSC 131 Fall 2006
Rance Cleaveland
©2006 University of Maryland
14
How To Implement Heaps?
The heap can also be stored
in a, along with sorted portion
Root is stored in a[0]
Level 1 is stored in a[1], a[2]
Level 2 is stored in a[3] – a[6]
etc.
Using this scheme:
Children of node stored in a[i]
are a[2i+1], a[2i+2]
Parent of node stored in a[i] is
a[(i-1) / 2]
17
14 11
1015
242219
2422191114101517
heap sorted