1. Design and Analysis of Algorithms
Dr. Maheswari Karthikeyan 25/02/2012, Lecture5

2. Quick Sort
Sorting in linear time

4. Sort an array A[p…r] Divide
A[p…q]
A[q+1…r] ≤
Sort an array A[p…r]
Divide
Partition the array A into 2 subarrays A[p..q] and A[q+1..r], such that each element of A[p..q] is smaller than or equal to each element in A[q+1..r]
Need to find index q to partition the array

5. Conquer Combine A[p…q] A[q+1…r] ≤
Recursively sort A[p..q] and A[q+1..r] using Quicksort
Combine
Trivial: the arrays are sorted in place
No additional work is required to combine them
The entire array is now sorted

6. Alg.: QUICKSORT(A, p, r) if p < r then q  PARTITION(A, p, r) QUICKSORT (A, p, q-1) QUICKSORT (A, q+1, r)
Initially: p=1, r=n
Recurrence:
T(n) = T(q) + T(n – q) + f(n)

7. PARTITION (A,p,r)
x ← A[r]
i ← p-1
for j ← p to r-1
do if A[j] <= x
then i ← i +1
exchange A[i] <-> A[j]
exchange A[i+1] <-> A[r]
return i+1

8. p j r i 23 6 45 9 11 34 12 5

9. p j r i 23 6 45 9 11 34 12 5 r p j i 23 6 45 9 11 34 12 5

10. p j r i 23 6 15 9 11 4 12 5 r p j i 23 6 15 9 11 4 12 5 p j r i 23 6 15 9 11 4 12 5

11. p j r i 23 6 15 9 11 4 12 5 r p j i 23 6 15 9 11 4 12 5 p j r i 23 6 15 9 11 4 12 5 p j r i 23 6 15 9 11 4 12 5

12. p j r i 23 6 15 9 11 4 12 5

13. p j r i 23 6 15 9 11 4 12 5 j p r i 23 6 15 9 11 4 12 5

14. p j r i 23 6 15 9 11 4 12 5 j p r i 23 6 15 9 11 4 12 5 j p i r 23 6 15 9 11 4 12 5

15. p j r i 23 6 15 9 11 4 12 5 j p r i 23 6 15 9 11 4 12 5 j p i r 23 6 15 9 11 4 12 5 j p i r 4 6 15 9 11 23 12 5

16. Quicksort( A,1, 1) Quicksort( A,3,8)p i j r 4 6 15 9 11 23 12 5 p i q j r 4 5 15 9 11 23 12 6
Quicksort( A,1, 1) Quicksort( A,3,8)

17. Sort the array 2,10,3,15,6,9,4,13,25,11

18. Sort the array 2,10,3,15,6,9,4,13,25,11 2 10 3 15 6 9 4 13 25 11

19. Sort the array 2,10,3,15,6,9,4,13,25,11 2 10 3 15 6 9 4 13 25 11 2 10 3 6 9 4 11 13 25 15 2 10 3 6 9 4

20. Sort the array 2,10,3,15,6,9,4,13,25,11 2 3 4 6 9 10 6 9 10 2 3 2 3 4 6 9 10

21. Sort the array 2,10,3,15,6,9,4,13,25,11 2 3 4 6 9 10 11 13 25 15 13 15 25 2 3 4 6 9 10 11 13 15 25

22. Worst Case Partitioning2 1 3
(n2)
Worst-case partitioning
One region has one element and the other has n – 1 elements
Maximally unbalanced
Recurrence: q=1
T(n) = T(1) + T(n – 1) + n,
T(1) = (1)
T(n) = T(n – 1) + n =

23. Best Case Partitioning
Partitioning produces two regions of size n/2
Recurrence: q=n/2
T(n) = 2T(n/2) + (n)
T(n) = (nlgn) (Master theorem)

24. Binary Tree ADT Tree: A directed, acyclic structure of linked nodes.
directed : Has one-way links between nodes.
acyclic : No path wraps back around to the same node twice.
Binary tree: One where each node has at most two children.
Recursive definition: A tree is either:
empty (null), or
a root node that contains:
data,
a left subtree, and
a right subtree.
(The left and/or right subtree could be empty.) 7 6 3 2 1 5 4
root

25. Binary Tree ADT Applications of Binary Tree
Expression Tree ( Parse Tree in Compilers)
Decision trees (AI)
Huffman Coding Tree

26. Types of Binary Tree
Full - Every node has exactly two children in all levels, except the last level. Nodes in last level have 0 children Complete - Full up to second last level and last level is filled from left to right Other - not full or complete

27. Types of Binary Tree Full, Complete or other? not binary A C D F I E BG H T M L O K J Q N
not binary

28. Types of Binary Tree A D F I B R S P G H T M L O K N

29. Types of Binary Tree A D F I B Q S R G H T M L P K N O

30. Types of Binary Tree A D F I B Q R G H M L P K N O

31. Terminology
node: an object containing a data value and left/right children
root: topmost node of a tree
leaf: a node that has no children
branch: any internal node; neither the root nor a leaf
parent: a node that refers to this one
child: a node that this node refers to
sibling: a node with a common parent
subtree: the smaller tree of nodes on the left or right of the current node
height: length of the longest path from the root to any node
level or depth: length of the path from a root to a given node 7 6 3 2 1 5 4
root
height = 3
level 1
level 2
level 3

32. Binary Tree Traversal
Traversal: An examination of the elements of a tree.
A pattern used in many tree algorithms and methods
Preorder Traversal
Each node is processed before any node in either of its subtrees
Inorder Traversal
Each node is processed after all nodes in its left subtree and before any node in its right subtree
Postorder Traversal
Each node is processed after all nodes in both of its subtrees

33. Preorder Traversal A D F I B R S P G H T M L O K N 1 2 11 3 8 12 13 14
Visit the root
Visit Left subtree
Visit Right subtree 2 11 3 8 12 13 14 4 5 9 10 6 7 15 16

34. Inorder Traversal A D F I B R S P G H T M L O K N 10 1 1 6 12 1 2 8 11
Visit Left subtree
Visit the root
Visit Right subtree 1 6 12 1 2 8 11 16 14 1 4 7 9 3 5 13 15

35. Postorder Traversal A D F I B R S P G H T M L O K N 16 9 15 2 5 8 10
Visit Left subtree
Visit Right subtree
Visit the root 9 15 2 5 8 10 14 13 1 3 4 2 6 7 2 3 11 12

36. Represent ion of Binary Tree ADT
A binary tree can represented using
- Linked List
- Array
Note : Array is suitable only for full and complete binary trees

37. Height and the number of nodes in a Binary Tree
If the height of a binary tree is h then the maximum number of nodes in the tree is 2h -1
If the number of nodes in a complete binary tree is n, then
2h - 1 = n
2h = n + 1
h = log(n+1) O(logn)