The document provides a detailed overview of the merge sort algorithm and queue data structure as part of a data structures and algorithms course. It explains the divide and conquer method used in merge sort, including the algorithm's implementation and analysis, achieving a time complexity of O(n log n). Additionally, it discusses the queue data structure, its operations, implementation using arrays and linked lists, and applications in various contexts.

1. CS261
DATA STRUCTURES & ALGORITHMS
(WEEK-8)
LECTURE-15 & 16
INTRODUCTION TO DATA STRUCTURES &
ALGORITHMS
Lecturer
Azka Aziz
Azka.a@scocs.edu.pk

2. Data structures & Algorithms
Lecture#15 Merge Sort

3. Lecture Contents
Divide & Conquer
Divide & Conquer ….. Application (Merge Sort)
mergeSort()
Merge()
Analysis

4. Divide & Conquer
Idea
Distribute a larger problem into many small similar problems
Solve every smaller problem recursively
Combine results of smaller solutions to form the overall solution
Usage
A typical idea be to split a list of n items into two lists of n/2 size and try
to sort them
Consequence
Running time of sorting can be improved from O(n2) to O(n log n)

5. Divide & Conquer ….. Application (Merge
Sort)
Divide and conquer strategy is applicable in a huge number of computational
problems.
The first application of divide and conquer strategy is a simple and efficient
sorting procedure called Merge Sort
Input: a list of n numbers is provided as an array A[1…n]
Output: a permutation of this sequence sorted in increasing order
Method: Split Recursively, Sort, Merge

6. Merge Sort Algorithm
proc mergeSort(arr[1…n], p, r)
if p < r then
q  (p+r)/2
mergeSort(arr, p, q)
mergeSort(arr, q+1, r)
merge(arr, p, q, r)
end if
end proc
Every pass of mergeSort()
procedure consists of two
recursive function calls followed by
one call to merge

7. Merge Sort Algorithm
proc mergeSort(arr[1…n], p, r)
if p < r then
q  (p+r)/2
mergeSort(arr, p, q)
mergeSort(arr, q+1, r)
merge(arr, p, q, r)
end if
end proc
mergeSort(arr, p, r) function checks
whether p (index) is less than r (index)
If it is so, it makes two calls to itself
splitting the array into two halves
Every further call splits array into further
two halves with available indexes until p is
not less than r (when will this happen?)

8. Merge Sort Algorithm
proc mergeSort(arr[1…n], p, r)
if p < r then
q  (p+r)/2
mergeSort(arr, p, q)
mergeSort(arr, q+1, r)
merge(arr, p, q, r)
end if
end proc
Every further call splits array into further
two halves with available indexes until p is
not less than r (when will this happen?)
At this point a merge(arr, p, q, r) call will
me made to merge two halves into a
sorted chunk

9. mergeSort() calls … Visually

10. merge() call
merge(arr, p, q, r) assumes that two halves of array [p….q], [q+1….r]
are sorted (Why is it? We have not sorted anything yet? Is there
some statement in mergeSort() procedure that does the sorting
part?)
proc mergeSort(arr[1…n], p, r)
if p < r then
q  (p+r)/2
mergeSort(arr, p, q)
mergeSort(arr, q+1, r)
merge(arr, p, q, r)
end if
end proc

11. mergeSort() calls … Visually
We have not sorted
anything yet?
Is there some
statement in
mergeSort()
procedure that does
the sorting part?
We just relied on a simple fact that splitting an array
into equal halves will result into smaller and smaller
arrays till there is only one element in array
Is single element array sorted? Always?

12. merge(arr, p, q, r) calls …
Every merge() call receives three parameters (except the data
array) i.e. p, q, r
Values between indexes p….q and q+1……r are sorted
(independent of each other)
Having two sorted sub-arrays, merging part is easier:::

13. merge(arr, p, q, r) calls …
proc merge(arr, p, q, r)
ip, kp, jq+1
temp[p…r] // using a temporary array
while i ≤ q and j ≤ r do
if arr[i] < arr[j] then
temp[k++]  arr[i++]
else
temp[k++]  arr[j++]
end if
next

14. merge(arr, p, q, r) calls … continued
while i ≤ q do
temp[k++]  arr[i++]
next
while j ≤ r do
temp[k++]  arr[j++]
next
for i p to r do
arr[i]  temp[i]
next
end proc

15. merge() calls … Visually

16. Merge Sort Analysis
First we analyze merge() procedure
First let us consider the running
time of the procedure merge(A, p, q,
r). Let n = r - p + 1 denote the total
length of both the left and right sub-
arrays. What is the running time of
Merge as a function of n?

17. Merge Sort Analysis
Cost of mergeSort(with n elements) = cost of mergeSort(n/2 elements)
+ cost of mergeSort(n/2 elements) + cost of merge(n elements)
MS(n) = MS(n/2) + MS(n/2)+M(n)
MS(n) = 2MS(n/2) + M(n)

18. Merge Sort Analysis
T(1) = 1
T(2) = T(1) + T(1) + 2 = 1 + 1 + 2 = 4
T(3) = T(2) + T(1) + 3 = 4 + 1 + 3 = 8
T(4) = T(2) + T(2) + 4 = 8 + 8 + 4 = 12
T(5) = T(3) + T(2) + 5 = 8 + 4 + 5 = 17
. . .
T(8) = T(4) + T(4) + 8 = 12 + 12 + 8 = 32
. . .
T(16) = T(8) + T(8) + 16 = 32 + 32 + 16 =
80
T(32) = T(16) + T(16) + 32 = 80 + 80+ 32
= 192

19. Merge Sort Analysis T(1) = 1
T(2) = T(1) + T(1) + 2 = 1 + 1 + 2 = 4
T(3) = T(2) + T(1) + 3 = 4 + 1 + 3 = 8
T(4) = T(2) + T(2) + 4 = 8 + 8 + 4 = 12
T(5) = T(3) + T(2) + 5 = 8 + 4 + 5 = 17
. . .
T(8) = T(4) + T(4) + 8 = 12 + 12 + 8 = 32
. . .
T(16) = T(8) + T(8) + 16 = 32 + 32 + 16 =
80
T(32) = T(16) + T(16) + 32 = 80 + 80+ 32
= 192
Is there some pattern?
Let’s try
T(n)/n = log n + 1
T(n) = n log n + n
O(n log n)

20. Merge Sort Analysis … Recurrence Relation
T(n) = 2 T(n/2) + n
Expanding we get
T(n) = 2 (2T(n/4)+n/2)+n ….. ? How?
= 4 T(n/4) + n + n
= 8 T(n/8) + n + n + n
= 16 T(n/16) + n + n + n + n
= …………….
Now to solve it assuming that n = 2k,
T(n) = 2k T(n/2k) + (n+n+n+n k items)
T(n) = 2k T(n/2k) + kn

21. Merge Sort Analysis … Recurrence Relation
T(n) = 2 T(n/2) + n
Expanding we get
T(n) = 2 (2T(n/4)+n/2)+n
= 4 T(n/4) + n + n
= 8 T(n/8) + n + n + n
= 16 T(n/16) + n + n +
n + n
= …………….
Now to solve it assuming that n =
2k,
T(n) = 2k T(n/2k) + (n+n+n+n k
Since n = 2k, i.e. k = log n
T(n) = 2log n T(n/2log n) + (log n)n
T(n) = n T(1) + n log n
T(n) = n + n log n
i.e. T(n) = O (n)

22. Merge Sort Analysis … Recurrence Relation

23. Data structures & Algorithms
Lecture#16 Queue

24. Queue Data Structure
Queue is a data structure that is used to store data items.
Data items are stored in Queue one by one. Stored data
items when removed from Queue follow First In First Out
order.

25. Queue
Queue maintains two control variables i. rear ii. front
Data items are removed from Queue (deque() operation) using control
variable ‘rear’
Data items are added to the Queue (enque() operation) using control
variable ‘front’
Data item being enqued to Queue earliest will be removed from Queue
earliest following First In First Out order (FIFO)
Queue can be implemented using an Array as well as a Linked List

26. Queue Operations …
Operation Description Pseudocode
Queue() Initialize ‘rear’ and ‘front’
with -1
 Assign -1 to ‘rear’
 Assign -1 to ‘front’
int is_empty() Returns TRUE if Queue is
empty
Returns FALSE if Queue is
not empty
 If ‘rear’ is less than ‘front’
 Return FALSE
 Otherwise
 Return TRUE
int is_full() Returns TRUE if Queue is
empty
Returns FALSE if Queue is
not empty
 If ‘rear’ is equal to -1 AND ‘front’ is equal to MAX-1
then
 Return TRUE
 Otherwise
 Return FALSE

27. Queue Operations …
Operation Description Pseudocode
int deque() Removes one item from
queue using ‘rear’
 If Queue is empty
 It returns -1
 Otherwise
 Increment ‘rear’ by 1
 Return value at current location of ‘rear’
void refresh() Relocates the queue
elements so that empty
positions are available post
‘front’ pointer
 All elements are relocated to the smaller index
positions making empty positions available post the
‘front’ locations
 Assign -1 to ‘rear’ and value of (‘front’-’rear’-1) to
‘front’
void enque() Added an new data item to
the Queue using ‘front’
pointer
 If Queue is full, Nothing can be added
 Else if ‘front’ is equal to MAX-1 then
 Refresh is called, ‘front’ is incremented by 1,
New element is added at ‘front’ location
 Otherwise
 ‘front’ is incremented by 1, New element is
added at ‘front’ location

28. Queue using array … implementation
Implementation details and issues have been discussed during
class

29. Queue ADT … (implementing with linked list)

30. Queue ADT … (implementing with linked list)
QueueNode *Front;
QueueNode *Rear;
Queue();
int is_empty();
void enque(int);
QueueNode* deque();

31. Queue Operations …
Operation Description Pseudocode
Queue() Initialize ‘rear’ and ‘front’ with
NULL
 Assign NULL to ‘rear’
 Assign NULL to ‘front’
int is_empty() Returns TRUE if Queue is
empty
Returns FALSE if Queue is not
empty
 If ‘front’ is equal to NULL
 Return TRUE
 Otherwise
 Return FALSE
QueueNode* deque() Removes one item from queue
using ‘rear’
 If Queue is empty
 It returns NULL
 Else if ‘rear’ and ‘front’ point to same node
 Assign the node to QueueNode pointer ‘temp’
 Assign NULL to ‘rear’ and ‘front’
 Return ‘temp’
 Else
 Assign QueueNode pointed by ‘rear’ to ‘temp’
 Assign ‘prev’ of ‘rear’ to ‘rear’
 Set NULL to ‘next’ of ‘rear’
 Return ‘temp’

32. Queue Operations …
Operation Description Pseudocode
void enque(int) Added an new data item to
the Queue using ‘front’
pointer
 Create new node using the parameter of
function and point it by ‘temp’
 If Queue is empty
 Assign ‘temp’ to ‘front’
 Assign ‘temp’ to ‘rear’
 Else
 Assign ‘front’ to ‘next’ of ‘temp’
 Assign ‘temp’ to ‘prev’ of ‘front’
 Assign ‘temp’ to ‘front’

33. Queue using array … implementation
Implementation details and issues have been discussed during
class

34. Queue applications
Level order tree traversal
Any job management software following FIFO order
Operating System maintains a queue of processes that are ready to
execute
Many computer based systems maintains a queue of messages to be
communicated among their components
Queue of people at any service point
Queue of packets in data communication
Queue of planes waiting for landing