1. Data Structures

2. DATA STRUCTURES
The logical or mathematical model of a particular organization of data is called a data structure

3. A primitive data type holds a single piece of data
DATA STRUCTURES
A primitive data type holds a single piece of data
e.g. in Java: int, long, char, boolean etc.
Legal operations on integers: * / ...
A data structure structures data!
Usually more than one piece of data
Should provide legal operations on the data
The data might be joined together (e.g. in an array): a collection

4. Static vs. Dynamic Structures
A static data structure has a fixed size
This meaning is different than those associated with the static modifier
Arrays are static; once you define the number of elements it can hold, it doesn’t change
A dynamic data structure grows and shrinks as required by the information it contains

5. Abstract Data Type
An Abstract Data Type (ADT) is a data type together with the operations, whose properties are specified independently of any particular implementation.

6. In computing, we view data from three perspectives:
Abstract Data Type
In computing, we view data from three perspectives:
Application level
View of the data within a particular problem
Logical level
An abstract view of the data values (the domain) and the set of operations to manipulate them
Implementation level
A specific representation of the structure to hold the data items and the coding of the operations in a programming language

7. Problem Solving: Main Steps
Problem definition
Algorithm design / Algorithm specification
Algorithm analysis
Implementation
Testing
[Maintenance]

8. Problem Definition What is the task to be accomplished?
Calculate the average of the grades for a given student
What are the time / space / speed / performance requirements?

9. . Algorithm Design / Specifications
Algorithm: Finite set of instructions that, if followed, accomplishes a particular task.
Describe: in natural language / pseudo-code / diagrams / etc.
Criteria to follow:
Input: Zero or more quantities (externally produced)
Output: One or more quantities
Definiteness: Clarity, precision of each instruction
Finiteness: The algorithm has to stop after a finite (may be very large) number of steps
Effectiveness: Each instruction has to be basic enough and feasible

10. Implementation, Testing, Maintenances
Decide on the programming language to use
C, C++, Lisp, Java, Perl, Prolog, assembly, etc. , etc.
Write clean, well documented code
Test, test, test
Integrate feedback from users, fix bugs, ensure compatibility across different versions  Maintenance

11. Algorithm Analysis Space complexity How much space is required
Time complexity
How much time does it take to run the algorithm
Often, we deal with estimates!

12. Space Complexity
Space complexity = The amount of memory required by an algorithm to run to completion
[Core dumps = the most often encountered cause is “memory leaks” – the amount of memory required larger than the memory available on a given system]
Some algorithms may be more efficient if data completely loaded into memory
Need to look also at system limitations
E.g. Classify 2GB of text in various categories [politics, tourism, sport, natural disasters, etc.] – can I afford to load the entire collection?

13. Space Complexity (cont’d)
Fixed part: The size required to store certain data/variables, that is independent of the size of the problem:
- e.g. name of the data collection
- same size for classifying 2GB or 1MB of texts
Variable part: Space needed by variables, whose size is dependent on the size of the problem:
- e.g. actual text
- load 2GB of text VS. load 1MB of text

14. Space Complexity (cont’d)
S(P) = c + S(instance characteristics)
c = constant
Example:
float sum (float* a, int n) {
float s = 0;
for(int i = 0; i<n; i++) {
s+ = a[i]; }
return s;
Space? one word for n, one for a [passed by reference!], one for i  constant space!

15. Time Complexity Often more important than space complexity
space available (for computer programs!) tends to be larger and larger
time is still a problem for all of us
3-4GHz processors on the market
researchers estimate that the computation of various transformations for 1 single DNA chain for one single protein on 1 TerraHZ computer would take about 1 year to run to completion
Algorithms running time is an important issue

16. Running Time Problem: prefix averages Given an array X
Compute the array A such that A[i] is the average of elements X[0] … X[i], for i=0..n-1
Sol 1
At each step i, compute the element X[i] by traversing the array A and determining the sum of its elements, respectively the average
Sol 2
At each step i update a sum of the elements in the array A
Compute the element X[i] as sum/I

17. Running time
Suppose the program includes an if-then statement that may execute or not:  variable running time
Typically algorithms are measured by their worst case

18. Experimental Approach
Write a program that implements the algorithm
Run the program with data sets of varying size.
Determine the actual running time using a system call to measure time (e.g. system (date) );
Problems?

19. Experimental Approach
It is necessary to implement and test the algorithm in order to determine its running time.
Experiments can be done only on a limited set of inputs, and may not be indicative of the running time for other inputs.
The same hardware and software should be used in order to compare two algorithms. – condition very hard to achieve!

20. Use a Theoretical Approach
Based on high-level description of the algorithms, rather than language dependent implementations
Makes possible an evaluation of the algorithms that is independent of the hardware and software environments

21. Algorithm Description
How to describe algorithms independent of a programming language
Pseudo-Code = a description of an algorithm that is
more structured than usual prose but
less formal than a programming language
(Or diagrams)
Example: find the maximum element of an array.
Algorithm arrayMax(A, n):
Input: An array A storing n integers.
Output: The maximum element in A.
currentMax  A[0]
for i 1 to n -1 do
if currentMax < A[i] then currentMax  A[i]
return currentMax

22. Properties of Big-Oh Expressions: use standard mathematical symbols
use  for assignment ( ? in C/C++)
use = for the equality relationship (? in C/C++)
Method Declarations: Algorithm name(param1, param2)
Programming Constructs:
decision structures: if ... then ... [else ..]
while-loops while ... do
repeat-loops: repeat ... until ...
for-loop: for ... do
array indexing: A[i]
Methods
calls: object method(args)
returns: return value
Use comments
Instructions have to be basic enough and feasible!

23. Asymptotic analysis - terminology
Special classes of algorithms:
logarithmic: O(log n)
linear: O(n)
quadratic: O(n2)
polynomial: O(nk), k ≥ 1
exponential: O(an), n > 1
Polynomial vs. exponential ?
Logarithmic vs. polynomial ?

24. Some Numbers

25. Relatives of Big-Oh “Relatives” of the Big-Oh
 (f(n)): Big Omega – asymptotic lower bound
 (f(n)): Big Theta – asymptotic tight bound
Big-Omega – think of it as the inverse of O(n)
g(n) is  (f(n)) if f(n) is O(g(n))
Big-Theta – combine both Big-Oh and Big-Omega
f(n) is  (g(n)) if f(n) is O(g(n)) and g(n) is  (f(n))
Make the difference:
3n+3 is O(n) and is  (n)
3n+3 is O(n2) but is not  (n2)

26. More “relatives”
Little-oh – f(n) is o(g(n)) if for any c>0 there is n0 such that f(n) < c(g(n)) for n > n0.
Little-omega
Little-theta
2n+3 is o(n2)
2n + 3 is o(n) ?

27. Example Remember the algorithm for computing prefix averages
compute an array A starting with an array X
every element A[i] is the average of all elements X[j] with j < i
Remember some pseudo-code … Solution 1
Algorithm prefixAverages1(X):
Input: An n-element array X of numbers.
Output: An n -element array A of numbers such that A[i] is the average of elements X[0], ... , X[i].
Let A be an array of n numbers.
for i 0 to n - 1 do
a  0
for j  0 to i do
a  a + X[j]
A[i]  a/(i+ 1)
return array A

28. Example (cont’d) Algorithm prefixAverages2(X):
Input: An n-element array X of numbers.
Output: An n -element array A of numbers such that A[i] is the average of elements X[0], ... , X[i].
Let A be an array of n numbers.
s 0
for i  0 to n do
s  s + X[i]
A[i]  s/(i+ 1)
return array A

29. Back to the original question
Which solution would you choose?
O(n2) vs. O(n)
Some math …
properties of logarithms:
logb(xy) = logbx + logby
logb (x/y) = logbx - logby
logbxa = alogbx
logba= logxa/logxb
properties of exponentials:
a(b+c) = aba c
abc = (ab)c
ab /ac = a(b-c)
b = a logab
bc = a c*logab

30. Important Series Sum of squares: Sum of exponents: Geometric series:
Special case when A = 2
… + 2N = 2N+1 - 1

31. Analyzing recursive algorithms
function foo (param A, param B) {
statement 1;
statement 2;
if (termination condition) {
return;
foo(A’, B’); }

32. Solving recursive equations by repeated substitution
T(n) = T(n/2) + c substitute for T(n/2)
= T(n/4) + c + c substitute for T(n/4)
= T(n/8) + c + c + c
= T(n/23) + 3c in more compact form
= …
= T(n/2k) + kc “inductive leap”
T(n) = T(n/2logn) + clogn “choose k = logn”
= T(n/n) + clogn
= T(1) + clogn = b + clogn = θ(logn)

33. Solving recursive equations by telescoping
T(n) = T(n/2) + c initial equation
T(n/2) = T(n/4) + c so this holds
T(n/4) = T(n/8) + c and this …
T(n/8) = T(n/16) + c and this … …
T(4) = T(2) + c eventually …
T(2) = T(1) + c and this …
T(n) = T(1) + clogn sum equations, canceling the terms appearing on both sides
T(n) = θ(logn)

34. RECURSION
Suppose P is a procedure containing either a CALL statement to itself or a CALL statement back to original procedure P .Then P is called a recursive procedure
Properties:
1. There must be certain criteria called basic criteria, for which the procedure does not call itself.
2. Each time the procedure does call itself (directly or indirectly), it must be closer to the base criteria.

35. FACTORIAL WITHOUT RECURSION
FACTORIAL(FACT,N)
This procedure calculates N! and return the vale in the variable FACT .
If N ==0,then :Set FACT:=1, and Return.
Set FACT:=1[Initialize FACT for loop]
Repeat for K:=1 to N
Set FACT:=K*FACT
[END of loop]
4. Return.

36. FACTORIAL WITH RECURSION
FACTORIAL(FACT,N)
This procedure calculates N! and return the vale in the variable FACT .
If N ==0,then :Set FACT:=1, and Return.
Call FACTORIAL(FACT,N-1).
3. Set FACT:=N*FACT.
4. Return.

37. FACTORIAL EXAMPLE USING RECURSION

38. FACTORIAL EXAMPLE USING RECURSION

39. FACTORIAL EXAMPLE USING RECURSION

40. FACTORIAL EXAMPLE USING RECURSION

41. FACTORIAL EXAMPLE USING RECURSION

42. FACTORIAL EXAMPLE USING RECURSION

43. FACTORIAL EXAMPLE USING RECURSION

44. FACTORIAL EXAMPLE USING RECURSION

45. FACTORIAL EXAMPLE USING RECURSION

46. FACTORIAL EXAMPLE USING RECURSION

47. FACTORIAL EXAMPLE USING RECURSION

48. Stack
A stack is a list that has addition and deletion of items only from one end.
It is like a stack of plates:
Plates can be added to the top of the stack.
Plates can be removed from the top of the stack.
This is an example of “Last in, First out”, (LIFO).
Adding an item is called “pushing” onto the stack.
Deleting an item is called “popping” off from the stack.

49. STACK OPERATION (PUSH)
PUSH(STACK,TOP,MAXSTK,ITEM)
This procedure pushes an ITEM onto a stack.
1.[Stack already filled]
If TOP== MAXSTK, then: Print:OVERFLOW, and Return.
2. Set TOP:=TOP+1.[ Increases TOP by 1]
3. Set STACK[TOP]:=ITEM. [Inserting ITEM in new TOP position]
Return.

50. STACK OPERATION (POP) POP(STACK,TOP,ITEM)
This procedure deletes the top element of STACK and assigns it to the variable ITEM .
1.[Stack has an item to be to removed]
If TOP== 0, then: Print:UNDERFLOW, and Return.
2. Set ITEM:=STACK[top].[ Assigns TOP element to ITEM ]
3. Set TOP:=TOP-1. [Decreases TOP by 1]
Return.

51. STACK EXAMPLES Implementing a stack : MAXSTK=7. Data push 43 push 23
pop
push 100

52. STACK EXAMPLES 7 6 5 4 2 1
TOP=0
MAXSTK=7

53. STACK EXAMPLES 7 6 5 4 2 1
TOP=1
MAXSTK=7
PUSH(43) 43

54. STACK EXAMPLES 7 6 5 4 2 1
TOP=2
MAXSTK=7
PUSH(23) 23 43

55. STACK EXAMPLES 7 6 5 4 2 1
TOP=3
MAXSTK=7
PUSH(53) 53 23 43

56. STACK EXAMPLES 7 6 5 4 2 1
TOP=2
MAXSTK=7
POP( ) 23 43

57. STACK EXAMPLES 7 6 5 4 2 1
TOP=0
MAXSTK=7
POP( ) 43

58. STACK EXAMPLES 7 6 5 4 2 1
TOP=0
MAXSTK=7
PUSH(100)
100 43

59. STACK EXAMPLES INFIX to POSTFIX POSTFIX expression solving
3. N-QUEEN’S problem

60. INFIX to POSTFIX
Properties while transforming infix to postfix expression
besides operands and operators, arithmetic expression contains left and right parentheses
We assume that the operators in q consist only of
1. Exponent
2. Multiplication
3. Division
4. Addition
5. Subtraction

61. INFIX to POSTFIX We have three levels of precedence
precedence operators
high right parentheses
exponent
multiplication, division
low addition, subtraction

62. INFIX to POSTFIX
POLISH(Q, P) Suppose Q is an arithmetic expression written in infix notation. This algorithm finds the equivalent postfix expression P.
Push “(“ into STACK and add “)” to the end of Q.
Scan Q from left to right and repeat Step 3 to 6 for each element of Q until the STACK is empty:
If an operand is encountered add it to P.
If a left parenthesis is encountered, push it onto STACK.
If an operator is encountered then:
(a) repeatedly pop from STACK and to P each operator (on the top of STACK) which has the same precedence as or higher precedence this operator
(b) Add operator to STACK

63. INFIX to POSTFIX 6. If a right parenthesis is encountered then:
(a) Repeatedly pop from STACK and add to P each operator( on the top of STACK) until a left parenthesis is encountered
(b) Remove the left parenthesis .[Do not add the left parenthesis to P]
7. Exit

64. INFIX to POSTFIX 7 6 5 4 2 1 INFIX Expression: 3 + 2 * 4
POSTFIX Expression: 7 6 5 4 2 1

65. INFIX to POSTFIX 7 6 5 4 2 1 INFIX Expression: + 2 * 4
POSTFIX Expression: 7 6 5 4 2 1

66. INFIX to POSTFIX 7 6 5 4 2 1 INFIX Expression: 2 * 4
POSTFIX Expression: 7 6 5 4 2 1 +

67. INFIX to POSTFIX 7 6 5 4 2 1 INFIX Expression: * 4
POSTFIX Expression: 7 6 5 4 2 1 +

68. INFIX to POSTFIX 7 6 5 4 2 1 INFIX Expression: 4
POSTFIX Expression: 7 6 5 4 2 1 * +

69. INFIX to POSTFIX 7 6 5 4 2 1 INFIX Expression:
POSTFIX Expression: 7 6 5 4 2 1 * +

70. INFIX to POSTFIX 7 6 5 4 2 1 INFIX Expression:
POSTFIX Expression: * 7 6 5 4 2 1 +

71. INFIX to POSTFIX 7 6 5 4 2 1 INFIX Expression:
POSTFIX Expression: * + 7 6 5 4 2 1

72. POSTFIX EXPRESSION SOLVING
This algorithm finds the VALUE of an arithmetic expression P written in postfix notation
Add a right parenthesis “)” at the end of P
Scan P from left to right and repeat step 3 and 4 for each element of P until the sentinel “)” is encountered.
If an operand is encountered, put it on STACK.
If an operator is uncounted then:
a. Remove the two elements of Stack, where A is the top element and B is the next top element
b. Evaluate B operator A
c. Pace the result of (b) back on STACK

73. POSTFIX EXPRESSION SOLVING
Set VALUE equal to the top element on STACK
Exit

74. POSTFIX EXPRESSION SOLVING
Push the numbers until operator comes 7 6 5 4 2 1

75. POSTFIX EXPRESSION SOLVING7 6 5 4 2 1 3

76. POSTFIX EXPRESSION SOLVING7 6 5 4 2 1 2 3

77. POSTFIX EXPRESSION SOLVING
Here we pop two time and perform
multiplication on elements (4*2) and
push the Result in to the stack 7 6 5 4 2 1 4 2 3

78. POSTFIX EXPRESSION SOLVING7 6 5 4 2 1 8 3

79. POSTFIX EXPRESSION SOLVING7 6 5 4 2 1 11

80. The N-Queens Problem
Can the queens be placed on the board so that no two queens are attacking each other in chess board

81. The N-Queens Problem
Two queens are not allowed in the same row

82. The N-Queens Problem
Two queens are not allowed in the same row, or in the same column...

83. The N-Queens Problem
Two queens are not allowed in the same row, or in the same column, or along the same diagonal.r along the same diagonal.

84. N Queens The N-Queens Problem
The number of queens, and the size of the board can vary.
N Queens
The number of queens, and the size of the board can vary.
N columns

85. The 3-Queens Problem
The program uses a stack to keep track of where each queen is placed.

86. The 3-Queens Problem
Each time the program decides to place a queen on the board, the position of the new queen is stored in a record which is placed in the stack.
ROW 1, COL 1

87. The 3-Queens Problem
We also have an integer variable to keep track of how many rows have been filled so far.
ROW 1, COL 1 1
filled

88. The 3-Queens Problem
Each time we try to place a new queen in the next row, we start by placing the queen in the first column
ROW 2, COL 1
ROW 1, COL 1 1
filled

89. The 3-Queens Problem
if there is a conflict with another queen, then we shift the new queen to the next column.
ROW 2, COL 2
ROW 1, COL 1 1
filled

90. The 3-Queens Problem
When there are no conflicts, we stop and add one to the value of filled.
ROW 2, COL 3
ROW 1, COL 1 1
filled

91. The 3-Queens Problem
When there are no conflicts, we stop and add one to the value of filled.
ROW 2, COL 3
ROW 1, COL 1 2
filled

92. The 3-Queens Problem
Let's look at the third row. The first position we try has a conflict
ROW 3, COL 1
ROW 2, COL 3
ROW 1, COL 1 2
filled

93. The 3-Queens Problem
so we shift to column 2. But another conflict arises
ROW 3, COL 2
ROW 2, COL 3
ROW 1, COL 1 2
filled

94. The 3-Queens Problem 2 and we shift to the third column.
Yet another conflict arises
ROW 3, COL 3
ROW 2, COL 3
ROW 1, COL 1 2
filled

95. The 3-Queens Problem
and we shift to column 4. There's still a conflict in column 4, so we try to shift rightward again
ROW 3, COL 4
ROW 2, COL 3
ROW 1, COL 1 2
filled

96. but there's nowhere else to go.
The 3-Queens Problem
but there's nowhere else to go.
ROW 3, COL 4
ROW 2, COL 3
ROW 1, COL 1 2
filled

97. The 3-Queens Problem 1 When we run out of room in a row:
pop the stack,
reduce filled by 1
and continue working on the previous row.
ROW 2, COL 3
ROW 1, COL 1 1
filled

98. The 3-Queens Problem
Now we continue working on row 2, shifting the queen to the right.
ROW 2, COL 4
ROW 1, COL 1 1
filled

99. The 3-Queens Problem
This position has no conflicts, so we can increase filled by 1, and move to row 3.
ROW 2, COL 4
ROW 1, COL 1 2
filled

100. The 3-Queens Problem 2 In row 3, we start again at the first column.
ROW 3, COL 1
ROW 2, COL 4
ROW 1, COL 1 2
filled

101. The 3-Queens Problem 3 In row 3, we start again at the first column.
ROW 3, COL 2
ROW 2, COL 4
ROW 1, COL 1 3
filled

102. QUEUES
A queue is a data structure that maintains a “first-in first-out” (FIFO) ordering.
In contrast, a stack maintains a “last-in first-out” (LIFO) ordering.
A queue adds new elements at the end. An element can only be removed at the front.
This is an abstraction of the “first-come first-served” practice.

103. QUEUE OPERATIONS A queue has two operations: QINSERT QDELETE
An enqueue operation adds new elements at the end of the queue or its tail. This is similar to the stack operation push; only that push now is done at the end of the array instead of at the front (or top).
A dequeue operation removes an element from the front of the array or its head.

104. QUEUE INSERTION QINSERT(rear, item)
1. IF rear == MAX_QUEUE_SIZE_1 then Print queue_full Return;
2. INFO [++rear] = item;

105. QUEUE DELETION
QDELETE(front, rear) IF front == rear then Return queue_empty( ); Return queue [++ front];

106. QUEUE Insert A Insert B delete OFFSET 1 2 3 front rear DATA OFFSET 1 21 2 3
front
rear
DATA
OFFSET 1 2 3
front
rear
Insert A
DATA A 1 1
OFFSET 1 2 3
front
rear
Insert B
DATA A B 1 2
OFFSET 1 2 3
front
rear
delete
DATA B 2 2

107. CIRCULAR QUEUE
When a new item is inserted at the rear, the to rear moves upwards.
Similarly, when an item is deleted from the queue the front arrow moves downwards.
After a few insert and delete operations the rear might reach the end of the queue and no more items can be inserted although the items from the front of the queue have been deleted and there is space in the queue.

108. CIRCULAR QUEUE
To solve this problem, queues implement wrapping around. Such queues are called Circular Queues.
Both the front and the rear wrap around to the beginning of the array when they reached the MAX size of the queue.
It is also called as “Ring buffer”.

109. CIRCULAR QUEUE INSERTION
QINSERT (QUEUE, N, FRONT, ITEM) This procedure insert an element ITEM into a queue.
[Queue already filled?]
IF ( FRONT==1 and REAR==N ) or FRONT ==REAR + 1,then: write: overflow, and Return
2.[Find new value of REAR] IF FRONT==NULL then [Queue initially empty.] Set FRONT=1 and REAR=1
ELSE IF REAR ==N then Set REAR=1
ELSE set REAR=REAR+1
[End of if structure]

110. CIRCULAR QUEUE INSERTION
Set QUEUE[REAR]=ITEM.[This inserts new element]
4. Return.

111. CIRCULAR QUEUE DELETION
QDELETE(QUEE, N, FRONT, REAR, ITEM)
This procedure deletes an element from a queue and assigns it to the variable ITEM
1.[Queue already empty]
if FRONT=NULL then write UNDERFLOW, and Return
2. Set ITEM=QUEUE[FRONT]
[Find new value of FRONT]
If FRONT =REAR then [Queue has only one element to start]
Set FRONT=NULL and REAR=NULL

112. CIRCULAR QUEUE DELETION
Else if FRONT ==N then
Set FRONT=1
Else
Set FRONT=FRONT +1
[End of if statement]
4.Return

113. CIRCULAR QUEUE DELETION
EMPTY QUEUE
[2]
[3] [2] [3] J2 J3
[1] [4] [1] [4] J1
[0] [5] [0] [5]
front = front = 0
rear = rear = 3

114. CIRCULAR QUEUE DELETION
[2] [3] [2] [3]
J J3
J J9
[1] [4][1] [4]
J J4 J7
J J5 J5
[0] [5] [0] [5]
front =0
rear = 5
front =4
rear =3

115. PRIORITY QUEUE
A priority queue is a collection of elements such that each element has been assigned a priority and such that the order in which elements are deleted and processed comes from the following rules
An element of a higher priority is processed before any elements of lower priority
Two elements with the same priority are processed according to the order in which they were added to the queue

116. ARRAY REPRESENTATION PRIORITY QUEUE
Use a separate queue for each level of priority
Each such queue will appear in its own circular array and must have its own pair of pointers FRONT and REAR
In fact, if each queue is allocated the same amount of space in two-dimensional array QUEUE can be used instead of linear array

117. DELETION ON PRIORITY QUEUE
This algorithm deletes and processes the first element in a priority queue maintained by a two-dimensional array QUEUE
1.[Find the first nonempty queue] Find the smallest k that FRONT!=NULL
2.Delete and process the front element in row K of QUEUE

118. INSERTION ON PRIORITY QUEUE
This algorithm adds an ITEM with priority number M to a priority queue maintained by a two-dimensional array QUEUE
Insert ITEM as the rear element in row M of QUEUE
Exit

119. LINKED LIST Linked list
Linear collection of self-referential class objects, called nodes
Connected by pointer links
Accessed via a pointer to the first node of the list
Subsequent nodes are accessed via the link-pointer member of the current node
Link pointer in the last node is set to null to mark the list’s end
Use a linked list instead of an array when
You have an unpredictable number of data elements
Your list needs to be sorted quickly

120. TYPES OF LINKED LIST Types of linked lists: Singly linked list
Begins with a pointer to the first node
Terminates with a null pointer
Only traversed in one direction
Circular, singly linked
Pointer in the last node points back to the first node
Doubly linked list
Two “start pointers” – first element and last element
Each node has a forward pointer and a backward pointer
Allows traversals both forwards and backwards
Circular, doubly linked list
Forward pointer of the last node points to the first node and backward pointer of the first node points to the last node

121. linked list
Start Node
A placeholder node at the beginning of the list, used to simplify list processing. It doesn’t hold any data
Tail Node
A placeholder node at the end of the list, used to simplify list processing

122. Singly linked list
Single Linked List
Consists of data elements and reference to the next Node in the linked list
First node is accessed by reference
Last node is set to NULL
Tail
Start A B C

123. SINGLE LINKED LIST INSERTION
INSFIRST(INFO, LINK, START, AVAIL, ITEM)
This algorithm inserts ITEM as the first node in the list
[OVERFLOW?] If AVAIL=NULL then :Write OVERFLOW and Exit
[Remove first node from AVIL list] Set NEW =AVAIL and AVAIL=LINK[AVAIL]
Set INFO[NEW]=ITEM [Copies new data into new node]
Set LINK[NEW]=START [new node now points to original first node]
Set START=NEW [Changes START so it points to the node]
Exit

124. SINGLE LINKED LIST INSERTION
start
current a d c s X 1

125. SINGLE LINKED LIST INSERTION
start
current a d c s X 1

126. SINGLE LINKED LIST INSERTION
start
current a d c s X 1

127. SINGLE LINKED LIST INSERTION
start
current a d c s X 1

128. SINGLE LINKED LIST INSERTION
start
current a d c s X 1

129. SINGLE LINKED LIST INSERTION
INSLOC(INFO, LINK,START, AVAIL, LOC, ITEM) This algorithm inserts ITEM so that ITEM follows the node with location LOC or inserts ITEM as the first node when LOC=NULL
[OVERFLOW ?] If AVAIL==NULL then write OVERFLOW and EXIT
[Remove first node from AVAIL list]
Set INFO[NEW]=ITEM[copies new data into new node]
If LOC==NULL then :[insert as first node] Set LINK[NEW]=Start and START=NEW
else [insert after node with location LOC] Set LINK[NEW]=LINK[LOC] and LINK[LOC]=NEW
end of if structure
5. exit

130. SINGLE LINKED LIST INSERTION
current
start 2 3 5 6 X 9
temp

131. SINGLE LINKED LIST INSERTION
current
start 2 3 5 6 X 9
temp

132. SINGLE LINKED LIST INSERTION
current
start 2 3 5 6 X 9
temp

133. SINGLE LINKED LIST INSERTION
current
start 2 3 5 6 X 9
temp

134. SINGLE LINKED LIST DELETION
DELETE(INFO, LINK,START, AVAIL, ITEM) This algorithm deletes from a linked list the first node N which contains the given ITEM of information
[Use procedure FIND given after this algorithm] FIND(INFO, LINK, START, ITEM, LOC, LOCP)
If LOC==NULL then: Write ITEM not in list and exit
If LOCP==NUL then Set START=LINK[START]
else : Set LINK[LOCP]=LINK[LOC]
3. {Return deleted node to the AVAIL list] Set LINK[LOC]=AVAIL and AVAIL=LOC
4. Exit

135. SINGLE LINKED LIST DELETION
FINDB(INFO, START, ITEM, LOC, LOCP) This procedure finds the location LOC of first node N which contains ITEM and the location LOVP of the node preceding N. if ITEM does not appear in the list then the procedure sets LOC=NULL and if ITEM appears in the first node then it sets LOCP=NULL
[list empty?] if START==NULL then Set LOC =NULL and LOCP==NULL and return
[ITEM in first node ] if INFO[START]==ITEM then Set LOC=START and LOCP=NULL and return
Set SAVE=START and PRT=LINK[START]
Repeat steps 5 and 6 while PTR!=NULL
If INFO[PTR]==ITEM then Set LOC=PTR and LOCP=NULL

136. SINGLE LINKED LIST DELETION
6. Set SAVE=PTR and PTR=LINK[PTR]
7. Set LOC=NULL
8. Return

137. SINGLE LINKED LIST DELETION
LOC
LOCP
start 2 3 5 6 X

138. SINGLE LINKED LIST DELETION
LOC
LOCP
start 2 3 5 6 X

139. SINGLE LINKED LIST DELETION
LOC
LOCP
start 2 3 5 6 X

140. Double Linked Lists A B C
Consists of nodes with two linked references one points to the previous and other to the next node
Maximises the needs of list traversals
Compared to single list inserting and deleting nodes is a bit slower as both the links had to be updated
It requires the extra storage space for the second list
Tail
start A B C

141. Insertion Double Linked Lists
INSERTWL(INFO,FORW, BACK, STACK,AVAIL,LOCA,LOCB, ITEM)
[OVERFLOW?] If AVAIL==NULL then write OVER FLOW and exit
[Remove node from AVAIL list and copy new data into node] Set NEW=AVIAL, AVIAL=FORW[AVAIL] INFO[NEW]=ITEM
[Insert node into list] Set FORW[LOCA]=NEW, FORW[NEW]=LOCB BACK[LOCB]=NEW BACK[NEW]=LOCA
Exit

142. Insertion Double Linked Lists
LOC B
LOC A
Tail
Node A
Node B
start 3 4 7
NEW 7

143. Insertion Double Linked Lists
LOC B
LOC A
Tail
Node A
Node B
start 3 4 7
NEW 7

144. Insertion Double Linked Lists
LOC B
LOC A
Tail
Node A
Node B
start 3 4 7
NEW 7

145. Insertion Double Linked Lists
LOC B
LOC A
Tail
Node A
Node B
start 3 4 7
NEW 7

146. Insertion Double Linked Lists
LOC B
LOC A
Tail
Node A
Node B
start 3 4 7
NEW 7

147. Deletion Double Linked Lists
DELTWL(INFO, FORW, BACK, START, AVAIL, LOC)
[Delete node]
Set FORE[BACK[LOC]]=FORW[LOC]
Set BACK[FORW[LOC]]=BACK[LOC]
[Return node to AVAIL list]
Set FORW[LOC]=AVAIL and AVAIL=LOC
3. Exit

148. Deletion Double Linked Lists
LOC
Tail
start A B C

149. Deletion Double Linked Lists
LOC
Tail
start A B C

150. Deletion Double Linked Lists
Tail
start A B C

151. Deletion Double Linked Lists
Tail
start A B C