Introduction to Data Structures Arrays, Stacks and Queues

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Introduction to Data StructuresArrays, Stacks
and Queues
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What is an Array?

• An array is a data structure used for storing a
  collection of data items that are all the same
  type.
• By using an array, we can associate a single
  variable name with an entire collection of data.
• This enables us to save the entire collection of
  data in adjacent cells of main memory and to
  easily reference individual items by an Index
  number.
• To process an individual item, we need to specify
  the array name and indicate which array element
  is being manipulated.

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Storage comparison

• Using normal variables
• Using an array

5
8
3
6
2
Var1
Var2
Var3
Var4
Var5
Index
(0)
5
8
(1)
3
(2)
(3)
6
(4)
2
MyArray
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Array Example 1
Size
Declaration Declare Size(6) as Single Will allow
7 Single numbers to be held (0 6)
Index number
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Array Example 2
Name
Declaration Declare Name(6) as String Will allow
7 Strings to be held (0 6)
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Array exercise

• We are going to use an array to calculate the
  average of a number of integer values that are
  entered by the user
• To do this we must declare the array.
• Note - It is important to put a value in the
  brackets to give the upper bound index number the
  array has to hold. The lower bound number
  defaults to 0
• We are going to accept 10 numbers so we declare
  the array
• VAR Number(9) as array of Integers

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Set up a loop

• We need to set up a loop to ask for ten numbers.
• We will use the For structure so that we can stop
  after the user enters the tenth number
• The code will be
• For x 1 to 10
• Number(X) Number input by user
• End For

For x 1 to 10
Number(X) UserNum
Does x 10?
No
Yes
End
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Next Bit

• We should now have all the numbers we entered
  stored in the array.
• The best way to add them up is by creating
  another loop to add them together a pass at a
  time.
• Total 0
• For Pass 1 To 10
• Total Total Number(Pass)
• Next

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Finally

• We could calculate the average by dividing this
  total by ten
• Average Total / 10

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Table Look-Up

• A common application of arrays is to use a value
  to look up another value in a table
• For example find the number of days in a month
  given the months number
• The array element numbers are given on the right
  these correspond to the month number
• Algorithm for look-up
• Enter MonthNum
• Num_of_Days Days(MonthNum)
• Display Num_of_Days

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Sequential Search

• We may not know the element number but know the
  value of the element we are looking for
• e.g. when we know someones name and want to look
  them up in the phone book
• The easiest way to do this would be to search
  sequentially through the entire array looking for
  a match

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Sequential Search Algorithm

• X 1
• While SearchName ltgt Array(X) and X lt ArraySize
• X X 1
• End While
• If X lt ArraySize
• Then Display Search name not found
• Else Display Array(X)

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Binary Search

• A faster method for searching the array is the
  binary search method
• This only works if the contents of the array have
  been sorted we will discuss algorithms for
  sorting later
• This algorithm is a bit more complex than the
  ones we have considered so far so I will use
  three flowcharts to describe it
• Each flowchart could be translated into a
  procedure in the eventual program

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Binary Search Flowchart 1 of 3
LB 1
Enter From Flowchart No. 3
UB N
Flag 0
Go to Flowchart No. 2
No
Does Flag 0 ?
Yes
R Int ((LB UB)/2)
Go to Flowchart No. 3
No
Does Array (R) S
Yes
Flag 1
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Analysis of Flowchart No. 1
LB 1
UB N
Flag 0
Does Flag 0 ?
R Int ((LB UB)/2)
Does Array (R) S
Flag 1
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Flowchart No 2 of 3

• We get to this flowchart because the flag is no
  longer 0 this means we have either found the
  search term or failed to find it after an
  exhaustive search

Does Flag 2 ?
Yes
No
Display Search Successful
Display Search Term not Found
End
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Flowchart No 3 of 3
Is S gt Array (R) ?
Yes
No
UB R - 1
LB R 1
Return to Flowchart 1 at indicated entrance
point
Is LB gt UB
No
Yes
Flag 2
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Analysis of Flowchart No 3

• This flowchart how we determine where to look
  next
• If the search term is bigger (or comes
  alphabetically later) than the array element we
  are currently considering then the lower boundary
  is set to one more than the current element
  number
• This means that we only consider the elements in
  the top-half of the array
• Similarly if the search term is smaller (or comes
  alphabetically earlier) than the current array
  element we set the upper bound to one less than
  the current element number
• This means that we will look for the desired
  element in the bottom-half of the array

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A Dry Run with the Algorithm

• We are searching for 331
• The number of comparisons made
• will be
• Check element 7 (UB 13 LB 1 Element No 7)
  
• Check element 10 (UB 13, LB 8 Element No
  10)
• Check element 12 (UB 13 LB 11 Element No
  12)
• We have found 331 after three
• comparisons compared to the 12 that
• would be needed in a sequential search

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Parallel Arrays

• We can store multiple pieces of related
  information across a number of parallel arrays
• For example - Student No. 3 has the name Emily
  Wade, lives at 12 Saxton Gardens and is aged 23

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Multidimensional Arrays

• We can work with arrays of arrays
• So if we want to hold data for 10 homework scores
  for 15 students we can create a structure like
  this

• And refer to elements in the structure like this
• ClassHomework (3,2)
• To find the mark that student number 3 has been
  awarded for homework number 2

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StacksA Special Type of Array
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What is a stack?

• An ordered collection of homogeneous items.
• Like a stack of coins

We need algorithms to add things to the stack
and to take things off it
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Stack Specification

• Procedures to be carried out on stacks
• Push
• Add something to the stack
• Pop
• Take something off the stack

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Algorithm for Pushing

• If we assume an array called MyStack and a
  variable
• called Top that holds the index of the last
  data item
• pushed on to the stack
• Obtain data item X
• IF stack is full
• THEN reply (Unable to Push)
• ELSE Top Top 1
• MyStack(Top) X
• Reply (X has been pushed onto stack)
• ENDIF

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Algorithm for Popping

• IF stack is empty
• THEN reply (Unable to Pop)
• ELSE X MyStack(Top)
• Top Top -1
• Reply (X has been obtained from the
  stack)
• ENDIF

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Exercise recognizing palindromes

• A palindrome is a string that reads the same
  forward and backward.
• Able was I ere I saw Elba 
• Develop an algorithm to push this line of text a
  character at a time onto a stack and then pop it
  to reverse the string for comparison purposes
• Would your algorithm work for a palindrome like
  this
• Stella won no wallets.

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Interesting application of a stack

• When you were at school
• You were taught BODMAS or some other way of
  remembering that
• Arithmetic Calculations are done in a particular
  order.
• Work out contents of Brackets
• Work out Orders, e.g. R2
• Work out Divisions and or Multiplications
• Work out Additions and/or Subtractions

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Why is it important?

• Programming languages follow the same rules as we
  have just seen. It is called Arithmetical Syntax
  or the order of calculation.
• Imagine this situation.
• A customer orders two items costing 20 and 80
• We need to calculate the VAT at 17.5
• So we write a line that says VAT 20 80
  17.5
• This will give us a VAT of 34 which is too much

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What went wrong?

• It came to 34 by working out the VAT on 80
  (14) then adding the 20 because multiplication
  occurs before addition.
• We should ensure that the two prices are added
  together before the VAT is calculated by
  enclosing the two figures in brackets.
• The correct line should be
• VAT (20 80) 17.5
• This will give 17.50 which is correct

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Another example

• Work out the average of 3, 7, 9 and 5
• Average 3795/4
• This gives 20.25 wrong again
• It has divided 5 by 4 first as division comes
  before addition in the rules.
• We should have written
• Average (3795)/4
• This gives an answer of 6 which is correct.
• Expressions represented this way are known as
  infix expressions

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Postfix Expressions

• Postfix notation is an alternative way of writing
  arithmetic expressions.
•  
• In postfix notation, the operator is written
  after the two operands.
•  
• infix 25 postfix 2 5
• Expressions are evaluated from left to right.
•  
• Precedence rules and parentheses are never
  needed!!

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Handling Postfix ExpressionsAn Algorithm using
Stacks

• WHILE more input items exist
• Get an item
• IF item is an operand
• stack.Push(item)
• ELSE
• stack.Pop(operand2)
• stack.Pop(operand1)
• Compute result
• stack.Push(result)
• stack.Pop(result)
• END IF

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Another Application

• Another application of the stack data structure
  is the history in a web browser
• Each time a user visits a new web page the
  address of the page the user is leaving is pushed
  onto the top of a stack
• Thus the stack ends up containing all the pages
  the user has previously visited
• When the user eventually clicks on the back
  button an address is popped off the top of the
  stack and the browser navigates to that address
• The address popped off will be the one the user
  has most recently visited

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An Illustrative Example

• Use a stack to reverse a string of text entered
  by a user.
• The string should have a maximum length of 500
  characters.
• The reversed string should be displayed on the
  screen.

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Possible Algorithm

• MakeSureStackisEmpty
• Obtain string from the user
• While there are still characters in the string do
• Read a character
• Push it on to the stack
• Move to next character
• EndWhile
• While there are characters on the stack do
• Pop a character from the stack
• Write it to the screen
• EndWhile

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Possible Pascal CodeData Declarations

• program ReverseStringwithStack
• uses crt
• const maxstacksize 500
• type
• stack record
• elements array1..maxstacksize of char
• NumItemsOnStack integer
• end
• var
• thestack stack
• i integer
• ch char

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Possible Pascal CodeProcedures

• procedure startstack
• begin
• thestack.NumItemsOnStack 0
• end
• procedure push
• begin
• with thestack do
• begin
• if NumItemsOnStack maxstacksize then
• writeln('ERROR Trying to place
  element on full stack!')
• else
• begin
• NumItemsOnStack NumItemsOnStack
  1
• elementsNumItemsOnStack ch
• end
• end
• end

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Possible Pascal CodeProcedures

• procedure pop
• begin
• with thestack do
• begin
• if NumItemsOnStack 0 then
• writeln('ERROR Trying to remove
  element from empty stack!')
• else
• begin
• ch elementsNumItemsOnStack
• NumItemsOnStack NumItemsOnStack
  -1
• end
• end
• end

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Possible Pascal CodeMain Body

• begin
• clrscr
• startstack
• write('Enter a string ')
• while ch ltgt 10 do
• begin
• read(ch)
• push
• end
• writeln
• writeln
• write('The string reversed is ')
• while thestack.NumItemsOnStack ltgt 0 do
• begin
• pop
• write(ch)
• end
• end.

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Two Example Outputs
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Queues
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What is a queue?

• An ordered group of homogeneous items of
  elements.
• Unlike stacks queues have two ends
• Elements are added at one end.
• Elements are removed from the other end.
• The element added first is also removed first
• FIFO First In, First Out

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Enqueue

• Function Adds newItem to the rear of the queue.
• Preconditions Queue is not full.
• Postconditions newItem is at rear of queue.

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Dequeue

• Function Removes front item from queue
• Preconditions Queue is not empty.
• Postconditions Front element has been removed
  from queue

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Applications of Queues

• Queues are useful in any situation where a
  clients demands for services can exceed the rate
  at which those services can be supplied
• For example in a local area network where a
  number of computers share a printer or
  communications channel

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Review Questions

• What is meant by an array element?
• Why are stacks called LIFO structures?
• What is infix notation and prefix notation?
• Why are queues called FIFO structures?

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Review Question

• Trace the following pseudocode showing the
  contents of the stack
• after each invocation
• Declare MyStack as Stack
• MyStack.Push (1)
• MyStack.Push (2)
• MyStack.Push (3)
• MyStack.Push (4)
• MyStack.Pop
• MyStack.Pop
• MyStack.Push (5)
• MyStack.Push (6)
• MyStack.Pop
• MyStack.Pop
• MyStack.Pop
• MyStack.Pop