4. Pseudo-Code
Pseudo-code is a semi-formal, English-like language with a limited
vocabulary that can be used to design and describe algorithms.
The main purpose of a pseudo-code is to define the procedural logic
of an algorithm in a simple, easy-to-understand manner for its
readers, who may or may not be proficient in computer programming.
5. Pseudo-Code
Used in designing algorithms.
Used in communicating to users.
Used in implementing algorithms as programs.
Used in debugging logic errors in programs.
Used in documenting programs for future maintenance and
expansion purposes.
6. Pseudo-Code
Must have a limited vocabulary.
Must be easy to learn.
Must produce simple, English-like narrative notation.
Must be capable of describing all algorithms, regardless of their
complexity.
8. Sequence
Series of steps or statements that are executed in the order they are
written.
Example:
Read taxable income
Read filing status
Compute income tax
Print income tax
9. Selection
Defines one or two courses of action depending on the evaluation of
a condition.
A condition is an expression that is either true or false.
Example
if condition (is true)
then-part
else
else-part
end_if
10. Nested Selection
if status is equal to 1
print “Single”
else
if status is equal to 2
print “Married filing jointly”
else
if status is equal to 3
print “Married filing separately
end_if
end_if
end_if
11. Repetition
Specifies a block of one or more statements that are repeatedly
executed until a condition is satisfied.
Example:
while condition
loop-body
end_while
12. Conventions
Each pseudo-code statement consists of keywords that describe
operations and some appropriate, English-like description of
operands.
Each statement should be written on a separate line. Continuation
lines should be indented
13. Conventions II
Sequence statements should begin with unambiguous words
(compute, set, initialize).
Selection statements then-part and else-part should be indented.
Selection statements end with the keyword end_if.
14. Convention III
Repetition statements end with end_while.
Loop-bodies are indented.
All words in a pseudo-code statement must be chosen to be
unambiguous, and as easy as possible to understand by non-
programmers.
Enclose comments between /* and */
15. Example 1
If student's grade is greater than or equal to 60
Print "passed"
else
Print "failed“
End_if
16. Example 2
Set total to zero
Set grade counter to one
While grade counter is less than or equal to ten
Input the next grade
Add the grade into the total
End_while
Set the class average to the total divided by ten
Print the class average.
17. Example 3
initialize passes to zero
initialize failures to zero
initialize student to one
while student counter is less than or equal to ten
input the next exam result
if the student passed
add one to passes
else
add one to failures
End_if
add one to student counter
End_while
print the number of passes
print the number of failures
if eight or more students passed
print “good result“
End_if
18. Set asterik counter to one
Set outer while counter to one
While out counter is less than or equal to 4
set spaces counter to 1
while spaces counter is less than 4
print “ ”
increment space by one
end_while
set asterik counter to one
while asterik counter is less than or equal to odd multiple of outer counter
print “*”
increment asterik counter by one
end_while
print new line
End_while
19. Basic Symbols
Rounded box - use it to represent an event which occurs
automatically. Such an event will trigger a subsequent action, for
example `receive telephone call’, or describe a new state of
affairs.
Rectangle or box - use it to represent an event which is
controlled within the process. Typically this will be a step or
action which is taken. In most flowcharts this will be the most
frequently used symbol.
Diamond - use it to represent a decision point in the process.
Typically, the statement in the symbol will require a `yes' or `no'
response and branch to different parts of the flowchart
accordingly.
Circle - use it to represent a point at which the flowchart
connects with another process. The name or reference for the
other process should appear within the symbol.
20. Flowchart
A flowchart is a diagrammatic representation that illustrates the
sequence of operations to be performed to get the solution of a
problem
21. Guide Lines
Flowcharts are usually drawn using some standard symbols; however, some
special symbols can also be developed when required
Start or end of the program. Computational steps or processing
function of a program
Input or output operation, Decision making and branching
Connector or joining of two parts of program
Magnetic Tape
Off-page connector
Flow line
Annotation
Display
22.
23. Example
Draw a flowchart to find the largest of three numbers A,B, and C.
24. Example
Draw a flowchart to find the sum of first 50 natural numbers.
25. Imagination is more important than knowledge
Knowledge is Limited.
Imagination encircles the world
Einstein
26. Efficiency of an Algorithm
The Amount of Computer Time and Space that an Algorithm
Requires
We are more interested in time complexity. Why?
Efficiency of algorithms helps comparing different methods of
solution for the same problem
Algorithm Analysis Should be Independent of
Specific Implementation languages,
Computers and
Data
Efficiency is a concern for large problems (large data size)
27. Algorithm Growth Rate
Algorithm growth rate is a measure of it’s efficiency
Growth Rate: Measure an Algorithm’s Time Requirement as a
Function of the Problem Size (N)
How quickly the Algorithm’s Time Requirement Grows as a Function
of the Problem Size
The Way to Measure a Problem’s Size Depends on the Application
Number of Nodes in a Linked List
The Size of an Array
Number of Items in the Stack
Etc.
28. Example
Suppose we have two algorithms A and B for solving the same
problem. Which one do we use?
Suppose
Algorithm A Requires Time Proportional to N2 (expressed as
O(N2 ). This is Big O Notation.)
Algorithm B Requires Time Proportional to N , i.e. B is O(N)
Obviously B is a better choice
29. Big O Notation
If Algorithm A Requires Time Proportional to f(N), Algorithm A is Said
to be Order f(N), Which is Denoted as O(f(N));
f(N) is Called Algorithm’s Growth-Rate Function
32. Common Growth Rates
Constant : O(1)
Time Requirement is Constant
Independent of Problem Size
Example: Accessing an array element
Logarithmic : O(log N)
Time Requirement Increases Slowly for a Logarithmic Function
Typical algorithm: Solves a Problem by Solving a Smaller Constant Fraction
of the Problem
Example: Binary search
Linear : O(N)
Time Requirement Increases Directly with the size of the Problem
Example: Retrieving an element from a linked list
33. Common Growth Rates
O(N log N)
Time Requirement Increases More Rapidly Than a Linear Algorithm
Typical algorithms : Divide a Problem into Smaller Problems That are
Each Solved Separately
Quadratic : O(N2)
Increases Rapidly with the Size of the Problem
Typical Algorithms : Algorithms that Use Two Nested Loops
Practical for Only Small Problems
Cubic : O(N3)
Increases Rapidly with the Size of the Problem
Typical Algorithms : Algorithms that Use Three Nested Loops
34. Common Growth Rates
Exponential time: O( 2N)
Very, very, very bad, even for small N
Usually Increases too Rapidly to be Practical
35. Sorting Algorithms
Sorting is the process of rearranging your data elements/Item in
ascending or descending order
Unsorted Data
4 3 2 7 1 6 5 8 9
Sorted Data (Ascending)
1 2 3 4 5 6 7 8 9
Sorted Data (Descending)
9 8 7 6 5 4 3 2 1
36. Sorting Algorithms
They are many
Bubble Sort
Selection Sort
Insertion Sort
Shell sort
Comb Sort
Merge Sort
Heap Sort
Quick Sort
Counting Sort
Bucket Sort
Radix Sort
Distribution Sort
Time Sort
Source: Wikipedia
37. Bubble Sort
Compares Adjacent Items and Exchanges Them if They are Out of
Order
When You Order Successive Pairs of Elements, the Largest Element
Bubbles to the Top(end) of the Array
Bubble Sort (Usually) Requires Several Passes Through the Array
39. Selection Sort
To Sort an Array into Ascending Order, First Search for the Largest Element
Because You Want the Largest Element to be in the Last Position of the Array,
You Swap the Last Item With the Largest Item to be in the Last Position of the
Array, You Swap the Last Item with the Largest Item, Even if These Items Appear
to be Identical
Now, Ignoring the Last (Largest) Item of the Array, search Rest of the Array For
Its Largest Item and Swap it With Its Last Item, Which is the Next-to-Last Item in
the original Array
You Continue Until You Have Selected and Swapped N-1 of the N Items in the
Array
The Remaining Item, Which is Now in the First Position of the Array, is in its
Proper Order
41. Insertion Sort
Divide Array Into Sorted Section At Front (Initially Empty), Unsorted
Section At End
Step Iteratively Through Array, Moving Each Element To Proper
Place In Sorted Section
Sorted Unsorted
….. …..
0 i N-1
After i Iterations
43. Shell Sort
Improved and efficient version of insertion sort
It iterates the data elements/items like insertion sort, but instead of
shifting it makes the swaps.
45. Merge Sort
Divide and Conquer Algorithm
Recursively split the input in half
Then recursively merge pairs of pieces
Recursive Steps:
Divide the Array in Half
Sort the Halves
Merge The Halves inside a Temporary Array
Copy Temporary Array to the appropriate locations in original array
46. Merge Sort
The Recursive calls Continue to divide the Array into Pieces Until
Each Piece Contains Only One Item
An Array of One Item is Obviously Sorted
The Algorithm Then Merges These small Pieces Until One Sorted
Array Results
48. Quick Sort
Divide and Conquer algorithm
Quicksort Works by Partitioning the Array into Two Pieces Separated
by a Single Element That is Greater Than all the Elements in the Left
part and Smaller Than all the Elements in the right part
This Guarantees That, the Single Element , Called the Pivot
Element, is in its Correct position
Then the Algorithm Proceeds, Applying the Same Method to the Two
parts Separately
49. Quick Sort
Partition (Divide)
Choose a pivot
Find the position for the pivot so that
all elements to the left are less
all elements to the right are greater
< pivot pivot >= pivot
50. Quick Sort
Conquer
Apply the same algorithm to each half
< pivot >= pivot
< p’ p’ > p’ pivot < p” p” >= p”
51. Partitioning Method
Must Arrange Items Into Two regions S1, the Set of Items Less Than the
Pivot, and S2, the Set of Items Greater Than or Equal to Pivot
Different algorithms for Choice of a Pivot
Retain Pivot in A[F] position
The Items That await Placement are in Another Region , Called the
Unknown Region
S1 S2 Unknown
P <P >=P ?
F LastS1 FirstUnknown L
At Each Step of the partition Algorithm you Examine One Item from
Unknown Region and Place it in S1 or S2
52. To Move A[FirstUnknown] into S1
Swap
S1 Unknown
S2
P <P >=P >=P <P ?
F LastS1 LastS1+1 FirstUnknown L
Swap A[FirstUnknown] With the First Item of S2, Which is A[LastS1+1], and Then
Increment S1 by 1
Thus the Item That Was in A[FirstUnknown] will be at the Rightmost Position of S1
Item of S2 That was Moved to A[FirstUnknown]: If you Increment FirstUnknown by 1,
That Item Becomes the Rightmost Member of S2
Thus, Statements for the Following Steps are Required
Swap A [FirstUnknown] with A[lastS1+1]
Increment LastS1
Increment FirstUnknown
53. To Move A[FirstUnknown] into S2
S1 S2 Unknown
P <P >=P ?
F LastS1 FirstUnknown L
Region S2 and Unknown are Adjacent
Simply Increment FirstUnknown by 1, S2 Expands to the Right
To Move Pivot Between S1 and S2
Interchange A[LastS1], the Rightmost Item in S1 with Pivot
Thus, Pivot Would be in its Actual Location
54. Quick Sort Example (only one step)
Pivot
Choose Pivot, keep it in A[F]
27 38 12 39 27 16
Pivot Unknown
FirstUnknown = 1(Points to 38
27 38 12 39 27 16 38 Belongs in S2
Pivot S2 Unknown
S1 is Empty
27 38 12 39 27 16 12 Belongs in S1, swap38 & 12
Pivot S1 S2 Unknown
27 12 38 39 27 16 39 Belongs in S2
Pivot S1 S2 Unknown
27 12 38 39 27 16 27 Belongs in S2
Pivot S1 S2 Unknown
27 12 38 39 27 16 16 Belongs in S1, Swap 38 & 16
Pivot S1 S2
27 12 16 39 27 38 No more Unknown
S1 Pivot S2
16 12 27 39 27 38 Place Pivot between S1 and S2
56. Comparison of Sorting Algorithms
Worst Case
Selection Sort N2
Bubble Sort N2
Insertion Sort N2
Mergesort N * log N
Quicksort N2
Radix Sort N
57. Searching Algorithms
Searching is the process of determining whether or not a given value
exists in a data structure or a storage media.
We will study two searching algorithms
Linear Search
Binary Search
58. Linear Search: O(n)
The linear (or sequential) search algorithm on an array is:
Start from beginning of an array/list and continues until the item is
found or the entire array/list has been searched.
Sequentially scan the array, comparing each array item with the
searched value.
If a match is found; return the index of the matched element;
otherwise return –1.
Note: linear search can be applied to both sorted and unsorted
arrays.
59. Linear Search
bool LinSearch(double x[ ], int n, double item)
{
for(int i=0;i<n;i++)
{
if(x[i]==item)
{
return true;
}
else
{
return false;
}
}
return false;
}
60. Linear Search Tradeoffs
Benefits
Easy to understand
Array can be in any order
Disadvantages
Inefficient for array of N elements
Examines N/2 elements on average for value in array, N
elements for value not in array
61. Binary Search: O(log2 n)
Binary search looks for an item in an array/list using
divide and conquer strategy
62. Binary Search
Binary search algorithm assumes that the items in the array being
searched is sorted
The algorithm begins at the middle of the array in a binary
search
If the item for which we are searching is less than the item in the
middle, we know that the item won’t be in the second half of the
array
Once again we examine the “middle” element
The process continues with each comparison cutting in half the
portion of the array where the item might be
63. Binary Search
Binary search uses a recursive method to search an array to find a
specified value
The array must be a sorted array:
a[0]≤a[1]≤a[2]≤. . . ≤ a[finalIndex]
If the value is found, its index is returned
If the value is not found, -1 is returned
Note: Each execution of the recursive method reduces the search
space by about a half
67. Key Points in Binary Search
1. There is no infinite recursion
• On each recursive call, the value of first is increased, or the
value of last is decreased
• If the chain of recursive calls does not end in some other
way, then eventually the method will be called with first larger
than last
2. Each stopping case performs the correct action for that case
• If first > last, there are no array elements between a[first] and
a[last], so key is not in this segment of the array, and result is
correctly set to -1
• If key == a[mid], result is correctly set to mid
68. Key Points in Binary Search
3. For each of the cases that involve recursion, if all recursive calls
perform their actions correctly, then the entire case performs
correctly
• If key < a[mid], then key must be one of the elements a[first]
through a[mid-1], or it is not in the array
• The method should then search only those elements, which it
does
• The recursive call is correct, therefore the entire action is
correct
69. Key Points in Binary Search
• If key > a[mid], then key must be one of the elements
a[mid+1] through a[last], or it is not in the array
• The method should then search only those elements, which it
does
• The recursive call is correct, therefore the entire action is
correct
The method search passes all three tests:
Therefore, it is a good recursive method definition
70. Efficiency of Binary Search
The binary search algorithm is extremely fast compared to an
algorithm that tries all array elements in order
About half the array is eliminated from consideration right at the
start
Then a quarter of the array, then an eighth of the array, and so
forth
71. Efficiency of Binary Search
Given an array with 1,000 elements, the binary search will only need
to compare about 10 array elements to the key value, as compared
to an average of 500 for a serial search algorithm
The binary search algorithm has a worst-case running time that is
logarithmic: O(log n)
A serial search algorithm is linear: O(n)
If desired, the recursive version of the method search can be
converted to an iterative version that will run more efficiently
