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QUIZ on Ch.6:
Write a PEP assembly program that reads in a character, and then outputs ‘A’ if the character was A, or ‘not A’ otherwise.
1
What is the hex ASCII code for A?
Use this code as template:
Problem Solving
How to Solve It: A New Aspect of Mathematical Method by George Polya, 1945
The book is written within the context of solving mathematical problems, but the methods described are applicable to problem solving in general.
3
We can use the methods described
by Polya to solve any computer-
related problem!
1. Understand the problem
– What are the hypotheses? Data? Unknowns?
2. Devise a plan
– Can we find a related problem? A sub-problem?
– Can we strengthen or relax the hypotheses to obtain an easier problem?
3. Carry out the plan
— Prove that each step is correct!
4. Look back
– Check result
– Find shortcuts and alternate solutions
– Generalize to related problems4
Strategies for step 2 (devise a plan)
Ask questions!
– What do I know about the problem?
– What is the information that I have to process in order the find the solution?
– What does the solution look like?
– What sort of special cases exist?
– How will I recognize that I have found the solution?
5
Strategies
Never reinvent the wheel!
Similar problems come up again and again in different guises
A good programmer recognizes a task or subtask that has been solved before and plugs in the solution
Can you think of two similar problems we solved in Python?
6
StrategiesDivide and Conquer!
Break up a large problem into smaller sub-problems and solve each separately
– Applies the concept of abstraction
– The divide-and-conquer approach can be applied over and over again until each subtask is manageable
7
See funny image!
Polya’s 4 steps can be applied to Computer Problem Solving
8
Analysis and Specification Phase
Ask questions to understand all the requirements
Explain in detail what needs to be achieved
Algorithm Development Phase
Develop algorithm
Test algorithm
Implementation Phase
Code algorithm
Test the code in various ways
Maintenance Phase
Use the code, find bugs
Fix bugs
Code new features, as requested by users
QUIZ: Match the steps in Polya’s method to the ones in the computer method for
problem solving
9
Analysis and Specification
Implementation
Algorithm Development
Maintenance
Devise a plan
Look back
Understand
Carry out the plan
Phase Interactions
10
Algorithms
Algorithm = A set of unambiguous instructions for solving a problem in a finite amount of time, using a finite amount of data
11
Algorithm = A set of unambiguous instructions for solving a problem in a finite amount of time, using a finite amount of data
12Image source: http://my-online-log.com/tech/archives/1214
Algorithm = A set of unambiguous instructions for solving a problem in a finite amount of time, using a finite amount of data
Abstract Step
An algorithmic step containing unspecified details
Concrete Step
An algorithm step in which all details are specified
13
Whether the step is abstract or concrete depends on the
programming language we’re using!
Algorithm = A set of unambiguous instructions for
solving a problem in a finite amount of time,using a finite amount of data
Douglas Hofstadter
“Godel, Escher, Bach”
The story of Alladin and G.O.D.
(G.O.D. Over Djinn –
recursive acronym!)
Algorithm = A set of unambiguous instructions for solving a problem in a finite amount of time, using a
finite amount of data
Douglas Hofstadter
“Godel, Escher, Bach”
The story of Alladin and G.O.D.
7.2 Algorithms with simple variables
Variable = a means of storing intermediate results from one task to the next.
At the hardware level, a simple variable is just one or several adjacent Bytes in the computer memory.
Q: How many Bytes does a simple variable have in PEP/8?
16
Algorithms with Selection Statements (a.k.a. decision or if)
17
Flowchart of if statement
Figure not in text
18
Algorithm with Selection
Problem: Write the appropriate dress for a given
temperature.
Write "Enter temperature"
Read temperature
Determine Dress
Which statements are concrete?
Which statements are abstract?
Algorithm Determine Dress v.1
Computer language is Python from now on!
19
Write “Enter temperature”
Read temperature
IF (temperature > 90)
Write “Texas weather: wear shorts”
ELSE IF (temperature > 70)
Write “Ideal weather: short sleeves are fine”
ELSE IF (temperature > 50)
Write “A little chilly: wear a light jacket”
ELSE IF (temperature > 32)
Write “Philadelphia weather: wear a heavy coat”
ELSE
Write “Stay inside”
Algorithm Determine Dress v.2
Is this concrete enough for implementation in Python?
Algorithms with Loops (a.k.a. repetition)
20
Flow of control of while statement
Figure not in text
QUIZ: There are loops that can execute 0 times and loops that must execute at least 1 time.Which type is this?
21
Answer: It depends on whether the decision (diamond) is at the beginning or at the end of the loop!
22
Not in text
Loops in Python
23
Not in text
Extra-credit question:
24
Event-controlled Loops, a.k.a. WHILE loops
25
They’re the most general type of loops!
Set sum to 0
Set allPositive to true
WHILE (allPositive)
Read number
IF (number > 0)
Set sum to sum + number
ELSE
Set allPositive to false
Write "Sum is " + sum
Counter-controlled Loops
26
They are a particular case of event-controlled
loops: the event is that a counter variable has
reached a predetermined limit
Set sum to 0
Set limit to 42
Set count to 1
While (count <= limit)
Read number
Set sum to sum + number
Increment count
Write "Sum is " + sum
Counter-controlled Loops
27
They are a particular case of event-controlled
loops: the event is that a counter variable has
reached a predetermined limit
Set sum to 0
Set limit to 42
Set count to 1
While (count <= limit)
Read number
Set sum to sum + number
Increment count
Write "Sum is " + sum
For loops are counter-
controlled!
28
Important application of looping: Successive approximation
algorithms
Read in square
Calculate the square root
Write out square and the square root
Algorithm Calculate Square Root v.1
Which steps are abstract and which concrete?
29
Set epsilon to 1
WHILE (epsilon > 0.001)
Calculate new guess
Set epsilon to abs(square - guess * guess)
Which steps are abstract and which concrete?
Algorithm Calculate Square Root v.2
In Python usemath.fabs(…)
A more appropriate name for guess would be approximation
30
Set newGuess to
(guess + (square/guess)) / 2.0
Set guess to square/4
Set epsilon to 1
What’s the mathematical formula for the new approximation?
How do we get the loop started?
Algorithm Calculate Square Root - Refinements in v.2
31
Read in square
Set guess to square/4
Set epsilon to 1
WHILE (epsilon > 0.001)
Set guess to (guess + (square/guess)) / 2.0
Set epsilon to abs(square - guess * guess)
Write out square and guess
Which steps are abstract and which concrete?
Algorithm Calculate Square Root v.3
32
Set newGuess to
(guess + (square/guess)) / 2.0
QUIZ: Square root algorithms
(approximations)
We want to calculate = 2.236…
Set your initial guess x0 = 1 and show the
next 3 approximations x1, x2, x3
5
33
Set newGuess to
(guess + (square/guess)) / 2.0
QUIZ: Square root algorithm
(approximations)
We want to calculate = 2.236…
Set your initial guess x0 = 42 and show the
next 3 approximations x1, x2, x3
5
EOL1
QUIZ
• 16: Count-controlled loops repeat a specific number of times.
• 17: Event-controlled loops repeat a specific number of times
• 18: Count-controlled loops are controlled by a counter
• Count-controlled loops are more general than event-controlled loops
34
35
Set newGuess to
(guess + (square/guess)) / 2.0
QUIZ: Square root algorithms
(approximations)
We want to calculate = 3.162…
Set your initial guess x0 = 10/4 and show
the next 3 approximations x1, x2, x3
10
Set newGuess to
(guess + (square/guess)) / 2.0
QUIZ: Square root algorithms
(approximations)
We want to calculate = 3.16227…
Set your initial guess x0 = 10/4 and show
the next 3 approximations x1, x2, x3
10
x0 = 2.5 x1 = 3.25 x2 = 3.16346… x3 = 3.16227…
37
QUIZ: Square root algorithm
We want to calculate = 2.6457…
Set your initial guess to x0 = 7/4 = 1.75
and show the next 3 approximations x1, x2, x3
7
38
QUIZ: Square root algorithm
We want to calculate = 2.645751…7
39
Remember: The
algorithm converges
irrespective of the
initial guess x0!
= 2.645751…7
40
QUIZ:
Re-write this program
using a for loop!
QUIZ:
Re-write this program
using a for loop!
Individual work for
next time:
Re-write this program
using a while loop!
7.3 Composite Data Types
They can be classified according to many criteria:
• homogeneous vs. heterogeneous
• accessed by index vs. accessed by name
• mutable vs. immutable
• linear vs. non-linear
• etc.
43
Composite Data Types
Records
A named heterogeneous collection of items in which individual items are accessed by name. For example, we could bundle name, age and hourly wage items into a record named Employee
Arrays
A named homogeneous collection of items in which an individual item is accessed by its position (index) within the collection
44
Are these the lists from Python? Why not?
Python strings are arrays of charactersCan implement in Python w/lists …
Composite Data Types
Lists (will be covered in next chapter)
A named heterogeneous collection of items in which individual items are accessed by position (index).
We have them in Python, e.g.
>>> myList = [“dog”, 42, 51.375, [1,2]]
>>> myList[1]
42
45
Composite Data Types - Records
Employee
name
age
hourly/Wage
Algorithm to store values into the fields of record:
Employee employee // Declare an Employee variableSet employee.name to “Frank Jones”Set employee.age to 32Set employee.hourlyWage to 27.50
46
Composite Data Types - Arrays
47
numbers[0]
numbers[4]
numbers
Some items in an array may be unused at a given time
48
first
last
numbers
??
??
??
??
Useful Algorithms on Arrays
• Initializing all items
• Printing all items
• Searching for an item
• Sorting the array
49
50
Initializing arrays
integer data[20]
Write “How many values?”
Read length
Set index to 0
WHILE (index < length)
Read data[index]
Set index to index + 1
Fill an array numbers with length values
that are being input from the keyboard
51
QUIZ
integer data[20]
Write “How many values?”
Read length
Set index to 0
WHILE (index < length)
Read data[index]
Set index to index + 1
Modify this pseudocode to print the values
after initializing them.
An Unsorted Array
52
data
A Sorted Array
53
data
Sorted Arrays
• The values stored in an array have unique keysof a type for which the relational operators are defined.
• Sorting rearranges the elements into either ascending or descending order within the array.
54
Reality check: In a real-life problem it’s very common
to encounter repeated keys!
7.4 Search algorithms
55
Sequential Search inUnsorted Array
A sequential search examines each item in turn and compares it to the one we are searching.
If it matches, we have found the item. If not, we look at the next item in the array.
We stop either when we have found the item or when we have looked at all the items and not found a match.
Thus, we have a loop with two ending conditions.
56
Set position to 0
Set found to FALSE
WHILE (position < length AND NOT found )
IF (numbers[position] equals searchItem)
Set found to TRUE
ELSE
Set position to position + 1
57
The array’s name is numbersThe value we’re searching for is stored in searchItem
Set position to 0
Set found to FALSE
WHILE (position < length AND NOT found )
IF (numbers[position] equals searchItem)
Set found to TRUE
ELSE
Set position to position + 1
58
QUIZ: When the loop exits, what do we need to do?
Set index to 0
Set found to FALSE
WHILE (index < length AND NOT found )
IF (data[index] equals searchItem)
Set found to TRUE
ELSE
Set index to index + 1
59
QUIZ: Desk-check this algorithm for the array
The item searched is:
• 35
• 43
42
100
35
1
EOL 2
QUIZ
-- Define arrays
-- What are the 4 fundamental types of algorithms
used to manipulate arrays?
-- What control structure is normally used to
access the elements of an array?
-- Explain sequential search in your own words
(unsorted array)
60
Sequential Search in Sorted Array
Idea:
If items in an array are sorted, we can stop looking when we pass the place where the item would be if it were present in the array
61
Is this better?
62
Sequential Search in Sorted Array
Set index to 0
Set found to FALSE
WHILE (index < length AND NOT found)
IF (data[index] equals searchItem)
Set found to TRUE
ELSE IF (data[index] > searchItem)
Set index to length
ELSE
Set index to index + 1
This is the new part!
(Compare with previous alg. for unsorted array)
This alg. is called sequential search with early termination
QUIZ:End-of-chapter question 66 a
63
SearchItem = 4
SearchItem = 49
SearchItem = 50
Binary Search in Sorted Array
Search begins at the middle and finds the item or eliminates half of the unexamined items; the process is then repeated on the half where the item might be
64
24 30 31 42 44 90 92 94 99
Example: searchItem = 42
Binary Search Algorithm
65
Set first to 0
Set last to length-1
Set found to FALSE
WHILE (first <= last AND NOT found)
Set middle to (first + last)/ 2
IF (item equals data[middle]))
Set found to TRUE
ELSE
IF (item < data[middle])
Set last to middle – 1
ELSE
Set first to middle + 1
RETURN found
IntegerDivision!
Binary Search example
66Figure 7.10 Trace of the binary search
rat
QUIZ Binary Search
67
Search for deer
8.5 Sorting algorithms
68
Sorting
Sorting
Arranging items in a collection so that there is an ordering on one (or more) of the fields in the items
Sort Key
The field (or fields) on which the ordering is based
Sorting algorithms
Algorithms that order the items in the collection based on the sort key
69
Selection Sort
70
Figure 7.11 Example of a selection sort (sorted elements are shaded)
71
100
10
11
12
35
200
13
2
Show the swapped elements with arrows.
Show the sorted elements with shading.
QUIZ Selection Sort
solution
72
73
24
10
11
12
35
20
1
2
Show the swapped elements with arrows.
Show the sorted elements with shading.
Individual work for next time
74
solution
75
Selection Sort Pseudocode
Selection Sort
Set firstUnsorted to 0
WHILE (not sorted yet)
Find smallest unsorted item
Swap firstUnsorted item with the smallest
Set firstUnsorted to firstUnsorted + 1
Not sorted yet
current < length – 1
Red steps
are
abstract
(not
concrete)
76
Find smallest unsorted item
Set indexOfSmallest to firstUnsorted
Set index to firstUnsorted + 1
WHILE (index <= length – 1)
IF (data[index] < data[indexOfSmallest])
Set indexOfSmallest to index
Set index to index + 1
Set index to indexOfSmallest
Swap firstUnsorted with smallest
Set tempItem to data[firstUnsorted]
Set data[firstUnsorted] to data[indexOfSmallest]
Set data[indexOfSmallest] to tempItem
SKIP Bubble Sort and Insertion Sort, just remember that they are as fast
(efficient) as Selection Sort
77
Is it possible to devise a more efficient sorting algorithm?
78
Is it possible to devise a more efficient sorting algorithm?
Yes, but, in order to do this, we have to understand subprograms and
recursion.
7.6 Recursive Algorithms
To understand Quick Sort, we need:
• a new control structure: subprogram
• a new concept: recursion
79
Subprogram Statements
We can give a section of code a name and use that name as a statement in another part of the program
When the name is encountered, the processing in the other part of the program halts while the named code is executed
When execution is finished, the first part of the program resumes execution
That section of code is called a subprogram80
Subprogram Statements
81Figure 7.14 Subprogram flow of control
We already used subprograms in Python!
… we called them
• Functions
– int() float() ord() chr() str() etc.
• Methods
– list.append() string.upper() string.find() etc.
82
Do you remember what each of them
does?
Subprogram Statements
What if the subprogram needs data from the calling unit? This data is called input.
Parameters
Identifiers listed in parentheses beside the subprogram declaration; sometimes called formal parameters
Arguments
Identifiers listed in parentheses on the subprogram call; sometimes called actual parameters
83
Parameters and arguments in Python
def disp(a, b):
print a, 'and', b
x, y = 1, 2
disp(x, y)
84
What if the subprogram needs to give data back to the calling unit? This data is called output.
Void subprograms
They do not return a value, just perform certain actions
>>> print(“Enter a positive integer”)
>>> printer(3, 4)
Value-returning subprograms
>>> a = input(“Enter a positive integer”)
The keyword RETURN is used in many programming languages
85
What if the subprogram needs to give data back to the calling unit? This data is called output.
Void subprograms
They do not return a value, just perform certain actions
>>> print(“Enter a positive integer”)
>>> printer(3, 4)
Value-returning subprograms
>>> a = input(“Enter a positive integer”)
The keyword RETURN is used in many programming languages
86
Do not confuse returning values with printing or displaying values. Returning means that a value is
given back to the main program!
Value-returning function in Python
def sum(a, b):
return a + b
x, y = 1, 2
print sum(x, y)
87
Compare to the previous function, which is void
(does not return anything)
Value-returning and void Subprograms
88
Subprogram Statements
Subprograms are very important tools for abstraction.
Other popular names for subprograms:
• sub
• subroutine
• function
• procedure
• module
• method
89
Recursion
Recursion
The ability of a subprogram to call itself
Base case
The case to which we have an answer
General case
The case that expresses the solution in terms of a call to itself with a smaller version of the problem
90
Recursion The factorial of a positive integer is defined as the integer times the product of all the integers between itself and 0:
N! = 123 … N
But an alternate recursive definition is possible:
N! = N(N 1)!
Base case
Fact(0) = 1 (0! is 1)
General Case
Fact(N) = N Fact(N-1) (for N ≥ 1)
91EOL 4
QUIZ Binary Search
How many comparisons are needed to find the key 9 (or
decide that it is not in the array)?
Show the left, right and middle at each step.
QUIZ Binary Search
How many comparisons are needed to find the key 9 (or
decide that it is not in the array)?
Show the left, right and middle at each step.
Exactly! The array must be sorted first! …
QUIZ Binary Search
-50 -28 -23 -10 -4 0 3 41 28 37 60 155 200
How many comparisons are needed to find the key 60 in
this array?
Show first, last, and middle for each iteration.
QUIZ Binary Search
-50 -28 -23 -10 -4 0 3 41 28 37 60 155 200
How many comparisons are needed to find the key 60 in
this array?
Show first, last, and middle for each iteration.
Array must be sorted first for Binary Search!
Two definitions for the factorial Iterative: N! = 123 … N
Recursive: N! = N(N 1)!
Base case
Fact(0) = 1 (0! is 1)
General Case
Fact(N) = N Fact(N-1) (for N ≥ 1)
96
Is this a valid recursion?
Source: http://www.urbandictionary.com/define.php?term=recursion97
Base case
Fact(0) = 1
General Case
Fact(N) = N Fact(N-1)
98
How does the
genie calculate
Fact(3)?
Your turn!N! = N * (N 1)!
Base case
Facto(0) = 1 (0! is 1)
General Case
Fact(N) = N * Fact(N-1) (for N ≥ 1)
Calculate:
0! =
1! =
2! =
5! = 99
100
Recursive Factorial algorithm
Write “Enter n”Read nSet result to Factorial(n)Write result + “ is the factorial of “ + n
Factorial(n)IF (n equals 0)
RETURN 1ELSE
RETURN n * Factorial(n-1)
101
QUIZ: Write this recursive function in Python
Write “Enter n”Read nSet result to Factorial(n)Write result + “ is the factorial of “ + n
Factorial(n)IF (n equals 0)
RETURN 1ELSE
RETURN n * Factorial(n-1)
solution
102
103
Recursive Binary Search BinarySearch (first, last)
IF (first > last)
RETURN FALSE
ELSE
Set middle to (first + last)/ 2
IF (item equals data[middle])
RETURN TRUE
ELSE
IF (item < data[middle])
BinarySearch (first, middle – 1)
ELSE
BinarySearch (middle + 1, last)
SKIP QUICKSORT …Just remember that it is faster
(more efficient) than Selection Sort
104
SKIP 7.7 Important Threads
105
Read and take notes:Ethical Issues
Open-Source Software Development
What are the advantages and disadvantages of open-source software?
What does the success of Linux suggest about the future of open-source software?
Should open-source software be licensed and subject to standard copyright laws?
106
Chapter review questions• Describe the computer problem-solving process and relate it
to Polya’s How to Solve It list
• Distinguish between a simple type and a composite type
• Distinguish between a void subprogram and a value-returning subprogram
• Recognize a recursive problem and write a recursive algorithm to solve it
• Distinguish between an unsorted array and a sorted array
• Describe the Quicksort algorithm
• Apply the linear search, binary search, selection sort and Quicksort to an array of items by hand
• What are the advantages and disadvantages of open-source software?
107
Who am I?
108
I am a
mathematician.
Why is my
picture in a
book about
computer
science?
Homework:
End-of-chapter questions 30, 31, 34, 35, 36, 66 b, c
Not from text: Calculate 7! by hand, showing all the recursive steps, as done in the lecture examples.
109Lecture 5 continues with ch.8
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