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Python代寫-MCD4710
時間:2021-08-25
MCD4710 Workbook
July 22, 2020
Contents
Preface 3
I Programming Fundamentals 4
1 Expressions 5
1.1 Numerical Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Boolean Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Control Flow 9
2.1 Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 While-Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 For-Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Sequences 20
3.1 Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Functions 24
5 Code Traces 31
6 Recursion 32
II Data Structures 36
7 Tables and Matrices 37
7.1 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
7.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
8 Graphs 45
8.1 Paths, cycles, and connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8.2 Spanning Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
8.3 Graph Traversals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
9 Stacks and Queues 52
III Algorithm Analysis 53
10 Invariants 54
10.1 Finding Loop Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1
11 Computational Complexity 59
11.1 Big-Oh Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
11.2 Computational Complexity of Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
IV Additional Resources 67
12 Practice Exam Questions 68
12.1 Discussion of Theoretical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
12.2 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
12.3 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
13 Solutions to Practice Exam Questions 72
13.1 Solutions: Discussion of Theoretical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
13.2 Solutions: Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
13.3 Solutions: Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2
Preface
This workbook is an additional learning resource for FIT1045 and FIT1053. This is our first semester offering a
workbook, so the workbook will grow as the semester progresses. This workbook is not sufficient for the unit. You
must also complete lectures, workshops, and tutorials.
The workbook will be useful in tutorials during time dedicated to self-directed-learning; as a tool for identifying
any gaps in your knowledge; and for providing materials with which to practice. You can use the workbook as much
or as little as is useful for you.
The questions in this workbook reflect the questions you can expect to see on the in-semester tests, and the final
exam. The difficulty of the questions can be understood like so:
1. Clarify questions allow you to identify any misunderstandings you may have with the fundamental tools you
will learn. These are very simple questions. If you can do the practice questions, there is no need to do these
questions.
2. Practice questions allow you to practice the kinds of questions you will see in your tests and exam. Some will
be knowledge-based, some will be problem solving. If you can do all the practice questions, you will be fully
prepared for the content of the tests and exam. Always start with the practice questions, and if you cannot
do them switch to clarify; if they are too easy switch to challenge.
3. Challenge questions allow you to practice questions that are more difficult than most of the questions you
will see in your tests and exam. It can be difficult to problem-solve under test conditions, so practicing on
questions that are more difficult than what you might expect to see can be useful. Some questions on the final
exam will be similar to challenge level questions. Solutions will not be provided for Challenege questions.
4. Extension questions are questions that contain non-assessable content. They may contain a taster of material
that will be taught in later units, or they may show you something cool you can do with the tool you are
being taught.
If you are ever confused by a workbook question, feel free to ask about it in your tutorial, on Moodle, or at a
consult.
Enjoy!
3
Part I
Programming Fundamentals
4
1 | Expressions
1.1 Numerical Expressions
Clarify
What do each of these numerical expressions yield?
1. 9+5
2. 7/2
3. 3**2
4. 10%3
5. 4/2
6. -9%2
7. -10%-4
8. 8//4
9. 14//5
10. 9.5//3
11. 6//3.0
12. 9.5%2
13. 8.0%3
14. 2**2**3
15. 4-3*2
16. 13/(5%3)
Practice
What do each of these numerical expressions yield? Make sure you understand when the expression yields a float
and when it yields an integer.
1. 10/(5%2)
2. 4+11%6//2**2
3. 2**(9%3+2)+12//(5.5-7.5)
4. 1*2**(3%2+2)+14//5
5. 3*2**2**(5//7+1)
6. 3+2*3**(10//5)-7
7. 7*-4**4
8. -8%-6//3
9. 3*3/-4
10. -1//-5*-8
11. 5**1-4
12. -2-7%4
13. 1+0%7-9%1
14. -8*6//2%2+-1
15. 6*8*4/-6//-4
16. 8+4+2*-3/-3
17. -9+-1%-7--8%-4
18. 9/1-7//7%9
19. 4--8*-4-5%-9
20. -8/-7*-7//-5*-9
21. 0-5-4%6//-2
22. -6%8+2+-8//-2
23. -6/8//8%8/4
24. 7--4//-4**6*9
25. -5+-2**-3**-9**2
26. -4**4%-1**2//8
27. 7**-10//2**-10/6
28. -4+2**2**3--9
29. -1**7//4*-2**4
30. -5+4+-3//-1*2
31. -2%4**3-2**9
32. -4--2+8**-7*0
5
Test 1
Test 2
Exam
Challenge
What do each of these numerical expressions yield?
1. int(False or True)*2**2**(14//5*2%3+2)+2
Answers
Write the expression in the Python shell and compare output.
6
1.2 Boolean Expressions
When answering the following questions, keep in mind Python operator precedence given in the lectures, or found in
the Python documentation.1 It can be helpful to think of operators with higher precedence as having brackets around
them. For example, to think of False or not True and True as being False or ((not True) and True).
Also keep in mind the behavior of the boolean operators given in the lectures, or found in the Python documen-
tation.2
Finally, be aware of when type coercion occurs.
Clarify
What do each of these function calls return or boolean expressions yield?3
1. bool(3)
2. bool('')
3. bool([0])
4. bool('cat')
5. not 'dog'
6. not []
What do each of these boolean expressions yield? Assume each expression is self-contained and identify which will
result in a NameError.
Boolean operators
1. True or False
2. True and False
3. False or False
4. not True or True
5. not (True or True)
6. not not True
Booleans - with short-circuiting
1. x or True
2. True or x
3. False and x
4. False or x
5. True and False or x
6. False and True or True or x
Comparison operators
1. 7 <= 3
2. 2 > 2
3. 1 != 0
4. 10 < 5 < 20
5. 3 > 2 < 20
6. 2 == 2 > -4
Comparisons - with booleans
1. True == 1
2. False < 3
3. False >= 0
4. True == 6
5. True == (not not 6)4
6. False == []
Practice
What do each of these boolean expressions yield? Assume each expression is self-contained and identify which will
result in a NameError.
1. '' or not 0
2. False or False and True or True or x
3. True or False or 2
4. 5 or True and False
5. 3>7 or 9>=15 and 4<5 or not 10==4
6. not [] and 2<2 and 'cat'or 7>4<12
1Operator precedence at url: http://docs.python.org/3/reference/expressions.html#operator-precedence
2Boolean Operator behavior at url: http://docs.python.org/3/library/stdtypes.html#boolean-operations-and-or-not.
3Truth value testing at url: http://docs.python.org/3/library/stdtypes.html#truth
4Note that == has higher precedence than not, which means brackets must be used to avoid a SyntaxError.
7
Test 1
Test 2
Exam
7. 'word'or range(1,2) or -6>=8 or -3<=-3
8. -3<=-3 or ''and -4==-1 or -4==-1
9. [] and 'word'or range(1,2) and ['']
10. -6>=8 and 0 and -3<=-3 or 'word'
11. '' and -4==-1 or -5 or range(0)
12. 2<0 and None and None or 2<0
13. not(-4>-1 or not [''] or not 'w'or 'word')
14. '0' and 4 or range(0) or -5
15. 0 and 3==4 or range(0) or ''
16. None and ''and 2<0 and range(1,2)
17. range(3,1) or None or [] or 'word'or ''
18. None and range(1,2) or None or -5 and -4==-1
19. 2<0 or [''] and [] or range(5,3) and 'word'
20. not([''] and -5 or -4==-1) or ''and 'word'
21. -3<=-3 and ''or -5 and '0'or range(-3, -9)
22. '0' or -4==-1 or not ('word'or 2<0 and None)
23. '' or 7 and -4==-1 and -6>=8 or []
24. 0 and not 'word'or 'string'or '0'and -4==-1
Answers
Write the expression in the Python shell and compare output.
8
2 | Control Flow
2.1 Conditionals
Clarify
Code trace
What is the value of x after the code sequence is executed?
1. x = 5
if x < 10:
x = 'small'
2. x = True
if x:
x = 'Yay!'
else:
x = 'No....'
3. x = 'dog'
if x == 'cat':
x = 'meow'
elif x == 'dog':
x = 'woof'
else:
x = 'animal sound'
4. x = 6
if x > 10:
x = 'big'
elif x < 5:
x = 'small'
5. x = 25
if x < 10:
x = 'cold'
elif x > 25:
x = 'hot'
elif x < 25:
x = 'average '
else:
x = 'perfect '
6. x = -10
if x < 0:
x = x * -1
7. a = 5
b = 8
x = 0
if a < b:
x = b
else:
x = a
8. x = 25
if x%2 == 0:
x = x // 2
if x%3 == 0:
x = x // 3
if x%5 == 0:
x = x // 5
Coding
Write a control sequence that ...
1. sets x to the smaller of two numbers, a and b.
9
Test 1
Test 2
Exam
2. sets x to the largest of three numbers, a, b, and c.
3. sets parity to even or odd, based on the value of num.
4. sets opt to 'different', 'pair', or 'triple', based on the value of a, b, and c.
5. sets message to 'Wear {option}' based on the value of temp. The value option should be 'a jacket' if
the temperature is cold, and 'sunscreen' if the temperature is hot. You may decide what temperature is
hot, and what is cold.
Practice
Write a control sequence that ...
1. sets shape to 'triangle' or 'not triangle' given the lengths of three lines.1
2. sets type to 'equilateral', 'isosceles', or 'scalene', given the lengths of the three sides of a triangle,
x, y, and z.
3. sets rounded to the decimal num rounded to the nearest integer. Follow the IEEE rounding rule ‘round to
nearest, ties to even’ (i.e. if rounding x.5, round to the nearest even integer).2
4. sorts a list of three numbers.3
Answers
Clarify
Code trace
1. x == 'small'
2. x == 'Yay!'
3. x == 'woof'
4. x == 6
5. x == 'perfect'
6. x == 10
7. x == 8
8. x == 5
Coding
1. a = ?
b = ?
x = a
if b < a:
x = b
2. a = ?
b = ?
c = ?
x = a
if x < b:
x = b
if x < c:
x = c
3. num = ?
parity = None
if num%2 == 0:
parity = 'even'
else:
parity = 'odd'
1For three lines to make a triangle, every line must be shorter than the sum of the remaining two.
2For more information on the IEEE rounding rules, see here: http://en.wikipedia.org/wiki/IEEE_754
3Attempt after completing section Lists.
10
4. a = ?
b = ?
c = ?
opt = 'different '
if a == b or b == c or a == c:
opt = 'pair'
if a == b and b == c:
opt = 'triple '
5. #15 or lower is cold; above 15 is hot
temp = ?
message = 'Wear '
if temp <= 15:
message = message + 'a jacket '
else:
message = message + 'sunscreen '
Practice
Note that there are many potential implementations. The answers given here are just an example of a possible
implementation.
2. x = ?
y = ?
z = ?
type = None
if x == y and y == z:
type = 'equilateral '
elif x==y or y==z or x==z:
type = 'isosceles '
else:
type = 'scalene '
4. lst = [?, ?, ?]
a = lst [0]
b = lst [1]
c = lst [2]
#correct position of a
if a > b or a > c:
if a > b and a > c:
lst [2] = a
else:
lst [1] = a
#correct position of b
if b < a and b < c:
lst [0] = b
elif:
if b > a and b > c:
lst [2] = b
#correct position of c
if c < a or c < b:
if c < a and c < b:
lst [0] = c
else:
lst [1] = c
11
2.2 Loops
2.2.1 While-Loops
All questions are Clarify. For Practice and higher, go to the Loops subsection.
Fill in the blanks
Fill in the blanks, _, so that each loop creates a list of the numbers 1 to 10.
1. x = _
lst = []
while x <= 10:
lst = lst + [x]
x = x + 1
2. x = 1
lst = []
while True:
lst = lst + [_]
x = x _ _
if x == _ :
_
3. x = 2
lst = []
while x < _:
lst = lst + [x//2]
x = x + _
4. x = _
lst = []
while True:
if x > _:
break
x = x _ _
lst = lst + [x]
Coding
Write a while-loop that ...
1. ... adds all the numbers from 1 to 100 (inclusive) and stores the total in the variable x.
2. ... adds all even numbers from 1 to 50 (inclusive) and stores the total in the variable x.
3. ... creates the list lst = [1,2,3,4,5,6,7,8,9].
4. ... creates the list lst = [10, 8, 6, 4, 2, 0].
5. ... creates the list lst = [1,1,1,1,1,1,1,1,1]. Use the length of the list in the while condition.
6. ... creates the list lst = [1,1,1,1,1,1,1,1,1]. Use a counter.
7. ... creates the list lst = [1,2,3,4,5,6,7,8,9]. Use a break statement.
8. ... creates the list lst = [2,2,2,2,2,2,2,2,2]. Use a break statement.
2.2.2 For-Loops
All questions are Clarify. For Practice and higher, go to the Loops subsection. This section should be done after
you have completed the Range subsection.
Fix the problem
Identify the problem with the given loop and implement a corrected version.
1. #should add the numbers 1 to 10
total = 0
for num in range(2, 10):
total = total + num
12
Test 1
Test 2
Exam
2. #should create the string n='12345'
n = ''
lst = [1,2,3,4,5]
for i in range(len(lst )):
n = n + str(i)
3. #should double each number in lst
lst = [1,2,3,4,5]
for n in lst:
n = n*2
4. #should replace all 'o's with 'a's
b = 'bonono '
for i in range (1,6,2):
b[i] = 'a'
Coding
Write a for-loop that ...
1. ... adds all the numbers from 1 to 100 (inclusive) and stores the total in the variable x.
2. ... adds all even numbers from 1 to 50 (inclusive) and stores the total in the variable x.
3. ... triples all the numbers in the given list lst.
4. ... creates a list containing ten 5s.
5. ... creates a list of all the even numbers that are greater than 1 and less than 100.
6. ... takes a string word and creates a new string identical word but with all 'a's removed.
2.2.3 Loops
Clarify
Choose
What kind of loop would you use for the following scenarios?
1. Halving a number until it is smaller than 100.
2. Doubling every number in a list.
3. Summing an infinite series to estimate an irrational number.
4. Summing every digit in a string.
5. Count the frequency of a character in a string.
6. Asks for input from the user, and keeps asking until input is valid.
Coding
Write a while-loop with the same behaviour as the given for-loop.
1. total = 0
for num in range(55, 105, 5):
total = total + num
2. x = 100
for num in range(-1, -10, -1):
x = x + num
3. lst = [1, 2, 3, 4, 5, 6]
total = 0
for num in lst:
total = total + num
4. lst = ['a', 'b', 'a', 'a', 'b', 'a']
for i in range(len(lst )):
if lst[i] == 'b':
lst[i] = 'aa'
13
Write a for-loop with the same behaviour as the given while-loop:
1. total = 0
num = 20
while num < 42:
total = total + num
num = num + 2
2. total = 0
num = 25
while num <= 100:
total = total + num
num = num + 25
3. lst = [2, 4, 3, 5, 2, 2, 3]
i = 0
while i < len(lst):
lst[i] = lst[i] // 2
i = i + 1
4. lst = ['P', 'y', 't', 'h', 'o', 'n']
word = ''
i = 0
while i < len(lst):
word = word + lst[i]
i = i + 1
Practice
Write loops with the following behaviours:
1. Generates a list containing the first 64 powers of 2.
2. Generates a list containing all the powers of 2 that are less than 1,000,000.
3. Generates a list containing the first 100 numbers of the Fibonacci sequence. I.e. [0,1,1,2,3,...].4
4. Determines whether a given list of numbers is in ascending order.
5. Finds the factorial of the given number.5
6. Finds the greatest common divisor of two given numbers.
7. Finds the lowest common multiple of two given numbers.
8. Finds the sum of all numbers in a given string. Assume numbers are positive integers. For example,
'4 and 20 blackbirds' sums to 24.
9. Finds the sum of all digits in a given number.
10. Finds the mean of a list of numbers.
11. Generates the reverse of a given string. Do not use string slicing or the function reversed().
12. Determines whether a given string is a palindrome. Do not use string slicing or the function reversed().
13. Generates a list of all factors of a given number.
14. Generates the complement of a given binary number. For example, the complement of '110101' is '001010'.
Challenge
Write loops with the following behaviours:
1. Generates a list containing all the perfect numbers between 1 and 1000. I.e. it should return this list:
[6, 28, 496]. A perfect number is a positive integer that is equal to the sum of its positive divisors,
excluding the number itself. For example, the divisors of 6 are 1, 2, 3, and 1 + 2 + 3 = 6.6
2. Generates a list of the first 100 primes.
4Read about Fibonacci numbers here: http://en.wikipedia.org/wiki/Fibonacci_number
5Read about factorials here: http://en.wikipedia.org/wiki/Factorial
6Read about perfect numbers here: http://en.wikipedia.org/wiki/Perfect_number
14
3. Checks whether the given number num is a happy number. A happy number is a number where repeatedly
calculating the sum of the squares of the number’s digits leads to 1. For example, if num=28 then when the
loop terminates is_happy = True; if num=22 then when the loop terminates is_happy = False.7
4. Generates a list containing all permutations of a given string. For example, 'cat' would generate
['cat', 'cta', 'act', 'atc', 'tca', 'tac'].
7Let s0 be our number. Let the sum of the squares of the digits of s0 be s1. Let the sum of the squares of the digits of s1 be s2.
Let the sum of the squares of the digits of sn be sn+1. A number x0 is happy if there exists an n whereby xn = 1. For instance, 28 is
happy as:
22 + 82 =68
62 + 82 =100
12 + 02 + 02 =1
Whereas 22 is unhappy as:
22 + 22 =8
82 =16
... and 16 is a known unhappy number. The following numbers are known unhappy numbers: 0, 4, 16, 20, 37, 42, 58, 89, or 145.
Read more about happy numbers here: http://mathworld.wolfram.com/HappyNumber.html. Note that the description given here is a
limited definition of happy numbers and not accurate if you are a number theory mathematician.
15
Answers
While-loops
Fill in the blanks
1. x = 1
lst = []
while x <= 10:
lst = lst + [x]
x = x + 1
2. x = 1
lst = []
while True:
lst = lst + [x]
x = x + 1
if x == 10:
break
3. x = 2
lst = []
while x < 21:
lst = lst + [x//2]
x = x + 2
4. x = 0
lst = []
while True:
if x > 9:
break
x = x + 1
lst = lst + [x]
Coding
2. x = 0
num = 1
while num <= 50:
if num%2 == 0:
x = x + num
num = num + 1
4. lst = []
num = 10
while num >= 0:
lst = lst + [num]
num = num - 2
6. lst = []
counter = 0
while counter < 9:
lst = lst + [1]
counter = counter + 1
8. lst = []
while True:
if len(lst) == 9:
break
lst = lst + [2]
For-loops
Fix the problem
1. #should add the numbers 1 to 10
total = 0
for num in range(1, 11):
total = total + num
2. #should create the string n='12345'
n = ''
lst = [1,2,3,4,5]
for i in range(len(lst )):
n = n + str(lst[i])
#alternative implementation
n = ''
lst = [1,2,3,4,5]
for num in lst:
n = n + str(num)
16
3. #should double each number in lst
lst = [1,2,3,4,5]
for i in range(len(lst )):
lst[i] = lst[i]*2
4. #should replace all 'o's with 'a's
b = 'bonono '
ban = ''
for char in b:
if char == 'o':
ban = ban + 'a'
else:
ban = ban + char
Coding
1. x = 0
for num in range(1, 101):
x = x + num
2. x = 0
for num in range(0, 51, 2):
x = x + num
3. lst = ??
for i in range(len(lst )):
lst[i] = lst[i]*3
4. lst = []
for _ in range (10):
lst = lst + [5]
5. lst = []
for num in range(2, 100, 2):
lst = lst + [num]
6. word = ??
new = ''
for char in word:
if word != 'a':
new = new + char
Loops
Clarify
1. As there is a condition to be met, should use a while-loop.
2. This can be done with a while-loop or for-loop. A for-loop will likely be simpler to implement.
3. Usually when estimating an irrational number, the intention is to get within a certain level of error. This
means there is a condition to be met, so should use a while-loop.
4. This can be done with a while-loop or for-loop. A for-loop will likely be simpler to implement.
5. This can be done with a while-loop or for-loop. A for-loop will likely be simpler to implement.
6. As there is a condition to be met, should use a while-loop.
For-loops to while-loops
1. total = 0
num = 55
while num <= 100:
total = total + num
num = num + 5
2. x = 100
num = -1
while num > -10:
x = x + num
num = num -1
17
3. lst = [1, 2, 3, 4, 5, 6]
total = 0
i = 0
while i < len(lst):
total = total + lst[i]
i = i + 1
4. lst = ['a', 'b', 'a', 'a', 'b', 'a']
i = 0
while i < len(lst):
if lst[i] == 'b':
lst[i] = 'aa'
i = i + 1
While-loops to for-loops
1. total = 0
for num in range(20, 42, 2):
total = total + num
2. total = 0
for num in range(25, 125, 25):
total = total + num
3. lst = [2, 4, 3, 5, 2, 2, 3]
for i in range(len(lst )):
lst[i] = lst[i] // 2
4. lst = ['P', 'y', 't', 'h', 'o', 'n']
word = ''
for letter in lst:
word = word + letter
Practice
Note that there are many potential implementations. The answers given here are just an example of a possible
implementation.
2. lst = []
index = 0
while 2** index < 1000000:
lst = lst + [2** index]
index = index + 1
4. lst = ??
in_order = True
for i in range(len(lst)-1):
if lst[i] > lst[i+1]:
in_order = False
6. big_num = ??
small_num = ??
while small_num != 0:
remainder = big_num % small_num
big_num = small_num
small_num = remainder
gcd = big_num
8. string = ??
numbers = \
['0','1','2','3','4',
'5','6','7','8','9']
i = 0
sum = 0
while i < len(string ):
if string[i] in numbers:
j = i
while j < len(string) and \
string[j] in numbers:
j = j+1
num = string[i:j]
sum = sum + int(num)
i = j
i = i+1
10. sum = 0
for num in lst:
sum = sum + num
mean = sum/len(lst)
12. string = ??
palindrome = True
for i in range(len(string )//2):
if string[i] != string[-(i+1)]:
palindrome = False
18
14. binary = ??
complement = ''
for digit in binary:
if digit == '0':
complement = complement + '1'
else:
complement = complement + '0'
19
3 | Sequences
Summary of sequence operations:
Operator/Function Result Type
x in seq whether x is contained in seq Boolean
x not in seq whether x is not contained in seq Boolean
len(seq) number of elements contained in seq integer
seq[i] the ith element of seq varies
seq[a:b]
the sub-sequence of seq from the ath
element (inclusive) to the bth element
(exclusive)
same as seq
3.1 Strings
Summary of useful string methods:1
Operation Description
str.count(sub) Returns the number of times substring sub is in string str
str.find(sub) Returns the lowest index sub is in str, or ?1 if not in str
str.format(*args) Formats the string str
str.index(sub)
Returns the lowest index sub is in str, throws error if not
in str
str.join(iter)
Returns a string which is the concatenation of the strings
in iter. Strings in iter are joined by str.
str.lower()
Returns a copy of str with all cased characters in lower
case
str.split(sep)
Returns a list of the words in str, using sep as the delimiter
string
str.strip() Removes all white space from beginning and end of str
str.upper()
Returns a copy of str with all cased characters in upper
case
Clarify
Basic Operators
What do each of these expressions yield? Ensure you include all information.
1. 'I have '+ str(2) + 'cats.'
2. len('animals')
3. 'ing' in 'Coding is fun!'
4. 'Coding is fun!'[0]
5. 'Coding is fun!'[10:13]
6. len('Python')
7. 'FIT1045'[:3]
8. 'Hello, world.'[3:5] + 'magic'[-3:len('magic')]
1For all string methods, see here: http://docs.python.org/3.8/library/stdtypes.html#string-methods
20
Test 1
Test 2
Exam
String Methods
What do each of these expressions yield? Ensure you include all information.
1. 'antelopes, ants, and alpacas are all animals'.count('an')
2. 'Python'.find('y')
3. 'Hello'.find('y')
4. ' and '.join(['lions','tigers','bears'])
5. 'Learn Python in FIT1045'.lower()
6. 'Do you know methods?'.split()
7. '1+3+5=2'.split('+')
8. ' Clean up... '.strip()
Practice
Evaluate
What do each of these expressions yield? Ensure you include all information. Which expressions throw errors or
do not behave as they ought, and how can they be fixed?
1. 'One wonders on a pond'.lower().count('on')
2. len('FIT1045,FIT1008,FIT2004,FIT3155'.split(',')[2])
3. 'and'.join('7 + 9 + 4'.split('+'))
4. 'oar'.upper().join('fat rat sat with bat and goat at math'.split('at')[2:5])
5. '{} {}s'.format('Emit'.lower()[::-1], 'Drawer'.lower()[::-1])
6. 'na'*8 + 'batman'.upper()
7. 'This question is number {}'.format('5 + 2'.split()[0] + '5 + 2'.split()[-1])
8. 'There are '+ 3+7 + 'movies with Ironman.'
Code
In the following questions the only strings you can use are those you are given. Write an expression that ...
1. ... evaluates to 'python' given the string string = 'Students in FIT1045 code in Python.'
2. ... evaluates to 'saidallinonebreath' given the string string = 'Said all in one breath!'
3. ... replaces all es in a string with os, given e='e' and o='o'. Assume the string is string.
4. ... evaluates to the given string, but with the words in reverse order. Assume the string is string. E.g.
'This is weird.' becomes 'weird. is This'
Answers
Write the expression in the Python shell and compare output.
21
3.2 Lists
Summary of useful string methods:2
Operation Description
lst.append(x) Appends item x to list lst
lst.extend(iter) Extends list lst by adding all items from iterable iter
lst.count(x) Returns the number of times item x is in list lst
lst.insert(k,x) Inserts item x at index k in list lst
lst.pop()
Returns the last item in list lst and removes it from the
list
lst.remove(x) Removes the first occurrence of item x in list lst
lst.reverse() Reverses all the items in list lst
Clarify
Basic Operators
What do each of these expressions yield? Ensure you include all information.
1. `I have '+ 2 + ` cats'
List Methods
What do each of these expressions yield? Ensure you include all information.
Practice
Evaluate
What do each of these expressions yield? Ensure you include all information. Which expressions throw errors or
do not behave as they ought, and how can they be fixed?
1. `I have '+ 2 + ` cats'
Code
Answers
Write the expression in the Python shell and compare output.
2For all string methods, see here: http://docs.python.org/3.8/library/stdtypes.html#string-methods
22
Test 1
Test 2
Exam
3.3 Range
Clarify
Evaluate
What sequence of numbers is equivalent to the given ranges?
1. range(6)
2. range(2,5)
3. range(4,10)
4. range(-3,5)
5. range(4,2)
6. range(2,8,2)
7. range(1,10,2)
8. range(25,5,-3)
9. range(-2,-20,-4)
10. range(10,-10,-5)
11. range(-14,-2,-3)
12. range(-4,15,6)
Coding
Express each of the following as a range.
1. [11,12,13,14,15]
2. [9,8,7,6,5,4]
3. [-2,-4,-6,-8,-10]
4. [-5,-4,-3,-2,-1,0,1,2,3,4,5]
5. [0,1,2,3,4,5,6,7,8,9]
6. [0,10,20,30,40,50,60,70,80]
7. [0,-4,-8,-12,-16,-20,-24]
8. -15,-10,-5,0,5,10,15,20
Practice
What do each of these expressions yield?
1. len(range(2,8))
2. 3 in range(-2,5)
3. range(-7,-2)[4]
4. range(4,20)[2:5]
5. range(12,18)[-2]
6. len(range(25,-5,-4))
7. list(range(-4,-18,-3)[2:8])
8. list(range(3,9)[-2:-5:-1])
9. list(range(10,20)[-5:7])
10. list(range(5,20,3)[-4:4])
11. len(range(17,-5,-6))
12. 18 in range(10,20,-2)
13. list(range(0,75,5)[2:-1:3])
14. list(range(10,60,4)[-2:1:-4]
15. range(3,12,4)[2]
16. len(range(2,-7,5))
Challenge
What do each of these expressions yield?
1. range(0,9,2)[:]
2. range(-2,20,5)[1:4]
3. range(3,18,2)[::2]
4. range(1,33,6)[1::2]
5. range(2,20)[-2:-10:-1]
6. range(-4,15,4)[-1:-5:-1]
7. range(-5,20,3)[-8:5:2]
8. range(100,50,-7)[1:-3:3]
Answers
Write the expression in the Python shell and compare output.
23
Test 1
Test 2
Exam
4 | Functions
Clarify
Fill in the blanks
1. #should return the sum of two numbers
def sum(x, y):
_ x + y
2. #should return the given string concatenated with itself
_ double_string(_):
_ my_string + my_string
3. #should return the sound the animal makes
_ animal_sound(_):
if animal == 'cat':
_ 'meow'
_ 'unknown animal '
4. #should return whether the number is even
_ is_even(_):
_ num%2 == 0
Exiting Early
Rewrite the following functions to use multiple return statements, and fewer elif and else statements.
1. def factors(num):
statement = ''
if num%2 == 0:
statement = '2 is the smallest non -1 factor '
elif num%3 == 0:
statement = '3 is the smallest non -1 factor '
else:
statement = 'neither 2 nor 3 are factors '
return statement
2. def contains_even(lst):
contains = False
for num in lst:
if num%2 == 0:
contains = True
return contains
24
Test 1
Test 2
Exam
3. def scissors_paper_rock(player1 , player2 ):
result = None
if player1 == player2:
result = 'Noone wins'
elif (player1 == 'scissors ' and player2 == 'paper' or
player1 == 'paper' and player2 == 'rock' or
player1 == 'rock' and player2 == 'scissors '):
result = 'Player1 wins'
else:
result = 'Player2 wins'
return result
4. def find_mode(x, y, z):
mode = None
if x==y or x==z:
mode = x
elif y==z:
mode = y
else:
mode = (x, y, z)
return mode
Print vs. Return
Identify which of the following code blocks will throw errors.
1. def sum(x, y):
print(x + y)
result = sum(2, 8)
double = result *2
2. def letter_in_word(word , letter ):
in_word = letter in word
return in_word
result = letter_in_word('apple', 'p')
print('Letter p in word apple: ' + str(result ))
3. def divide(x, y):
print(x/y)
result = divide (6,3)
print('6/3 is shown above.')
4. def double_string(string ):
print(string *2)
my_string = 'phone'
result = double_string(my_string)
print(my_string + ' doubled is ' + result)
25
Variable arguments
Identify which of the following code blocks will throw errors.
1. def sum(*nums):
total = 0
for n in nums:
total = total + n
return total
sum(3, 7, 2)
2. def multiply(start_val , *nums):
for n in nums:
start_val = start_val * n
return start_val
multiply (10, 2, 2, 3)
3. def divide(numer , denom , *nums):
total = numer/denom
for n in nums:
total = total / n
return total
divide (5)
4. def subtract (*nums):
total = 0
for n in nums:
total = total - n
return total
subtract ()
Sequence Unpacking
Identify which of the following code blocks will throw errors.
1. def find_range(lst):
minimum = min(lst)
maximum = max(lst)
return minimum , maximum
my_min , my_max = find_range ([2, 2, 7, 1, 3, 9, 3])
2. def move_point_up(x, y, z):
return x+1, y, z
x, y, z = move_point_up (2, 3, 1)
3. def move_point_right(x, y, z):
return x, y+1, z
new_point = move_point_right (2, 3, 1)
26
4. def move_point_down(x, y, z):
return x-1, y, z
x, y = move_point_down (2, 3, 1)
5. def move_point_left(x, y):
return x, y-1
x, y, z = move_point_left (2, 3)
6. def move_point_left(x, y):
return x, y-1
x, y, z = move_point_left (2, 3)
Coding
Write a function that ...
1. ... takes integers x and y as input, where x2. ... takes integers x and y as input, and returns the minimum integer (if they are the same size, than both are
the minimum).
3. ... takes as input a list lst containing only 0s and 1s, and returns True if there are more 1s than 0s, else
returns False.
4. ... takes as input a list lst containing only numbers, and returns the list with all numbers inside it doubled.
5. ... takes as input a list lst containing exactly five numbers, and returns True if there are three or more
identical numbers in a row, else returns False.
6. ... takes as input a string word, and returns a new string identical to word but with all vowels removed
(a,e,i,o,u).
Practice
Write a function that ...
1. ... takes as input three integers, date, month, year, and returns True if the three integers make a valid date,
else returns False.
2. ... takes as input a list numbers containing exactly five numbers, and returns an integer that is the maximum
number of identical numbers in a row.
3. ... takes as input a list numbers containing only numbers, and an integer n, and returns the sum of every nth
number in numbers.
Keep an eye on this section. It will likely grow over the semester.
27
Answers
Clarify
Fill in the blanks
1. #should return the sum of two numbers
def sum(x, y):
return x + y
2. #should return the given string concatenated with itself
def double_string(my_string ):
return my_string + my_string
3. #should return the sound the animal makes
def animal_sound(animal ):
if animal == 'cat':
return 'meow'
return 'unknown animal '
4. #should return whether the number is even
def is_even(num):
return num%2 == 0
Exiting Early
These are suggested alternative implementations. Note that the following implementations are not necessarily better
than the alternatives.
1. def factors(num):
if num%2 == 0:
return '2 is the smallest non -1 factor '
if num%3 == 0:
return '3 is the smallest non -1 factor '
return 'neither 2 nor 3 are factors '
2. def contains_even(lst):
for num in lst:
if num%2 == 0:
return True
return False
3. def scissors_paper_rock(player1 , player2 ):
if player1 == player2:
return 'Noone wins'
if (player1 == 'scissors ' and player2 == 'paper' or
player1 == 'paper' and player2 == 'rock' or
player1 == 'rock' and player2 == 'scissors '):
return 'Player1 wins'
return 'Player2 wins'
28
4. def find_mode(x, y, z):
if x==y or x==z:
return x
if y==z:
return y
return (x, y, z)
Print vs. Return
1. Code block throws an error, as the function call sum(2,8) returns None, and the assignment double = None*2
throws an error.
2. Code block does not throw an error. The function call letter_in_word('apple', 'p') returns True, and
the print statement print('Letter p in word apple: '+ str(True)) succeeds in printing.
3. Code block does not throw an error. The function call divide(6,3) prints 2.0 and returns None, and the
print statement print('6/3 is shown above.') succeeds in printing as it makes no reference to the function
call.
4. Code block throws an error, as the function call double_string('phone') returns None, and the print
statement print('phone'+ 'doubled is '+ None) throws an error.
Variable arguments
1. Code block does not throw an error, as the function definition sum(*nums) allows *nums to take any number
of arguments, thus sum(3, 7, 2) is a valid function call.
2. Code block does not throw an error, as the function definition multiply(start_val, *nums) requires at
least one argument for start_val, and any number of arguments for *nums, thus multiply(10, 2, 2, 3) is
a valid function call.
3. Code block throws an error, as the function definition divide(numer, denom, *nums) requires at least two
arguments, one for numer and one for denom, and any number of arguments for *nums, thus divide(5) is an
invalid function call as it has insufficient arguments.
4. Code block does not throw an error, as the function definition subtract(*nums) allows *nums to take any
number of arguments (including zero arguments), thus subtract() is a valid function call.
Sequence Unpacking
1. Code block does not throw an error, as the function returns two values, which are unpacked and assigned to
two variables.
2. Code block does not throw an error, as the function returns three values, which are unpacked and assigned
to three variables.
3. Code block does not throw an error, as the function returns three values as a tuple, and the tuple is assigned
to one variable.
4. Code block throws an error, as the function returns three values, thus they cannot be unpacked into two
variables.
5. Code block throws an error, as the function returns two values, thus they cannot be unpacked into three
variables.
6. Code block throws an error, as the function returns two values, thus they cannot be unpacked into three
variables.
29
Coding
1. def sum_range(x, y):
return sum(range(x,y+1))
2. def minimum(x, y):
if xreturn x
return y
3. def triples(lst):
ones = lst.count (1)
zeroes = lst.count (0)
return ones > zeroes
4. def double_values(lst):
for i in range(len(lst )):
lst[i] = lst[i]*2
return lst
5. def find_triple(lst):
return (lst[0] == lst [1] and lst [1] == lst[2] or
lst [1] == lst[2] and lst [2] == lst [3] or
lst [2] == lst[3] and lst [3] == lst [4] )
6. def remove_vowels(word):
vowels = 'aeiou'
new_word = ''
for char in word:
if char not in vowels:
new_word = new_word + char
return new_word
Practice
Ask for assistance in tutorials and consultations.
30
5 | Code Traces
To be completed.
31
6 | Recursion
Clarify
1. The following recursive function find_min returns the minimum of a list. Fill in the base case
def find_min(lst):
#base case
return min(lst[0], find_min(lst [1:]))
For example:
>>> find_min ([5, 7, 8 , -5, 10])
-5
2. The following recursive function first_upper returns the first upper case letter of a string. It returns an empty
string if it cannot find any. Fill in the base case.
def first_upper(string ):
#base case
else:
return first_upper(string [1:])
For example:
>>> first_upper('abc Xyz)
X
3. The following recursive function count_repeat counts how many times a target element appears in a list. Fill
in the recursive case.
def count_repeat(lst ,target ):
if len(lst) == 0:
return 0
#recursive case
For example:
>>> count_repeat ([15, 1, 10, 9, 10], 10)
2
4. Write a function which returns factorial of a given number n. For example:
>>> factorial (5)
120
32
Test 2
Exam
Practice
1. Write a recursive function digit_sum that sums the digits of an integer number, without converting the
number to string. For example:
>>> digit_sum (523)
10
2. Write a recursive function remove_neg to remove all negative numbers from a list. For example:
>>> remove_neg ([2, -5, 3, -7, 10])
[2, 3, 10]
3. Write a recursive function which returns the value of the Fibonacci series for the nth element. For example:
>>> fib(6)
8
>>> fib (10)
55
Challenge
1. Write a recursive function replace_between(string, deli, ch) to replace all characters bounded by a
delimiter deli (special word) in a string with a character ch. Note that the string contains zero or at most one
special word. For example:
>>> replace_between('zzz$1234$xx ', '$', '0')
'zzz$0000$xx '
>>> replace_between('aaa$bbb ', '$', '0')
'aaa$bbb '
Hints:
– try to reduce string by one or two characters (if possible) in each recursion.
– think about the smallest string you might have and what should be returned with such string.
– you might have more than one base case and recursive case.
2. Write a function that returns all the permutations of a given string. For example:
>>> permute('abc')
['abc', 'acb', 'bac', 'bca', 'cab', 'cba']
Hints:
– the permutations of a single character is itself.
– the permutations of ’abc’ is given by
permute('abc') == ['a' + permute('bc'),
'b' + permute('ac'),
'c' + permute('ab')]
Answers
Clarify
1. def find_min(lst):
if len(lst) == 1:
return lst[0]
return min(lst[0], find_min(lst [1:]))
33
2. def first_upper(string ):
if string =='':
return ''
elif string [0] != ' ' and string [0] == string [0]. upper ():
return string [0]
else:
return first_upper(string [1:])
3. def count_repeat(lst ,target ):
if len(lst) == 0:
return 0
if lst[0] == target:
return 1 + count_repeat(lst[1:], target)
else:
return count_repeat(lst[1:], target)
4. def factorial(n):
if n == 1 or n == 0:
return 1
else:
return n * factorial(n-1)
Practice
1. def digit_sum(number ):
if number == 0:
return 0
return number %10 + digit_sum(number //10)
2. def remove_neg(lst):
if len(lst) == 0:
return []
if lst [0] >=0:
return [lst [0]] + remove_neg(lst [1:])
else:
return remove_neg(lst [1:])
3. def fib(n):
if n == 0:
return 0
elif n == 1:
return 1
return fib(n-1) + fib(n-2)
34
Challenge
1. def replace_between(string , ch1 , ch2):
if len(string) <= 2:
return string
if string [0] == string [-1] == ch1:
return ch1 + ch2*(len(string )-2) + ch1
if string [0] == ch1:
return replace_special_word(string [:-1], ch1 , ch2) + string [-1]
if string [-1] == ch1:
return string [0] + replace_special_word(string [1:], ch1 , ch2)
return string [0] + replace_special_word(string [1:-1], ch1 , ch2) + string [-1]
2. def permute(s):
newlst = []
if len(s) == 1:
return s
else:
for i,j in enumerate(s):
for perm in permute(s[:i] + s[i+1:]):
newlst += [j + perm]
return newlst
35
Part II
Data Structures
36
7 | Tables and Matrices
7.1 Tables
Clarify
Code trace
What is the value of x after the code sequence is executed?
1. table = []
for _ in range (2):
row = [None , None]
table.append(row)
x = table
2. table = []
row = [1, 2, 3]
for _ in range (3):
table.append(row)
table [2][2] = 5
x = table [0]
3. table = []
row = [None ]*3
for _ in range (4):
table.append(row)
for i in range(len(table )):
table[i][0] = i+1
table[i][1] = i+2
table[i][2] = i+3
x = table [0]
4. table = []
for _ in range (3):
row = [1, 2]
table.append(row)
table [1][0] = 3
x = table [0]
5. table = []
for _ in range (3):
row = [None ]*3
table.append(row)
for i in range(len(table )):
table[i][0] = i+1
table[i][1] = i+2
table[i][2] = i+3
x = table [0]
6. table = []
for i in range (4):
table.append ([])
for j in range (2):
table[i]. append(j)
table [2][1] = 3
x = table
Accessing elements
Assuming table is an n×m table, how can the following items be accessed?
1. The element in the first row and first column.
2. The (entire) second row.
3. The element in the final row and first column.
4. The element in the ith row and second column.
5. The third, fourth, and fifth element from the nineth row.
37
Test 1
Test 2
Exam
6. The second-from-the-bottom element in the fourth column.
Coding
Write a function that returns a table that ...
1. ... is n× n, and all elements are None.
2. ... is 3× 3, and each row contains the letters 'a', 'b', and 'c' in alphabetical order.
3. ... is 6× 3, and each row is a different permutation of three given items.
4. ... is 4× 4, and contains the numbers 1 to 16.
5. ... is n× n, and contains the numbers 1 to n2.
Practice
Code tables
Write a function that returns a table that ...
1. ... is an n× n multiplication table. E.g. if n=3, the function returns [[1,2,3],[2,4,6],[3,6,9]].
2. ... concatenates each entry in two given sequences. E.g. if the given sequences are 'abc' and '12', the
function returns [['a1','a2'], ['b1','b2'], ['c1','c2']].
3. ... is a 2 × 2 truth table. The function should take a string version of a boolean operator as input. E.g. if
operator = 'and', the function returns [[True, False],[False, False]].
Access and alter tables
Unless otherwise stated, each of the following functions takes as input an n×m table of numbers.
1. Implement a function that returns the mean of the row with the largest mean.
2. Implement a function that returns the mean of the column with the smallest mean.
3. Implement a function that returns True if at least one row contains all the same values, else returns False.
4. Implement a function that returns True if at least one column contains all the same values, else returns False.
5. Implement a function that takes as input an n × n table of numbers, and returns True if both diagonals
contains all the same values, else returns False.
6. Implement a function that returns the mode of all values in the table.
7. Implement a function that replaces the highest and lowest value in each column with the median of that
column.
8. Implement a function that replaces the highest and lowest value in each row with the mean of the row.
9. Implement a function that appends the sum of the row to the end of each row.
Challenge
Tic-Tac-Toe
Tic-tac-toe, aka noughts and crosses, is a simple two-person game where both players, X and O, take turns placing
their mark, with the aim of placing three of their marks in a horizontal, vertical, or diagonal row.
Implement a function, ttt_winning_move(board), that takes an n × n board, represented as an n × n table,
where each row contains n items from: 'x', 'o', and None. The function should return a list of players with an
immediate winning move, where the aim is to place n marks in a horizonal, vertical, or diagonal row. The function
should return an empty list [] if no one has the winning move.
38
Knights
Chess is a two-person strategy board game where both players, black and white, take turns moving pieces, with
the aim of placing the other player’s king in check. One of the pieces in chess is the knight. The knight moves in
an L-shape: either two squares vertically and one square horizontally, or two squares horizontally and one square
vertically.
Implement a function, move_knight(board, start, move), that takes an 8× 8 board, represented as an 8× 8
table, where each row contains 8 items from: 'K', and '_'; a tuple start that contains the rank and file (row and
column) of the current position of the knight; and a tuple move that contains the rank and file where the player
wishes to move the knight. If the move is legal, the function should return the board with the knight moved to its
new position; else the function should return the original board.
39
Answers
Clarify
Code trace
1. x == [[None, None], [None, None]]
2. x == [1, 2, 5]
3. x == [4, 5, 6]
4. x == [1, 2]
5. x == [1, 2, 3]
6. x == [[0,1], [0,1], [0,3], [0,1]]
Accessing elements
1. table[0][0]
2. table[1]
3. table[-1][0]
4. table[i][1]
5. table[8][2:5]
6. table[2][3]
Coding
1. def n_table(n):
table = []
for _ in range(n):
table.append ([None]*n)
return table
2. def abc_table ():
table = []
for _ in range (3):
table.append (['a', 'b', 'c'])
return table
3. def perm_table(x, y, z):
table = []
table.append ([x,y,z])
table.append ([x,z,y])
table.append ([y,x,z])
table.append ([y,z,x])
table.append ([z,x,y])
table.append ([z,y,x])
return table
4. def sixteen_table ():
table = []
for i in range (4):
row = []
for j in range (4):
row = row + [i*4 + j + 1]
table.append(row)
return table
40
5. def n_squared_table(n):
table = []
for i in range(n):
row = []
for j in range(n):
row = row + [i*n + j + 1]
table.append(row)
return table
Practice
Ask for assistance in tutorials and consultations.
41
7.2 Matrices
Practice
Solve the following systems of linear equations.
2x2 Linear Systems
1. Easy Example (
2 1
6 4
)(
x1
x2
)
=
(
7
22
)
2. Easy Example (?1 2
4 4
)(
x1
x2
)
=
(
7
?4
)
3x3 Linear Systems
1. Standard Example ?? 4 1 0?4 ?2 2
8 1 3
????x1x2
x3
?? =
?? 6?6
15
??
2. Standard Example ??3 ?2 59 ?2 16
3 ?10 1
????x1x2
x3
?? =
???10
?5
??
3. Pivot Required ?? 2 ?3 1?4 6 2
6 ?8 4
????x1x2
x3
?? =
?? 2?4
8
??
42
Exam
Answers
Practice
2x2 Linear Systems
1. Easy Example (
2 1
6 4
)(
x1
x2
)
=
(
7
22
)
Perform row operation: R2← R2? 3×R1. This gives:(
2 1
0 1
)(
x1
x2
)
=
(
7
1
)
Back substitution gives x2 = 1, 2x1 + 1 = 7 =? x1 = 3
2. Easy Example (?1 2
4 4
)(
x1
x2
)
=
(
7
?4
)
Perform row operation: R2← R2 + 4×R1. This gives:(?1 2
0 12
)(
x1
x2
)
=
(
7
24
)
Back substitution gives 12x2 = 24 =? x2 = 2, ?x1 + 2x2 = 7 =? x1 = ?3
3x3 Linear Systems
1. Standard Example ?? 4 1 0?4 ?2 2
8 1 3
????x1x2
x3
?? =
?? 6?6
15
??
Perform row operations: R2← R2 +R1 and R3← R3? 2×R1. This gives:??4 1 00 ?1 2
0 ?1 3
????x1x2
x3
?? =
??60
3
??
Perform row operation: R3← R3?R2. This gives:??4 1 00 ?1 2
0 0 1
????x1x2
x3
?? =
??60
3
??
Back substitution gives x3 = 3, ?x2 + 2x3 = 0 =? x2 = 6, 4x1 + x2 = 6 =? x1 = 0.
2. Standard Example ??3 ?2 59 ?2 16
3 ?10 1
????x1x2
x3
?? =
???10
?5
??
Perform row operations: R2← R2? 3×R1 and R3← R3?R1. This gives:??3 ?2 50 4 1
0 ?8 ?4
????x1x2
x3
?? =
???13
?4
??
Perform row operation: R3← R3 + 2×R2. This gives:??3 ?2 50 4 1
0 0 ?2
????x1x2
x3
?? =
???13
2
??
Back substitution gives ?2x3 = 2 =? x3 = ?1, 4x2+x3 = 3 =? x2 = 1, 3x1?2x2+5x1 = ?1 =? x1 = 2
43
3. Pivot Required ?? 2 ?3 1?4 6 2
6 ?8 4
????x1x2
x3
?? =
?? 2?4
8
??
Perform row operations: R2← R2 + 2×R1 and R3← R3? 3×R1. This gives:??2 ?3 10 0 4
0 1 1
????x1x2
x3
?? =
??20
2
??
Swap rows 2 and 3 to get: ??2 ?3 10 1 1
0 0 4
????x1x2
x3
?? =
??22
0
??
Back substitution gives x3 = 0, x2 + x3 = 2 =? x2 = 2, 2x1 ? 3x2 + x3 = 2 =? x1 = 4.
44
8 | Graphs
8.1 Paths, cycles, and connectivity
Clarify
For each of the graphs in Figure 8.1:
1. How many cycles are in the graph? Here, we consider two cycles to be identical if they contain the same
vertices. That is, we do not care which vertex is the start/end vertex.
2. How many paths exist between points A and B?
3. What is the minimum number of edges which, if removed, would disconnect the graph?
0 1 2 3
4
A B
(a)
0 1 2
5 4 3
A
B
(b)
0 1 2
3 4
5 6 7
A
B
(c)
0 1 2
5 4 3
A
B
(d)
0 2
1
6 3
5 4
A
B
(e)
0 1 2 3
7 6 5 4
A
B
(f)
Figure 8.1: Graphs
45
Test 2
Exam
Answers
Clarify
Graph 8.1a
? Cycles: 0
? Edges to remove: 1
? Paths A→B: 1
– 0→ 1→ 2→ 3
Graph 8.1b
? Cycles: 1
– 0→ 1→ 2→ 3→ 4→ 5→ 0
? Edges to remove: 2
? Paths A→B: 2
– 0→ 1→ 2→ 3
– 0→ 5→ 4→ 3
Graph 8.1c
? Cycles: 3
– 0→ 3→ 5→ 0
– 1→ 3→ 6→ 4→ 1
– 2→ 4→ 7→ 2
? Edges to remove: 2
? Paths A→B: 8
– 0→ 3→ 1→ 4→ 2→ 7
– 0→ 3→ 1→ 4→ 7
– 0→ 3→ 6→ 4→ 2→ 7
– 0→ 3→ 6→ 4→ 7
– 0→ 5→ 3→ 1→ 4→ 2→ 7
– 0→ 5→ 3→ 1→ 1→ 7
– 0→ 5→ 3→ 6→ 4→ 2→ 7
– 0→ 5→ 3→ 6→ 4→ 7
Graph 8.1d
? Cycles: 3
– 0→ 5→ 4→ 1→ 0
– 1→ 4→ 3→ 2→ 1
– 0→ 1→ 2→ 3→ 4→ 5→ 0
? Edges to remove: 2
? Paths A→B: 4
– 0→ 1→ 2→ 3
– 0→ 1→ 4→ 3
– 0→ 5→ 4→ 1→ 2→ 3
– 0→ 5→ 4→ 3
Graph 8.1e
? Cycles: 3
– 1→ 0→ 6→ 1
– 1→ 5→ 4→ 1
– 1→ 3→ 2→ 1
? Edges to remove: 2
? Paths A→B: 4
– 0→ 1→ 2→ 3
– 0→ 1→ 3
– 0→ 6→ 1→ 2→ 3
– 0→ 6→ 1→ 3
Graph 8.1f
? Cycles: 15
– 0→ 1→ 7→ 0
– 3→ 2→ 4→ 3
– 6→ 1→ 7→ 6
– 5→ 2→ 4→ 5
– 0→ 1→ 6→ 7→ 0
– 0→ 1→ 2→ 5→ 6→ 7→ 0
– 0→ 1→ 2→ 3→ 4→ 5→ 6→ 7→ 0
– 1→ 2→ 5→ 6→ 1
– 1→ 2→ 3→ 4→ 5→ 6→ 1
– 2→ 3→ 4→ 5→ 2
– 1→ 7→ 6→ 5→ 4→ 2→ 1
– 1→ 7→ 6→ 5→ 2→ 1
– 1→ 7→ 6→ 5→ 4→ 3→ 2→ 1
– 2→ 4→ 5→ 6→ 1→ 2
– 2→ 4→ 5→ 6→ 7→ 0→ 1→ 2
? Edges to remove: 2
? Paths A→B: 18
– 0→ 7→ 6→ 5→ 4
– 0→ 7→ 6→ 5→ 2→ 4
– 0→ 7→ 6→ 5→ 2→ 3→ 4
– 0→ 7→ 6→ 1→ 2→ 5→ 4
46
– 0→ 7→ 6→ 1→ 2→ 4
– 0→ 7→ 6→ 1→ 2→ 3→ 4
– 0→ 7→ 1→ 6→ 5→ 4
– 0→ 7→ 1→ 6→ 5→ 2→ 4
– 0→ 7→ 1→ 6→ 5→ 2→ 3→ 4
– 0→ 7→ 1→ 2→ 4
– 0→ 7→ 1→ 2→ 5→ 4
– 0→ 7→ 1→ 2→ 3→ 4
– 0→ 1→ 6→ 5→ 2→ 3→ 4
– 0→ 1→ 6→ 5→ 2→ 4
– 0→ 1→ 6→ 5→ 4
– 0→ 1→ 2→ 3→ 4
– 0→ 1→ 2→ 4
– 0→ 1→ 2→ 5→ 4
47
8.2 Spanning Trees
Clarify
How many spanning trees are there for the graphs in Figures 8.1b and 8.1e?
Practice
Use Prim’s algorithm to find a minimum spanning tree for graph G in Figure 8.2. List the edges added to the tree
after each iteration of the algorithm, and state the weight of the final spanning tree.
0 1 2
3 4 5
6 7 8
1 1
1
3
2
3 3
2
4 2
4
3 3
Figure 8.2: Graph G
48
Exam
Answers
Clarify
? The graph in Figure 8.1b has six spanning trees. Each one is a path with five edges:
– a path from vertex 0 to vertex 5;
– a path from vertex 1 to vertex 0;
– a path from vertex 2 to vertex 1;
– a path from vertex 3 to vertex 2;
– a path from vertex 4 to vertex 3; and
– a path from vertex 5 to vertex 4.
? The graph in Figure 8.1e has 27 spanning trees. Each triangular subgraph must be covered by exactly two
edges in a spanning tree of the graph (any more will create a cycle; any fewer will result in a disconnected
graph, or one that doesn’t cover all the vertices). There are 3 different ways of choosing 2 edges in a triangle,
and 33 different ways of combining these choices for all three triangles.
Practice
After each iteration, the set of edges in the partial spanning tree will be as follows (note the ordering given here is
not a unique solution):
? {(0, 1)}
? {(0, 1), (0, 3)}
? {(0, 1), (0, 3), (1, 2)}
? {(0, 1), (0, 3), (1, 2), (0, 7)}
? {(0, 1), (0, 3), (1, 2), (0, 7), (4, 7)}
? {(0, 1), (0, 3), (1, 2), (0, 7), (4, 7), (4, 5)}
? {(0, 1), (0, 3), (1, 2), (0, 7), (4, 7), (4, 5), (6, 7)}
? {(0, 1), (0, 3), (1, 2), (0, 7), (4, 7), (4, 5), (6, 7), (7, 8)}
The total weight of the minimum spanning tree is 15.
49
8.3 Graph Traversals
Practice
Apply Breadth First Search and Depth First Search to each of the graphs in Figure 8.3. Start your traversal from
vertex 0, and write down the order in which vertices will be visited during the traversal.
0 1 2
3 4 5
6 7
8 9
(a)
0 1 2 3
4 5 6
7 8 9 10
(b)
Figure 8.3: Graphs for BFS and DFS
50
Test 2
Exam
Answers
Practice
Note that these solutions are not unique — there are other possible correct orderings for both the BFS and DFS
traversals.
? Graph 8.3a
– BFS traversal: 0→ 3→ 4→ 1→ 6→ 7→ 2→ 8→ 9→ 5
– DFS traversal: 0→ 3→ 6→ 8→ 9→ 7→ 5→ 2→ 1→ 4
? Graph 8.3b
– BFS traversal: 0→ 4→ 5→ 1→ 7→ 8→ 9→ 2→ 10→ 6→ 3
– DFS traversal: 0→ 4→ 7→ 5→ 8→ 9→ 10→ 2→ 6→ 3→ 1
51
9 | Stacks and Queues
to be written...
52
Part III
Algorithm Analysis
53
10 | Invariants
10.1 Finding Loop Invariants
Clarify
For the following Python functions, identify the loop exit condition, and the loop invariants that, together with the
loop exit condition, show that the function produces the correct output.
1. def factorial(n):
"""
Input : positive integer n
Output: n!
"""
res = 1
m = 1
while m <= n:
res *= m
m = m+1
return res
2. def reverse(lst):
"""
Input : list lst of elements
Output: list of elements lst in reverse order
"""
revlst = []
while len(lst) > 0:
revlst.append(lst.pop ())
return revlst
3. def exponentiation(x,n):
"""
Input : integers x, n
Output: x**n
"""
res = 1
for i in range(n):
res *= x
return res
54
Test 2
Exam
4. def quotient_and_remainder(x,y):
"""
Input : positive integers x, y
Output: x//y, x%y
"""
q=0
r=x
while y <= r:
r = r-y
q = 1+q
return q,r
Practice
For the following Python functions, identify the loop invariant(s), the loop exit condition, and the loop post-
condition(s) which show the algorithm’s correctness.
1. def sum_change(list_of_cents ):
"""
Input: A list list_of_cents of non -negative integers
Output: Two non -negative integers (total_dollars , total_cents),
such that the sum of all numbers in list_of_cents is
total_dollars *100 + total_cents , and total_cents < 100.
For example:
>>> sum_change ([50 ,35 ,165 ,20])
(2, 70)
"""
total_dollars , total_cents = 0, 0
while len(list_of_cents) > 0:
total_cents += list_of_cents.pop()
if total_cents >= 100:
total_dollars += 1
total_cents -= 100
return (total_dollars , total_cents)
2. def even_sum(lst):
"""
Input: A list lst of integers
Output: True if sum(lst) is even , False otherwise.
For example:
>>> even_sum ([1,2,3,4])
True
>>> even_sum ([1,2,3,4,5])
False
"""
is_even = True
while len(lst) > 0:
if lst.pop() % 2 == 1:
is_even = not is_even
return is_even
55
Answers
Clarify
1. def factorial(n):
"""
Input : positive integer n
Output: n!
"""
res = 1
m = 1
while m <= n:
#Invariant: res == (m-1)!
res *= m
m = m+1
#Invariant: res == (m-1)!
#Exit condition: m == n+1
#Post -condition: res == n!
return res
2. def reverse(lst):
"""
Input : list lst of elements
Output: list of elements lst in reverse order
"""
revlst = []
while len(lst) > 0:
#Invariant: lst + reverse(revlst) gives the original list
revlst.append(lst.pop ())
#Invariant: lst + reverse(revlst) gives the original list
#Exit condition: lst is empty
#Post -condition: revlst contains the entire original list in reverse order
return revlst
3. def exponentiation(x,n):
"""
Input : integers x, n
Output: x**n
"""
res = 1
for i in range(n):
#Invariant: x**i == res
res *= x
#Invariant: x**(i+1) == res
#Exit condition: i == n - 1
#Post -condition: res == x**n
return res
56
4. def quotient_and_remainder(x,y):
"""
Input : positive integers x, y
Output: x//y, x%y
"""
q=0
r=x
while y <= r:
#Invariant: x == y*q + r
r = r-y
q = 1+q
#Invariant: x == y*q + r
#Exit condition: r < y
#Post -condition: x == y*q + r and r < y.
In other words , q == x//y and r == x%y.
return q,r
Practice
1. def sum_change(list_of_cents ):
"""
Input: A list list_of_cents of non -negative integers
Output: Two non -negative integers (total_dollars , total_cents),
such that the sum of all numbers in list_of_cents is
total_dollars *100 + total_cents , and total_cents < 100.
For example:
>>> sum_change ([50 ,35 ,165 ,20])
(2, 70)
"""
total_dollars , total_cents = 0, 0
while len(list_of_cents) > 0:
total_cents += list_of_cents.pop()
if total_cents >= 100:
total_dollars += 1
total_cents -= 100
#INV1: sum of original list_of_cents == (total_dollars * 100) +
# total_cents + sum(list_of_cents)
#INV2: total_cents < 100
#EXC: len(list_of_cents) == 0
#POC: sum of original list_of_cents == (total_dollars * 100) +
# total_cents , and total_cents < 100
return (total_dollars , total_cents)
57
2. def even_sum(lst):
"""
Input: A list lst of integers
Output: True if sum(lst) is even , False otherwise.
For example:
>>> even_sum ([1,2,3,4])
True
>>> even_sum ([1,2,3,4,5])
False
"""
is_even = True
while len(lst) > 0:
if lst.pop() % 2 == 1:
is_even = not is_even
#INV: is_even is True if and only if the sum of
# all items removed from lst is even
#EXC: len(lst) == 0
#POC: is_even is True if and only if the sum of all
# items in the original list is even
return is_even
58
11 | Computational Complexity
11.1 Big-Oh Notation
The purpose of the big-oh notation is to succinctly describe the “order of growth” of a function. Recall the formal
definition from the lecture:
Definition 1. A function f(n) is in O(g(n)) (read “f is in big oh of g”) if there are positive numbers c and n0 such
that for all n ≥ n0 it holds that t(n) ≤ cg(n).
In computational complexity analysis we apply this idea to time complexity functions T (n), i.e., the functions
that describe the number of elementary steps that an algorithm needs to perform for an input of size n.
Clarify
Polynomial bounds
For each of the following concrete time complexity give the tightest bound in terms of a simple polynomial function
nc using big-Oh notation. That is, determine the smallest c such that T (n) ∈ O(nc).
1. T (n) = n+ 10
2. T (n) = 2n
3. T (n) = 5n+ 15
4. T (n) = n2
5. T (n) = 2n2 + n
6. T (n) = n2 ? 10n? 1
7. T (n) = n/2
8. T (n) = 10
9. T (n) = n mod 100
Poly-logarithmic bounds
For each of the following concrete time complexity give the tightest bound in terms of a simple poly-logarithmic
function nc logd n using big-Oh notation. That is, determine the smallest pair c and d such that T (n) ∈ O
(
nc logd n
)
.
That is, we consider n log2 n to be smaller1 than n2 log n. Generally, the sum log refers to the logarithm of base 2
and by log2 n we mean (log n)2.
1. T (n) = 10 log n+ 10
2. T (n) = n+ log n
3. T (n) = n log n
4. T (n) = log(10n+ 100)
5. T (n) = log10 n
6. T (n) = log1.5 n
7. T (n) = log n2
8. T (n) = log2 n
9. T (n) = log2 n+ n log n
Practice
For each of the following concrete time complexity give the tightest bound in terms of a simple poly-logarithmic
function nc logd n using big-Oh notation (see clarify part for more details).
1Technically, we could say that we use the lexicographic order to determine what is a smaller pair. That is, we consider (c, d) = (1, 2)
to be smaller than (c′, d′) = (2, 1).
59
Test 2
Exam
1. T (n) = 3, 700, 000
2. T (n) = 10n2 + 2
3. T (n) = sin(360n/(2pi))
4. T (n) = 10n2 + 2 log3 n
5. T (n) = n3/10? n2
6. T (n) = 25n cos(2pi(n mod 100)/100)
7. T (n) = n2/2 + 10n log n
8. T (n) = 10n log(2n) + 5
9. T (n) = n3 + n2 log(n2)
10. T (n) = 1 + pi2(log n)
11. T (n) = n+ sin(360n/(2pi)) log n
12. T (n) = (10n2 + 10)/n
Challenge
1. T (n) = n2 log n+ n log2 n 2. T (n) = 10

n+ log n
60
Answers
Clarify
Polynomial bounds
1. T (n) ∈ O(n)
2. T (n) ∈ O(n)
3. T (n) ∈ O(n)
4. T (n) ∈ O(n2)
5. T (n) ∈ O(n2)
6. T (n) ∈ O(n2)
7. T (n) ∈ O(n)
8. T (n) ∈ O(1)
9. T (n) ∈ O(1)
Poly-logarithmic bounds
1. T (n) ∈ O(log n)
2. T (n) ∈ O(n)
3. T (n) ∈ O(n log n)
4. T (n) ∈ O(log n)
5. T (n) ∈ O(log n)
6. T (n) ∈ O(log n)
7. T (n) ∈ O(log n)
8. T (n) ∈ O(log2 n)
9. T (n) ∈ O(n log n)
Practice
1. T (n) ∈ O(1)
2. T (n) ∈ O(n2)
3. T (n) ∈ O(1)
4. T (n) ∈ O(n2)
5. T (n) ∈ O(n3)
6. T (n) ∈ O(n)
7. T (n) ∈ O(n2)
8. T (n) ∈ O(n log n)
9. T (n) ∈ O(n3)
10. T (n) ∈ O(log n)
11. T (n) ∈ O(n)
12. T (n) ∈ O(n)
Challenge
1. T (n) ∈ O(n2 log n) 2. T (n) ∈ O(n0.5)
61
11.2 Computational Complexity of Loops
Clarify
Determine for each variant of the function the computational complexity per line and conclude the overall compu-
tational complexity of the function in terms of the input parameter n (in big-oh notation) .
1. def ones(n):
res = [1]
while len(res) < n:
res += [1]
return res
2. def ones(n):
res = [1]
while len(res) < n:
res = res + [1]
return res
3. def ones(n):
res = [1]
while len(res) < n:
res = res + res
return res
4. def ones(n):
res = [1]
while sum(res) < n:
res += [1]
return res
5. def ones(n):
res = '1'
while len(res) < n:
res += '1'
return res
Practice
For each of the following functions identify the tightest simple bound (using big-oh notation) of the worst-case
computational complexity in terms of the input size and justify your answer. Depending on the specific function
this justification could be based on one or more of:
? structural properties of the worst-case input for a given input size (e.g., "in the worst case the input list is
inversely ordered")
? what line(s) is/are dominating the overall computational cost and
? additional arguments as necessary.
If helpful to make your argument, you can copy+paste the code of the function into your response to annotate
individual lines with comments.
1. def loop_eg1(lst):
n = len(lst)
res = []
for i in range(n):
for j in range(n):
for k in range(n):
res += [lst[i]*lst[j]*lst[k]]
return res
2. def loop_eg2(lst):
n = len(lst)
res = []
for i in range(n):
for j in range(i+1,n):
res += [lst[i]*lst[j]]
return res
62
Test 2
Exam
3. def loop_eg3(lst):
n = len(lst)
res = []
for i in range(0,n,2):
for j in range(1,n,2):
res += [lst[i]*lst[j]]
return res
4. def loop_eg4(lst):
n = len(lst)
res = []
for i in range(n):
j=1
while jres += [lst[i]*lst[j]]
j = j*2
return res
5. def loop_eg5(lst ,target ):
n = len(lst)
for i in range(n):
for j in range(i,n):
if lst[i]+lst[j] == target:
return True
return False
6. def loop_eg6(lst , target ):
n = len(lst)
res = []
for i in range(1,n):
if sum(lst[0:i]) > target:
return i
return None
63
Answers
Clarify
1. Complexity is O(n). The loop iterates n times, and the complexity of the augmented assignment statement
within the loop is only constant time, as only a single item is added to the list each time.
def ones(n): # complexity O(n)
res = [1] # 1 = O(1)
while len(res) < n: # 1 + 1 ... + 1 (n times) = O(n)
res += [1] # 1 + 1 ... + 1 (n times) = O(n)
return res # 0 = O(0)
2. The loop iterates n times. Here, though, the concatenation statement inside the loop is creating a new list of
length len(res) + 1 each time, where len(res) ranges from 1..n? 1. So this statement is O(n). Since it is
executed n times, the total complexity is O(n2).
def ones(n): # complexity O(n^2)
res = [1] # 1 = O(1)
while len(res) < n: # 1 + 1 ... + 1 (n times) = O(n)
res = res + [1] # 2 + 3 ... + n = n(n+1)/2 - 1 = O(n^2)
return res # 0 = O(0)
3. The cost of the line inside the loop is roughly 1 + 2 + 4 + . . . + n2 + n. This is less than n times the infinite
geometric sum (1 + 12 +
1
4 +
1
8 . . .), so is O(n).
def ones(n): # complexity O(n)
res = [1] # 1 = O(1)
while len(res) < n: # 1 + 1 ... + 1 (log n times) = O(log n)
res = res + res # 2 + 4 ... + n = O(n)
return res # 0 = O(0)
4. The loop iterates n times, and the augmented assignment statement inside the loop is only constant time.
But the comparison operation occurring inside the loop condition contains a call to sum, which is O(n). Since
this comparison occurs n times, this gives an overall complexity of O(n2).
def ones(n): # complexity O(n^2)
res = [1] # 1 = O(1)
while sum(res) < n: # 1 + 2 ... + n = n(n+1)/2 = O(n^2)
res += [1] # 1 + 1 ... + 1 (n times) = O(n)
return res # 0 = O(0)
5. Since there are no in-place updates for strings, as they are immutable objects in Python, the statement inside
the loop requires recreating a new copy of the variable res each time, so its complexity is determined by the
length of the string that is created, that is, 2 + 3 + 4 + . . .+ n = O(n2).
def ones(n): # complexity O(n^2)
res = '1' # 1 = O(1)
while len(res) < n: # 1 + 1 ... + 1 (n times) = O(n)
res += '1' # 2 + 3 ... + n = n(n+1)/2 - 1 = O(n^2)
return res # 0 = O(0)
Practice
1. The computational complexity of the function is O(n3). This is because there are three nested loops, each of
which iterates n times. The assignment operation augmenting the list is constant time, and is executed O(n3)
times in the best and worst cases.
64
def loop_eg1(lst):
n = len(lst)
res = []
for i in range(n): # O(n)
for j in range(n): # O(n^2)
for k in range(n): # O(n^3)
res += [lst[i]*lst[j]*lst[k]] # O(n^3)
return res
2. The computational complexity of the function is O(n2). The outer loop iterates n times. The innermost
loop iterates (n ? 1) + (n ? 2) + . . . 2 + 1 times, which is n(n+1)2 , or 12n2 + 12n. The n2 term still dominates
the overall cost. The assignment operation inside the two nested loops is a constant time operation, and is
executed O(n2) times in the best and worst cases.
def loop_eg2(lst):
n = len(lst)
res = []
for i in range(n): # O(n)
for j in range(i+1,n): # O(n^2)
res += [lst[i]*lst[j]] # O(n^2)
return res
3. The computational complexity of the function is O(n2). The outer loop iterates n/2 times, and the inner loop
iterates n/2 times for every outer iteration, which is a total of (n/2)2 = 14n
2 times. The assignment operation
inside the two nested loops is a constant time operation, and is executed O(n2) times in the best and worst
cases.
def loop_eg3(lst):
n = len(lst)
res = []
for i in range(0,n,2): # O(n)
for j in range(1,n,2): # O(n^2)
res += [lst[i]*lst[j]] # O(n^2)
return res
4. The computational complexity of the function is O(nlogn). The outer loop iterates n times. The inner loop
iterates log n times for every outer iteration. The two assignments inside the nested loops are constant time,
and will be executed O(n log n) times in the best and worst cases.
def loop_eg4(lst):
n = len(lst)
res = []
for i in range(n): # O(n)
j=1
while jres += [lst[i]*lst[j]] # O(nlogn)
j = j*2 # O(nlogn)
return res
5. The computational complexity of the function is O(n2). In the worst case, no pair of elements in the list add
up to the target, so the two nested loops finish iterating completely before the function returns. In this case,
the outer loop iterates n times, and the innermost loop iterates (n ? 1) + (n ? 2) + . . . 2 + 1 times, which
is n(n+1)2 , or
1
2n
2 + 12n. The assignment operation inside the two nested loops is a constant time operation
which will be executed O(n2) times.
In the best case, the first two elements in the list add up to the target, so the function returns before a single
loop iteration has completed. This is O(1).
65
def loop_eg5(lst ,target ):
n = len(lst)
for i in range(n): # O(n)
for j in range(i,n): # O(n^2)
if lst[i]+lst[j] == target: # O(n^2)
return True
return False
6. The computational complexity of the function is O(n2). In the worst case, the sum of all elements in the list
is still less than the target, so the function runs through to the end before returning. In this case, the outer
loop iterates n times. The call to sum has a complexity of O(i) each time it is called, where i ranges from 1
to n-1. So this line requires 1 + 2 + 3 + . . .+ (n? 2) + (n? 1) operations in the worst case, which is O(n2).
In the best case, the first element in the list is greater than the target, so the function returns before a single
loop iteration has completed, and the call to sum only requires summing a single element. This is O(1).
def loop_eg6(lst , target ):
n = len(lst)
res = []
for i in range(1,n): # O(n)
if sum(lst[0:i]) > target: # O(n^2)
return i
return None
66
Part IV
Additional Resources
67
12 | Practice Exam Questions
The following collection of questions is an example for the middle part of the exam. It is preceded by a part where
you have to write simple Python expression and apply known algorithms to specific inputs, and it is succeeded by
a final part where you have to write Python programs to solve unknown problems (on the level of simple workshop
problems). The recommended writing time for this middle part is 40 minutes.
12.1 Discussion of Theoretical Concepts
For each of the questions in this section, we are looking for a short answer of around 3-5 sentences. Try to be precise
and coherent and use the technical terms covered in the unit where relevant.
Decrease-and-conquer
A decrease-and-conquer algorithm decreases its input in each iteration by a constant c using O(1) time per iteration.
That is, in the first iteration it goes from an input of size n to an input of size n? c. In the second iteration it goes
from an input of size n? c to an input of n? 2c and so on. Once the algorithm reaches an input of size less than
c it directly solves the problem in time O(1). What is the overall computational complexity of the algorithm and
why?
Vertex Cover
Assume a graph G = (V,E) with n vertices has a vertex cover X of size k. What can be said about the size of the
largest independent set of G and why?
Complexity Classes
Recall the greedy algorithm for the Knapsack Problem that constructs solution based on some score function:
def greedy_knapsack(values , weights , capacity , score ):
n = len(values)
items = sorted(range(n), key=score , reverse=True)
sel , value , weight = [], 0, 0
for i in items:
if weight + weights[i] <= capacity:
sel += [i]
weight += weights[i]
value += values[i]
return sel , value , weight
Assume you find a score function to be used in this algorithm that
? has a computational complexity of O(n5) and
? makes the algorithm always return an optimal solution.
Based on your knowledge about the complexity classes P, NP, and NP-complete, what does this imply for the
Hamiltonian Cycle Problem and why?
68
Exam
12.2 Computational Complexity
For each of the following functions identify the tightest simple bound (using big-oh notation) of the worst-case
computational complexity in terms of the input size and justify your answer. Depending on the specific function
this justification could be based on one or more of:
? structural properties of the worst-case input for a given input size (e.g., "in the worst case the input list is
inversely ordered")
? what line(s) is/are dominating the overall computational cost and
? additional arguments as necessary.
If helpful to make your argument, you can copy+paste the code of the function into your response to annotate
individual lines with comments.
Exponential Summary
For the following function that accepts as input a lst, identify the tightest simple bound of the worst-case compu-
tational complexity in terms of the input size n = len(lst) and justify your answer.
def exp_summary2(lst):
n = len(lst)
res = []
i = 1
while i <= n:
res += [lst[i-1]]
i *= 2
return res
Finding Common Elements
For the following function that accepts as input two lists lst1 and lst2, identify the tightest simple bound of the
worst-case computational complexity in terms of the combined input size n = len(lst1)+len(lst2) and justify
your answer.
def contained_in_both3(lst1 , lst2):
n1 , n2 = len(lst1), len(lst2)
lst1 = sorted(lst1)
lst2 = sorted(lst2)
res = []
i, j = 0, 0
while i < n1 and j < n2:
if lst1[i] < lst2[j]:
i += 1
elif lst1[i] > lst2[j]:
j += 1
else:
res += [lst1[i]]
i += 1
j += 1
return res
Median Square
For the following function that accepts as input two lists lst1 and lst2, identify the tightest simple bound of the
worst-case computational complexity in terms of the combined input size n = len(lst1)+len(lst2) and justify
your answer.
69
def median_square(lst1 , lst2):
squares = []
for x in lst1: # O(n)
for y in lst2: # O(n^2)
squares += [x*y] # O(n^2)
return sorted(squares )[len(squares )//2] # O(n^2 log n)
12.3 Invariants
For each of the following functions, identify the loop invariant(s), the loop exit condition, and the loop post-condition
which shows the algorithm’s correctness.
You can simply list the assertions, or you can copy and paste the function into the answer field and add assertion
annotations as comments, as shown in the lecture.
Inversions
For the following function, identify the loop invariant(s), the loop exit condition, and the loop post-condition which
shows the algorithm’s correctness. You can either use the invariant at the end of the inner or the outer loop.
def inversions(lst):
"""
Input : a list of comparable elements
Output: the number of 'inversions ' in lst , i.e., the number
of index pairs i, j that violate ascending order or
in other words: i < j and lst[i] > lst[j]
For example:
>>> inversions ([1, 3, 2])
1
>>> inversions ([3, 2, 1, 0])
6
"""
n = len(lst)
count = 0
for k in range(n):
for l in range(k+1, n):
count += lst[k] > lst[l]
return count
Longest Run
For the following function, identify the loop invariant(s), the loop exit condition, and the loop post-condition which
shows the algorithm’s correctness. You can either use the invariant at the end of the inner or the outer loop.
def longest_run(lst):
"""
Input : a list of comparable elements
Output: a longest sublist of lst that is sorted
For example:
>>> longest_run ([1, 2, 0, 1, 2, 1, 0, 1])
[0, 1, 2]
>>> longest_run ([10, 9, 8, 7, 11])
[7, 11]
"""
70
n = len(lst)
k, l = 0, 1
i, j = 0, 1
while j < n:
if lst[j] < lst[j-1]:
i = j
j += 1
if j - i > l - k:
k, l = i, j
return lst[k: l]
71
13 | Solutions to Practice Exam Questions
13.1 Solutions: Discussion of Theoretical Concepts
Decrease-and-conquer
Such an algorithm has an overall linear time computational complexity, because it requires n//c iterations to solve
the problem. Since each iteration costs O(1), this results in a complexity of O(n//c)=O(n).
Vertex Cover
All vertices that are not in the vertex cover X form an independent set (because if there was an edge between any
two of them this edge wouldn’t be covered by X). There are n-k vertices that are not contained in X. Thus, the
largest independent set of G must be of at least size n - k (but perhaps there are even larger independent sets).
Complexity Classes
Using this hypothetical score function would result in an overall O(n**6 log n) algorithm for the Knapsack Problem,
which is polynomial time. This would also imply a polynomial time algorithm for the decision variant of the
Knapsack problem, which is NP-complete. Since the Hamiltonian Cycle Problem is in NP, it can be polynomially
reduced to the Knapsack Problem and, hence, we would also have a polynomial time algorithm for that problem.
13.2 Solutions: Computational Complexity
Exponential Summary
The computational complexity of the function is O(log n) irrespectively of the structure of the input list.
This is because the iteration variable is doubled in each iteration resulting in O(log n) executions of the while
loop, and each individual operation inside the loop body as well as each individual check of the loop condition cost
O(1).
def exp_summary2(lst): # O(log n)
n = len(lst)
res = []
i = 1
while i <= n: # O(log n)
res += [lst[i-1]] # O(log n)
i *= 2 # O(log n)
return res
Finding Common Elements
The computational worst-case complexity of the function is O(n log n), which is attained for two unsorted inputs
lst1 and lst2 of roughly equal lengths n1 and n2.
In this case the two sorting steps dominate the computational cost, which are both of complexity O(n/2 log
n/2) = O(n log n). In contrast, the while loop only costs O(n1+n2) = O(n) because in every iteration at least one
of i or j is increased, which means that after at most n iterations either i == n1 or j == n2.
72
def contained_in_both3(lst1 , lst2): # O(n log n)
n1 , n2 = len(lst1), len(lst2)
lst1 = sorted(lst1) # O(n log n)
lst2 = sorted(lst2) # O(n log n)
res = []
i, j = 0, 0
while i < n1 and j < n2: # O(n)
if lst1[i] < lst2[j]: # O(n)
i += 1 # O(n)
elif lst1[i] > lst2[j]: # O(n)
j += 1 # O(n)
else:
res += [lst1[i]] # O(n)
i += 1 # O(n)
j += 1 # O(n)
return res
Median Square
The complexity of the function is O(n**2 log n) for the worst case of two input lists of roughly equal size n1=n2
(n=n1+n2).
This is because the nested for-loop create an unordered list of length n1*n2=O(n**2), which is then sorted in
time O(n**2 log n**2) = O(n**2 log n) dominating the overall computational cost.
def median_square(lst1 , lst2):
squares = []
for x in lst1: # O(n)
for y in lst2: # O(n^2)
squares += [x*y] # O(n^2)
return sorted(squares )[len(squares )//2] # O(n^2 log n)
13.3 Solutions: Invariants
Inversions
def inversions(lst):
n = len(lst)
count = 0
for k in range(n):
for l in range(k+1, n):
count += lst[k] > lst[l]
# INV: count equal to number of index pairs i, j
# with i < k < j < l and lst[i] > lst[j]
# EXC: k == l == n
# POC: count equal to number of index pairs i, j
# with i < j < n and lst[i] > lst[j]
return count
Longest Run
def longest_run(lst):
n = len(lst)
k, l = 0, 1
73
i, j = 0, 1
while j < n:
if lst[j] < lst[j-1]:
i = j
j += 1
if j - i > l - k:
k, l = i, j
# INV1: lst[i:j] and lst[k: l] are sorted
# INV2: if lst[a: b] is sorted for a= b - a
# EXC: j == n
# POC: if lst[a: b] is sorted for a= b - a
return lst[k: l]
74

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