# CMPSC 442: Homework 1

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## 1. Python Concepts (6 points) [6 points]

For each of the following questions, write your answers as triply-quoted strings using the indicated
variables in the provided file.

1. [2 points] [2 points] Explain what it means for Python to be both strongly and dynamically
typed, and give a concrete example of each.

2. [2 points] [2 points] You would like to create a dictionary that maps some 2-dimensional points
to their associated names. As a first attempt, you write the following code:
points_to_names = {[0, 0]: “home”, [1, 2]: “school”, [-1, 1]: “market”}
However, this results in a type error. Describe what the problem is, and propose a solution.

3. [2 points] [2 points] Consider the following two functions, each of which concatenates a list of
strings.
def concatenate1(strings):
result = “”
for s in strings:
result += s
return result
def concatenate2(strings):
return “”.join(strings)
One of these approaches is significantly faster than the other for large inputs. Which version is
better, and what is the reason for the discrepancy?

## 2. Working with Lists (15 points) [15 points]

1. [5 points] [5 points] Consider the function extract_and_apply(l, p, f) shown below, which
extracts the elements of a list l satisfying a boolean predicate p, applies a function f to each
such element, and returns the result.
def extract_and_apply(l, p, f):
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result = []
for x in l:
if p(x):
result.append(f(x))
return result
Rewrite extract_and_apply(l, p, f) in one line using a list comprehension.

2. [5 points] [5 points] Write a function concatenate(seqs) that returns a list containing the
concatenation of the elements of the input sequences. Your implementation should consist of a
single list comprehension, and should not exceed one line.
>>> concatenate([[1, 2], [3, 4]])
[1, 2, 3, 4]
>>> concatenate([“abc”, (0, [0])])
[‘a’, ‘b’, ‘c’, 0, [0]]

3. [5 points] [5 points] Write a function transpose(matrix) that returns the transpose of the input
matrix, which is represented as a list of lists. Recall that the transpose of a matrix is obtained by
swapping its rows with its columns. More concretely, the equality
matrix[i][j] == transpose(matrix)[j][i] should hold for all valid indices i and j. You may
assume that the input matrix is well-formed, i.e., that each row is of equal length. You may
further assume that the input matrix is non-empty. Your function should not modify the input.
>>> transpose([[1, 2, 3]])
[[1], [2], [3]]
>>> transpose([[1, 2], [3, 4], [5, 6]])
[[1, 3, 5], [2, 4, 6]]

## 3. Sequence Slicing (6 points) [6 points]

The functions in this section should be implemented using sequence slices. Recall that the slice
parameters take on sensible default values when omitted. In some cases, it may be necessary to use
the optional third parameter to specify a step size.

1. [2 points] [2 points] Write a function copy(seq) that returns a new sequence containing the
same elements as the input sequence.
>>> copy(“abc”)
‘abc’
>>> copy((1, 2, 3))
(1, 2, 3)
>>> x = [0, 0, 0]; y = copy(x)
>>> print(x, y); x[0] = 1; print(x, y)
[0, 0, 0] [0, 0, 0]
[1, 0, 0] [0, 0, 0]
2. [2 points] [2 points] Write a function all_but_last(seq) that returns a new sequence
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containing all but the last element of the input sequence. If the input sequence is empty, a new
empty sequence of the same type should be returned.
>>> all_but_last(“abc”)
‘ab’
>>> all_but_last((1, 2, 3))
(1, 2)
>>> all_but_last(“”)

>>> all_but_last([])
[]
3. [2 points] [2 points] Write a function every_other(seq) that returns a new sequence containing
every other element of the input sequence, starting with the first. Hint: This function can be
written in one line using the optional third parameter of the slice notation.
>>> every_other([1, 2, 3, 4, 5])
[1, 3, 5]
>>> every_other(“abcde”)
‘ace’
>>> every_other([1, 2, 3, 4, 5, 6])
[1, 3, 5]
>>> every_other(“abcdef”)
‘ace’

## 4. Combinatorial Algorithms (11 points) [11 points]

The functions in this section should be implemented as generators. You may generate the output in
any order you find convenient, as long as the correct elements are produced. However, in some cases,
you may find that the order of the example output hints at a possible implementation.
Although generators in Python can be used in a variety of ways, you will not need to use any of their
more sophisticated features here. Simply keep in mind that where you might normally return a list of
elements, you should instead yield the individual elements.

Since the contents of a generator cannot be viewed without employing some form of iteration, we
wrap all function calls in this section’s examples with the list function for convenience.

1. [6 points] [6 points] The prefixes of a sequence include the empty sequence, the first element,
the first two elements, etc., up to and including the full sequence itself. Similarly, the suffixes of
a sequence include the empty sequence, the last element, the last two elements, etc., up to and
including the full sequence itself. Write a pair of functions prefixes(seq) and suffixes(seq)
that yield all prefixes and suffixes of the input sequence.
>>> list(prefixes([1, 2, 3]))
[[], [1], [1, 2], [1, 2, 3]]
>>> list(suffixes([1, 2, 3]))
[[1, 2, 3], [2, 3], [3], []]
>>> list(prefixes(“abc”))
[”, ‘a’, ‘ab’, ‘abc’]
>>> list(suffixes(“abc”))
[‘abc’, ‘bc’, ‘c’, ”]
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2. [5 points] [5 points] Write a function slices(seq) that yields all non-empty slices of the input
sequence.
>>> list(slices([1, 2, 3]))
[[1], [1, 2], [1, 2, 3], [2], [2, 3],
[3]]
>>> list(slices(“abc”))
[‘a’, ‘ab’, ‘abc’, ‘b’, ‘bc’, ‘c’]

## 5. Text Processing (20 points) [20 points]

1. [5 points] [5 points] A common preprocessing step in many natural language processing tasks
is text normalization, wherein words are converted to lowercase, extraneous whitespace is
removed, etc. Write a function normalize(text) that returns a normalized version of the input
string, in which all words have been converted to lowercase and are separated by a single space.
No leading or trailing whitespace should be present in the output.
>>> normalize(“This is an example.”)
‘this is an example.’
>>> normalize(” EXTRA SPACE “)
‘extra space’

2. [5 points] [5 points] Write a function no_vowels(text) that removes all vowels from the input
string and returns the result. For the purposes of this problem, the letter ‘y’ is not considered to
be a vowel.
>>> no_vowels(“This Is An Example.”)
‘Ths s n xmpl.’
>>> no_vowels(“We love Python!”)
‘W lv Pythn!’

3. [5 points] [5 points] Write a function digits_to_words(text) that extracts all digits from the
input string, spells them out as lowercase English words, and returns a new string in which they
are each separated by a single space. If the input string contains no digits, then an empty string
should be returned.
>>> digits_to_words(“Zip Code: 19104”)
‘one nine one zero four’
>>> digits_to_words(“Pi is 3.1415…”)
‘three one four one five’

4. [5 points] [5 points] Although there exist many naming conventions in computer programming,
two of them are particularly widespread. In the first, words in a variable name are separated
using underscores. In the second, words in a variable name are written in mixed case, and are
strung together without a delimiter. By mixed case, we mean that the first word is written in
lowercase, and that subsequent words have a capital first letter. Write a function
to_mixed_case(name) that converts a variable name from the former convention to the latter.

Leading and trailing underscores should be ignored. If the variable name consists solely of
underscores, then an empty string should be returned.
>>> to_mixed_case(“to_mixed_case”)
‘toMixedCase’
>>> to_mixed_case(“__EXAMPLE__NAME__”)
‘exampleName’

## 6. Polynomials (37 points) [37 points]

In this section, you will implement a simple Polynomial class supporting basic arithmetic,
simplification, evaluation, and pretty-printing. An example demonstrating these capabilities is shown
below.
>>> p, q = Polynomial([(2, 1), (1, 0)]), Polynomial([(2, 1), (-1, 0)])
>>> print(p); print(q)
2x + 1
2x – 1
>>> r = (p * p) + (q * q) – (p * q); print(r)
4x^2 + 2x + 2x + 1 + 4x^2 – 2x – 2x + 1 – 4x^2 + 2x – 2x + 1
>>> r.simplify(); print(r)
4x^2 + 3
>>> [(x, r(x)) for x in range(-4, 5)]
[(-4, 67), (-3, 39), (-2, 19), (-1, 7), (0, 3), (1, 7), (2, 19), (3, 39), (4, 67)]

1. [2 points] [2 points] In this problem, we will think of a polynomial as an immutable object,
represented internally as a tuple of coefficient-power pairs. For instance, the polynomial 2x + 1
would be represented internally by the tuple ((2, 1), (1, 0)). Write an initialization method
__init__(self, polynomial) that converts the input sequence polynomial of coefficient-power
pairs into a tuple and saves it for future use. Also write a corresponding method
get_polynomial(self) that returns this internal representation.
>>> p = Polynomial([(2, 1), (1, 0)])
>>> p.get_polynomial()
((2, 1), (1, 0))
>>> p = Polynomial(((2, 1), (1, 0)))
>>> p.get_polynomial()
((2, 1), (1, 0))

2. [4 points] [4 points] Write a __neg__(self) method that returns a new polynomial equal to the
negation of self. This method will be used by Python for unary negation.
>>> p = Polynomial([(2, 1), (1, 0)])
>>> q = -p; q.get_polynomial()
>>> p = Polynomial([(2, 1), (1, 0)])
>>> q = -(-p); q.get_polynomial()
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((-2, 1), (-1, 0)) ((2, 1), (1, 0))

3. [3 points] [3 points] Write an __add__(self, other) method that returns a new polynomial
equal to the sum of self and other. This method will be used by Python for addition. No
simplification should be performed on the result.
>>> p = Polynomial([(2, 1), (1, 0)])
>>> q = p + p; q.get_polynomial()
((2, 1), (1, 0), (2, 1), (1, 0))
>>> p = Polynomial([(2, 1), (1, 0)])
>>> q = Polynomial([(4, 3), (3, 2)])
>>> r = p + q; r.get_polynomial()
((2, 1), (1, 0), (4, 3), (3, 2))

4. [3 points] [3 points] Write a __sub__(self, other) method that returns a new polynomial
equal to the difference between self and other. This method will be used by Python for
subtraction. No simplification should be performed on the result.
>>> p = Polynomial([(2, 1), (1, 0)])
>>> q = p – p; q.get_polynomial()
((2, 1), (1, 0), (-2, 1), (-1, 0))
>>> p = Polynomial([(2, 1), (1, 0)])
>>> q = Polynomial([(4, 3), (3, 2)])
>>> r = p – q; r.get_polynomial()
((2, 1), (1, 0), (-4, 3), (-3, 2))

5. [5 points] [5 points] Write a __mul__(self, other) method that returns a new polynomial
equal to the product of self and other. This method will be used by Python for multiplication.
No simplification should be performed on the result. Your result does not need to match the
examples below exactly, as long as the same terms are present in some order.
>>> p = Polynomial([(2, 1), (1, 0)])
>>> q = p * p; q.get_polynomial()
((4, 2), (2, 1), (2, 1), (1, 0))
>>> p = Polynomial([(2, 1), (1, 0)])
>>> q = Polynomial([(4, 3), (3, 2)])
>>> r = p * q; r.get_polynomial()
((8, 4), (6, 3), (4, 3), (3, 2))

6. [3 points] [3 points] Write a __call__(self, x) method that returns the result of evaluating the
current polynomial at the point x. This method will be used by Python when a polynomial is
called as a function. Hint: This method can be written in one line using Python’s exponentiation
operator, **, and the built-in sum function.
>>> p = Polynomial([(2, 1), (1, 0)])
>>> [p(x) for x in range(5)]
[1, 3, 5, 7, 9]
>>> p = Polynomial([(2, 1), (1, 0)])
>>> q = -(p * p) + p
>>> [q(x) for x in range(5)]

[0, -6, -20, -42, -72]

7. [8 points] [8 points] Write a simplify(self) method that replaces the polynomial’s internal
representation with an equivalent, simplified representation. Unlike the previous methods,
simplify(self) does not return a new polynomial, but rather acts in place. However, because
the fundamental character of the polynomial is not changing, we do not consider this to violate
the notion that polynomials are immutable.

The simplification process should begin by combining terms with a common power. Then,
terms with a coefficient of zero should be removed, and the remaining terms should be sorted in
decreasing order based on their power. In the event that all terms have a coefficient of zero after
the first step, the polynomial should be simplified to the single term 0 · x0, i.e. (0, 0).
>>> p = Polynomial([(2, 1), (1, 0)])
>>> q = -p + (p * p); q.get_polynomial()
((-2, 1), (-1, 0), (4, 2), (2, 1),
(2, 1), (1, 0))
>>> q.simplify(); q.get_polynomial()
((4, 2), (2, 1))
>>> p = Polynomial([(2, 1), (1, 0)])
>>> q = p – p; q.get_polynomial()
((2, 1), (1, 0), (-2, 1), (-1, 0))
>>> q.simplify(); q.get_polynomial()
((0, 0),)

8. [9 points] [9 points] Write a __str__(self) method that returns a human-readable string
representing the polynomial. This method will be used by Python when the str function is
called on a polynomial, or when a polynomial is printed.
In general, your function should render polynomials as a sequence of signs and terms each
separated by a single space, i.e. “sign1 term1 sign2 term2 … signN termN”, where signs can
be “+” or “-“, and terms have the form “ax^b” for coefficient a and power b. However, in
adherence with conventional mathematical notation, there are a few exceptional cases that
require special treatment:
The first sign should not be separated from the first term by a space, and should be left
blank if the first term has a positive coefficient.

The variable and power portions of a term should be omitted if the power is 0, leaving
only the coefficient.
The power portion of a term should be omitted if the power is 1.
Coefficients with magnitude 0 should always have a positive sign.
Coefficients with magnitude 1 should be omitted, unless the power is 0.
You may assume that all polynomials have integer coefficients and non-negative integer
powers.

>>> p = Polynomial([(1, 1), (1, 0)])
>>> qs = (p, p + p, -p, -p – p, p * p)
>>> for q in qs: q.simplify(); str(q)

‘x + 1’
‘2x + 2’
‘-x – 1’
‘-2x – 2’
‘x^2 + 2x + 1’
>>> p = Polynomial([(0, 1), (2, 3)])
>>> str(p); str(p * p); str(-p * p)
‘0x + 2x^3’
‘0x^2 + 0x^4 + 0x^4 + 4x^6’
‘0x^2 + 0x^4 + 0x^4 – 4x^6’
>>> q = Polynomial([(1, 1), (2, 3)])
>>> str(q); str(q * q); str(-q * q)
‘x + 2x^3’
‘x^2 + 2x^4 + 2x^4 + 4x^6’
‘-x^2 – 2x^4 – 2x^4 – 4x^6’

## 7. Feedback (5 points) [5 points]

1. [1 point] [1 point] Approximately how long did you spend on this assignment?

2. [2 points] [2 points] Which aspects of this assignment did you find most challenging? Were
there any significant stumbling blocks?

3. [2 points] [2 points] Which aspects of this assignment did you like? Is there anything you
would have changed?