Homework 8

Due by 11:59pm on Monday, 7/27

Instructions

Download hw08.zip. Inside the archive, you will find a file called hw08.py, along with a copy of the OK autograder.

Submission: When you are done, submit with python3 ok --submit. You may submit more than once before the deadline; only the final submission will be scored. See Lab 1 for instructions on submitting assignments.

Using OK: If you have any questions about using OK, please refer to this guide.

Readings: You might find the following references useful:

Required questions

Question 1

Implement __contains__ for the Link class, which allows us to use the in operator to check if a value is contained in a linked list.

class Link:
    empty = ()

    def __init__(self, first, rest=empty):
        assert rest is Link.empty or isinstance(rest, Link)
        self.first = first
        self.rest = rest

    def __contains__(self, value):
        "*** YOUR CODE HERE ***"

Use OK to test your code:

python3 ok -q contains

Generating natural numbers

The following questions use the naturals generator function, which yields an infinite sequence of integers starting at 1.

def naturals():
    """A generator function that yields the infinite sequence of natural
    numbers, starting at 1. 

    >>> m = naturals()
    >>> type(m)
    <class 'generator'>
    >>> [next(m) for _ in range(10)]
    [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
    """
    i = 1
    while True:
        yield i
        i += 1

Question 2

Implement an iterator class called ScaleIterator that scales elements in an iterable s by a number k.

class ScaleIterator:
    """An iterator the scales elements of the iterable s by a number k.

    >>> s = ScaleIterator([1, 5, 2], 5)
    >>> list(s)
    [5, 25, 10]

    >>> m = ScaleIterator(naturals(), 2)
    >>> [next(m) for _ in range(5)]
    [2, 4, 6, 8, 10]
    """
    def __init__(self, s, k):
        "*** YOUR CODE HERE ***"

    def __iter__(self):
        return self

    def __next__(self):
        "*** YOUR CODE HERE ***"

Use OK to test your code:

python3 ok -q ScaleIterator

Question 3

Implement the generator function scale(s, k), which yields elements of the given iterable s, scaled by k.

def scale(s, k):
    """Yield elements of the iterable s scaled by a number k.

    >>> s = scale([1, 5, 2], 5)
    >>> type(s)
    <class 'generator'>
    >>> list(s)
    [5, 25, 10]

    >>> m = scale(naturals(), 2)
    >>> [next(m) for _ in range(5)]
    [2, 4, 6, 8, 10]
    """
    "*** YOUR CODE HERE ***"

Use OK to test your code:

python3 ok -q scale

Question 4

Implement merge(s1, s2), which takes two iterables s1 and s2 whose elements are ordered. merge yields elements from s1 and s2 in sorted order, elimnating repetition. You may assume s0 and s1 themselves do not contain repeats. You may also assume s0 and s1 represent infinite sequences; that is, their iterators never raise StopIteration.

See the doctests for example behavior.

def merge(s0, s1):
    """Yield the elements of strictly increasing iterables s0 and s1, removing
    repeats. Assume that s0 and s1 have no repeats. You can also assume that s0
    and s1 represent infinite sequences.

    >>> twos = scale(naturals(), 2)
    >>> threes = scale(naturals(), 3)
    >>> m = merge(twos, threes)
    >>> type(m)
    <class 'generator'>
    >>> [next(m) for _ in range(10)]
    [2, 3, 4, 6, 8, 9, 10, 12, 14, 15]
    """
    i0, i1 = iter(s0), iter(s1)
    e0, e1 = next(i0), next(i1)
    "*** YOUR CODE HERE ***"

Use OK to test your code:

python3 ok -q merge

Question 5

A famous problem, first raised by Richard Hamming, is to enumerate, in ascending order with no repetitions, all positive integers with no prime factors other than 2, 3, or 5. These are called regular numbers. One obvious way to do this is to simply test each integer in turn to see whether it has any factors other than 2, 3, and 5. But this is very inefficient, since, as the integers get larger, fewer and fewer of them fit the requirement.

As an alternative, we can write a generator function of such numbers. Let us call the sequence of numbers s and notice the following facts about it.

  • s begins with 1.
  • The elements of scale(s, 2) are also elements of s.
  • The same is true for scale(s, 3) and scale(s, 5).
  • These are all of the elements of s.

Now all we have to do is combine elements from these sources. Use the merge function you defined previously to fill in the definition of make_s:

def make_s():
    """A generator function that yields all positive integers with only factors
    2, 3, and 5.

    >>> s = make_s()
    >>> type(s)
    <class 'generator'>
    >>> [next(s) for _ in range(20)]
    [1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36]
    """
    "*** YOUR CODE HERE ***"

Use OK to test your code:

python3 ok -q make_s