Due by 11:59pm on Monday, 8/7

Instructions

Download hw09.zip.

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. Check that you have successfully submitted your code on okpy.org. See Lab 0 for more instructions on submitting assignments.

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

Homework Questions

Iterators

Question 1: Link Iterable

To make the Link class iterable, implement the LinkIterator class.
class LinkIterator:
    """
    >>> lnk = Link(1, Link(2, Link(3)))
    >>> lnk_iter = LinkIterator(lnk)
    >>> next(lnk_iter)
    1
    >>> next(lnk_iter)
    2
    """
    def __init__(self, link):
        "*** YOUR CODE HERE ***"

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

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

Use OK to test your code:

python3 ok -q LinkIterator

Generators

Question 2: In Order

Write a function that yields the entries of a valid binary search tree in sorted order.

def in_order(t):
    """
    Yields the entries of a valid binary search tree in sorted order.

    >>> b = BTree(5, BTree(3, BTree(2), BTree(4)), BTree(6))
    >>> list(in_order(b))
    [2, 3, 4, 5, 6]
    >>> list(in_order(bst([1, 3, 5, 7, 9, 11, 13])))
    [1, 3, 5, 7, 9, 11, 13]
    >>> list(in_order(BTree(1)))
    [1]
    """
    "*** YOUR CODE HERE ***"

Hint: The yield from expression is helpful if you want to yield all the values from another sequence.

Use OK to test your code:

python3 ok -q in_order

Question 3: Generate Permutations

Given a list of unique elements, a permutation of the list is a reordering of the elements. For example, [2, 1, 3], [1, 3, 2], and [3, 2, 1] are all permutations of the list [1, 2, 3].

Implement permutations, a generator function that takes in a lst and outputs all permutations of lst, each as a list (see doctest for an example).

def permutations(lst):
    """Generates all permutations of sequence LST. Each permutation is a
    list of the elements in LST in a different order.

    The order of the permutations does not matter.

    >>> sorted(permutations([1, 2, 3]))
    [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]
    >>> type(permutations([1, 2, 3]))
    <class 'generator'>
    >>> sorted(permutations((10, 20, 30)))
    [[10, 20, 30], [10, 30, 20], [20, 10, 30], [20, 30, 10], [30, 10, 20], [30, 20, 10]]
    >>> sorted(permutations("ab"))
    [['a', 'b'], ['b', 'a']]
    """
    if not lst:
        yield []
        return
    "*** YOUR CODE HERE ***"

The order in which you generate permutations is irrelevant.

Hint: If you had the permutations of lst minus one element, how could you use that to generate the permutations of the full lst?

Use OK to test your code:

python3 ok -q permutations

Streams

Question 4: Scale Stream

Implement the function scale_stream, which returns a stream over each element of an input stream, scaled by k:

def scale_stream(s, k):
    """Return a stream of the elements of S scaled by a number K.

    >>> ints = make_integer_stream(1)
    >>> s = scale_stream(ints, 5)
    >>> stream_to_list(s, 5)
    [5, 10, 15, 20, 25]
    >>> s = scale_stream(Stream("x", lambda: Stream("y")), 3)
    >>> stream_to_list(s)
    ['xxx', 'yyy']
    >>> stream_to_list(scale_stream(Stream.empty, 10))
    []
    """
    "*** YOUR CODE HERE ***"

Use OK to test your code:

python3 ok -q scale_stream

Question 5: Regular Numbers

Acknowledgements. This exercise is taken from Structure and Interpretation of Computer Programs, Section 3.5.2.

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 build a stream of such numbers. Let us call the required stream of numbers s and notice the following facts about it.

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

Now all we have to do is combine elements from these sources. For this we define a merge function that combines two ordered streams into one ordered result stream, eliminating repetitions.

Fill in the definition of merge, then fill in the definition of make_s below:

def merge(s0, s1):
    """Return a stream over the elements of strictly increasing s0 and s1,
    removing repeats. Assume that s0 and s1 have no repeats.

    >>> ints = make_integer_stream(1)
    >>> twos = scale_stream(ints, 2)
    >>> threes = scale_stream(ints, 3)
    >>> m = merge(twos, threes)
    >>> stream_to_list(m, 10)
    [2, 3, 4, 6, 8, 9, 10, 12, 14, 15]
    """
    if s0 is Stream.empty:
        return s1
    elif s1 is Stream.empty:
        return s0

    e0, e1 = s0.first, s1.first
    "*** YOUR CODE HERE ***"

def make_s():
    """Return a stream over all positive integers with only factors 2, 3, & 5.

    >>> s = make_s()
    >>> stream_to_list(s, 20)
    [1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36]
    """
    def rest():
        "*** YOUR CODE HERE ***"
    s = Stream(1, rest)
    return s

Use OK to test your code:

python3 ok -q merge

Use OK to test your code:

python3 ok -q make_s

Question 6: Linear Congruential Generator

A common method of producing pseudo-random numbers is by means of the following recurrence relation:

  • R0 = seed value
  • Ri+1 = (a*Ri + c) % n where Ri denotes the ith pseudo-random number in the stream; a, c, and n are constant integers, and seed value is some initial value provided by the user or chosen automatically by the system.

Define a function that returns a stream of random numbers that uses this linear-congruential formula.

from operator import add, mul, mod

def make_random_stream(seed, a, c, n):
    """The infinite stream of pseudo-random numbers generated by the
    recurrence r[0] = SEED, r[i+1] = (r[i] * A + C) % N.

    >>> s = make_random_stream(25, 29, 5, 32)
    >>> stream_to_list(s, 10)
    [25, 26, 23, 0, 5, 22, 3, 28, 17, 18]
    >>> s = make_random_stream(17, 299317, 13, 2**20)
    >>> stream_to_list(s, 10)
    [17, 894098, 115783, 383424, 775373, 994174, 941859, 558412, 238793, 718506]
    """
    "*** YOUR CODE HERE ***"

Your solution must use only the functions defined in the skeleton, without defining any additional ones. Likewise, any lambda expressions should contain only calls to these functions.

Use OK to test your code:

python3 ok -q make_random_stream

Question 7: Stream of Streams

Write the function make_stream_of_streams, which returns an infinite stream, where the element at position i, counting from 1, is an infinite stream of integers that start from i. Your solution should have the form
result = Stream(..., lambda: ...)
return result

and should not require any additional helper functions (i.e., just use recursively defined streams, and any additional functions supplied in your starter file). You may find the map_stream function useful.

def make_stream_of_streams():
    """
    >>> stream_of_streams = make_stream_of_streams()
    >>> stream_of_streams
    Stream(Stream(1, <...>), <...>)
    >>> stream_of_streams.rest
    Stream(Stream(2, <...>), <...>)
    >>> stream_of_streams.rest.rest
    Stream(Stream(3, <...>), <...>)
    >>> stream_of_streams
    Stream(Stream(1, Stream(2, Stream(3, <...>))), Stream(Stream(2, Stream(3, <...>)), Stream(Stream(3, <...>), <...>)))
    """
    "*** YOUR CODE HERE ***"

Use OK to test your code:

python3 ok -q make_stream_of_streams