Due by 11:59pm on Thursday, October 17

Instructions

Download hw05.zip. Inside the archive, you will find a file called hw05.py, along with a copy of the ok autograder.

Submission: When you are done, submit the assignment by uploading all code files you've edited to Gradescope. 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 Gradescope. See Lab 0 for more 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:

• Section 4.2

Grading: Homework is graded based on correctness. Each incorrect problem will decrease the total score by one point. This homework is out of 2 points.

Required Questions

Q1: Infinite Hailstone

Write a generator function that yields the elements of the hailstone sequence starting at number n. After reaching the end of the hailstone sequence, the generator should yield the value 1 indefinitely.

Here is a quick reminder of how the hailstone sequence is defined:

1. Pick a positive integer n as the start.
2. If n is even, divide it by 2.
3. If n is odd, multiply it by 3 and add 1.
4. Continue this process until n is 1.

Try to write this generator function recursively. If you are stuck, you can first try writing it iteratively and then seeing how you can turn that implementation into a recursive one.

Hint: Since hailstone returns a generator, you can yield from a call to hailstone!

def hailstone(n):
    """
    Yields the elements of the hailstone sequence starting at n.
    At the end of the sequence, yield 1 infinitely.

    >>> hail_gen = hailstone(10)
    >>> [next(hail_gen) for _ in range(10)]
    [10, 5, 16, 8, 4, 2, 1, 1, 1, 1]
    >>> next(hail_gen)
    1
    """
    "*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q hailstone
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Q2: Merge

Definition: An infinite iterator is a iterator that never stops providing values when next is called. For example, ones() evaluates to an infinite iterator:

def ones():
    while True:
        yield 1

Write a generator function merge(a, b) that takes two infinite iterators, a and b, as inputs. Both iterators yield elements in strictly increasing order with no duplicates. The generator should produce all unique elements from both input iterators in increasing order, ensuring no duplicates.

Note: The input iterators do not contain duplicates within themselves, but they may have common elements between them.

def merge(a, b):
    """
    Return a generator that has all of the elements of generators a and b,
    in increasing order, without duplicates.

    >>> def sequence(start, step):
    ...     while True:
    ...         yield start
    ...         start += step
    >>> a = sequence(2, 3) # 2, 5, 8, 11, 14, ...
    >>> b = sequence(3, 2) # 3, 5, 7, 9, 11, 13, 15, ...
    >>> result = merge(a, b) # 2, 3, 5, 7, 8, 9, 11, 13, 14, 15
    >>> [next(result) for _ in range(10)]
    [2, 3, 5, 7, 8, 9, 11, 13, 14, 15]
    """
    a_val, b_val = next(a), next(b)
    while True:
        if a_val == b_val:
            "*** YOUR CODE HERE ***"
        elif a_val < b_val:
            "*** YOUR CODE HERE ***"
        else:
            "*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q merge
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Q3: Stair Ways

Imagine that you want to go up a staircase that has n steps, where n is a positive integer. You can take either one or two steps each time you move.

Write a generator function stair_ways that yields all the different ways you can climb the staircase.

Each "way" of climbing a staircase can be represented by a list of 1s and 2s, where each number indicates whether you take one step or two steps at a time.

For example, for a staircase with 3 steps, there are three ways to climb it:

• You can take one step each time: [1, 1, 1].
• You can take two steps then one step: [2, 1].
• You can take one step then two steps: [1, 2]..

Therefore, stair_ways(3) should yield [1, 1, 1], [2, 1], and [1, 2]. These can be yielded in any order.

def stair_ways(n):
    """
    Yield all the ways to climb a set of n stairs taking
    1 or 2 steps at a time.

    >>> list(stair_ways(0))
    [[]]
    >>> s_w = stair_ways(4)
    >>> sorted([next(s_w) for _ in range(5)])
    [[1, 1, 1, 1], [1, 1, 2], [1, 2, 1], [2, 1, 1], [2, 2]]
    >>> list(s_w) # Ensure you're not yielding extra
    []
    """
    "*** YOUR CODE HERE ***"

Use Ok to test your code:

python3 ok -q stair_ways
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Q4: Yield Paths

Write a generator function yield_paths that takes a tree t and a target value. It yields each path from the root of t to any node with the label value.

Each path should be returned as a list of labels from the root to the matching node. The paths can be yielded in any order.

Hint: If you are having trouble getting started, think about how you would approach this problem if it was not a generator function. What would the recursive steps look like?

Hint: Remember, you can iterate over generator objects because they are a type of iterator!

def yield_paths(t, value):
    """
    Yields all possible paths from the root of t to a node with the label
    value as a list.

    >>> t1 = tree(1, [tree(2, [tree(3), tree(4, [tree(6)]), tree(5)]), tree(5)])
    >>> print_tree(t1)
    1
      2
        3
        4
          6
        5
      5
    >>> next(yield_paths(t1, 6))
    [1, 2, 4, 6]
    >>> path_to_5 = yield_paths(t1, 5)
    >>> sorted(list(path_to_5))
    [[1, 2, 5], [1, 5]]

    >>> t2 = tree(0, [tree(2, [t1])])
    >>> print_tree(t2)
    0
      2
        1
          2
            3
            4
              6
            5
          5
    >>> path_to_2 = yield_paths(t2, 2)
    >>> sorted(list(path_to_2))
    [[0, 2], [0, 2, 1, 2]]
    """
    if label(t) == value:
        yield ____
    for b in branches(t):
        for ____ in ____:
            yield ____

Use Ok to test your code:

python3 ok -q yield_paths
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Check Your Score Locally

You can locally check your score on each question of this assignment by running

python3 ok --score

This does NOT submit the assignment! When you are satisfied with your score, submit the assignment to Gradescope to receive credit for it.

Submit Assignment

Submit this assignment by uploading any files you've edited to the appropriate Gradescope assignment. Lab 00 has detailed instructions.

Exam Practice

Homework assignments will also contain prior exam questions for you to try. These questions have no submission component; feel free to attempt them if you'd like some practice!

1. Summer 2016 Final Q8: Zhen-erators Produce Power
2. Spring 2018 Final Q4(a): Apply Yourself