Homework 5 Solutions

Solution Files

You can find the solutions in hw05.py.

Required Questions

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Q1: Infinite Hailstone

Write a generator function that yiels 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's 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're 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):
    """Q1: 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)
yield n if n == 1: yield from hailstone(n) elif n % 2 == 0: yield from hailstone(n // 2) else: yield from hailstone(n * 3 + 1)

Use Ok to test your code:

python3 ok -q hailstone

Q2: Merge

Write a generator function merge that takes in two infinite generators a and b that are in increasing order without duplicates and returns a generator that has all the elements of both generators, in increasing order, without duplicates.

def merge(a, b):
    >>> 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]
first_a, first_b = next(a), next(b) while True: if first_a == first_b: yield first_a first_a, first_b = next(a), next(b) elif first_a < first_b: yield first_a first_a = next(a) else: yield first_b first_b = next(b)

Use Ok to test your code:

python3 ok -q merge

Q3: Yield Paths

Define a generator function yield_paths which takes in a tree t, a value value, and returns a generator object which yields each path from the root of t to a node that has label value.

Each path should be represented as a list of the labels along that path in the tree. You may yield the paths in any order.

def yield_paths(t, value):
    """Q4: 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)
    >>> 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)
    >>> path_to_2 = yield_paths(t2, 2)
    >>> sorted(list(path_to_2))
    [[0, 2], [0, 2, 1, 2]]
    if label(t) == value:
yield [value]
for b in branches(t):
for path in yield_paths(b, value):
yield [label(t)] + path

Hint: If you're having trouble getting started, think about how you'd approach this problem if it wasn't a generator function. What would your recursive calls be? With a generator function, what happens if you make a "recursive call" within its body?

Hint: Try coming up with a few simple cases of your own! How should this function work when t is a leaf node?

Hint: Remember, it's possible to loop over generator objects because generators are iterators!

Note: Remember that this problem should yield paths -- do not return a list of paths!

Use Ok to test your code:

python3 ok -q yield_paths

If our current label is equal to value, we've found a path from the root to a node containing value containing only our current label, so we should yield that. From there, we'll see if there are any paths starting from one of our branches that ends at a node containing value. If we find these "partial paths" we can simply add our current label to the beinning of a path to obtain a path starting from the root.

In order to do this, we'll create a generator for each of the branches which yields these "partial paths". By calling yield_paths on each of the branches, we'll create exactly this generator! Then, since a generator is also an iterable, we can iterate over the paths in this generator and yield the result of concatenating it with our current label.

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 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 2018 Final Q7a,b: Streams and Jennyrators
  2. Spring 2019 Final Q1: Iterators are inevitable
  3. Spring 2021 MT2 Q8: The Tree of L-I-F-E
  4. Summer 2016 Final Q8: Zhen-erators Produce Power
  5. Spring 2018 Final Q4a: Apply Yourself