Homework 7
Due by 11:59pm on Friday, 4/15
Instructions
Download hw07.zip. Inside the archive, you will find a file called hw07.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 0 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:
Generators
Question 1: 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.
>>> 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: This problem yields to recursion. If you had the permutations of a subsequence of a that was missing one element, how could generate from it the permutations of the full sequence?
Use OK to test your code:
python3 ok -q permutations
Streams
The Stream
class defines a lazy sequence, a lazily
evaluated linked list. In other words, a Stream's elements (except for the
first element) are only evaluated as those values are needed.
class Stream:
"""A lazily computed linked list."""
class empty:
def __repr__(self):
return 'Stream.empty'
empty = empty()
def __init__(self, first, compute_rest=lambda: Stream.empty):
assert callable(compute_rest), 'compute_rest must be callable.'
self.first = first
self._compute_rest = compute_rest
@property
def rest(self):
"""Return the rest of the stream, computing it if necessary."""
if self._compute_rest is not None:
self._rest = self._compute_rest()
self._compute_rest = None
return self._rest
def __repr__(self):
return 'Stream({0}, <...>)'.format(repr(self.first))
Instead of specifying all of the elements in __init__
, we
provide a function, compute_rest
, that encapsulates the algorithm
used to calculate the remaining elements of the stream.
The code in the function body is not evaluated until it is called,
which lets us implement the desired evaluation behavior.
This implementation of streams also uses memoization. The first time
a program asks a Stream
for its rest
field, the Stream
code
computes the required value using compute_rest
, saves the resulting
value, and then returns it. After that, every time the rest
field is
referenced, the stored value is simply returned and it is not computed
again.
Here is an example (which you may use in solutions):
def make_integer_stream(first=1):
def compute_rest():
return make_integer_stream(first+1)
return Stream(first, compute_rest)
We start out with a stream whose first
element is 1, and whose compute_rest
function creates another stream.
So when we do compute the rest
, we get another stream whose first
element is one greater than the previous element, and whose
compute_rest
creates another stream. Hence, we effectively get an
infinite stream of integers, computed one at a time. This is almost
like an infinite recursion, but one which can be viewed one step at a
time, and so does not crash.
Another example:
def map_stream(fn, s):
if s is Stream.empty:
return s
return Stream(fn(s.first), lambda: map_stream(fn, s.rest))
Question 2: 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 3: Efficient Map
The map-stream
function:
def map_stream(fn, s):
if s is Stream.empty:
return s
return Stream(fn(s.first), lambda: map_stream(fn, s.rest))
creates a new lambda function each time the tail of the list is
expanded via the rest
attribute, because it calls map_stream
, which
then executes lambda: map_stream(fn, s.rest)
. The value of a lambda
expression is a new Python object, requiring space. But each function
that is given as a second argument to Stream
is only called once, and then
thrown away (self._compute_rest = None
). See if you can find a way to
avoid creating a new function, and instead re-use the same function for each
_compute_rest
value. Obviously, the crux of the problem is to devise a way
to get this (parameterless) function to return a different value each time.
def lst_to_stream(lst):
"""A utility for converting iterator or iterable LST into a corresponding
Stream value. (Use for testing only)"""
if not lst:
return Stream.empty
return Stream(lst[0], lambda: lst_to_stream(lst[1:]))
def efficient_map_stream(fn, s):
"""Compute the Stream resulting from applying FN to each value of Stream
S in turn. [This implementation should only create a finite number of
functions in total, regardless of the length of the stream S.]
>>> str = lst_to_stream([7, -9, -3, 0])
>>> ab = efficient_map_stream(abs, str)
>>> stream_to_list(ab)
[7, 9, 3, 0]
>>> stream_to_list(efficient_map_stream(abs, Stream.empty))
[]
>>> stream_to_list(efficient_map_stream(abs, Stream(-3)))
[3]
>>> stream_to_list(efficient_map_stream(lambda x: x*x, make_integer_stream(1)))
[1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
"""
"*** YOUR CODE HERE ***"
Use OK to test your code:
python3 ok -q efficient_map_stream
Question 4: Stream of Streams
Write the functionmake_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).
def make_stream_of_streams():
"""
>>> stream_of_streams = make_stream_of_streams()
>>> stream_of_streams
Stream(Stream(1, <...>), <...>)
>>> stream_to_list(stream_of_streams, 3)
[Stream(1, <...>), Stream(2, <...>), Stream(3, <...>)]
>>> stream_to_list(stream_of_streams, 4)
[Stream(1, <...>), Stream(2, <...>), Stream(3, <...>), Stream(4, <...>)]
"""
"*** YOUR CODE HERE ***"
Use OK to test your code:
python3 ok -q make_stream_of_streams
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 with1
.- The elements of
scale_stream(s, 2)
are also elements ofs
. - The same is true for
scale_stream(s, 3)
andscale_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