Homework 8
Due by 11:59pm on Tuesday, 4/18
Instructions
Download hw08.zip. Problems are in hw08.py and hw08.scm.
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.
Readings: You might find the following references useful:
Homework Questions
Some Review: Sets
One really convenient thing about Python sets is that many operations on sets (adding elements, removing elements, checking membership) run in θ(1) (constant) time (usually).
Some of the problems use a utility method called timeit
, which takes a
parameterless function as argument, executes it, and returns the time required
to do so. It's a variation on the function timeit.timeit
function in the
Python3 library.
Question 1: Missing Value
Write the following function so it (usually) runs in θ(n) time, where n is
the length of lst0
.
def missing_val(lst0, lst1):
"""Assuming that lst0 contains all the values in lst1, but lst1 is missing
one value in lst0, return the missing value. The values need not be
numbers.
>>> from random import shuffle
>>> missing_val(range(10), [1, 0, 4, 5, 7, 9, 2, 6, 3])
8
>>> big0 = [str(k) for k in range(15000)]
>>> big1 = [str(k) for k in range(15000) if k != 293 ]
>>> shuffle(big0)
>>> shuffle(big1)
>>> missing_val(big0, big1)
'293'
>>> timeit(lambda: missing_val(big0, big1)) < 1.0
True
"""
"*** YOUR CODE HERE ***"
Use OK to test your code:
python3 ok -q missing_val
Question 2: Find duplicates k
Write the following function so it runs in O(n) time.
Hint: Sets have an
add
and aremove
method.
def find_duplicates_k(k, lst):
"""Returns True if there are any duplicates in lst that are within k
indices apart.
>>> find_duplicates_k(3, [1, 2, 3, 4, 1])
False
>>> find_duplicates_k(4, [1, 2, 3, 4, 1])
True
>>> find_duplicates_k(4, [1, 1, 2, 3, 3])
True
"""
"*** YOUR CODE HERE ***"
Use OK to test your code:
python3 ok -q find_duplicates_k
More Review: Scheme
Note: Q3 and Q4 (Substitute, Sub All) were extra lab questions from Lab 9. You may check the solutions if you are stuck, but we highly recommend you work through the problem on your own for practice.
Question 3: Substitute
Write a procedure substitute
that takes three arguments: a list s
, an old
word, and a new
word. It returns a list with the elements of s
, but with
every occurrence of old
replaced by new
, even within sub-lists.
Hint: The built-in pair?
predicate returns True if its argument is a cons
pair.
(define (substitute s old new)
'YOUR-CODE-HERE
)
Use OK to test your code:
python3 ok -q substitute
Question 4: Sub All
Write sub-all
, which takes a list s
, a list of old
words, and a
list of new
words; the last two lists must be the same length. It returns a
list with the elements of s
, but with each word that occurs in the second
argument replaced by the corresponding word of the third argument.
(define (sub-all s olds news)
'YOUR-CODE-HERE
)
Use OK to test your code:
python3 ok -q sub-all
Streams
Question 5: 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 6: 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
Question 7: 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. Your solution
must not use any lambdas or def's that we have not supplied in the
skeleton.
>>> 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
Extra questions
Extra questions are not worth extra credit and are entirely optional.
Question 8: 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). 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_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
Differentiation
The following problems develop a system for
symbolic differentiation
of algebraic expressions. The derive
Scheme procedure takes an
algebraic expression and a variable and returns the derivative of the
expression with respect to the variable. Symbolic differentiation is of
special historical significance in Lisp. It was one of the motivating
examples behind the development of the language. Differentiating is a
recursive process that applies different rules to different kinds of
expressions:
; derive returns the derivative of EXPR with respect to VAR
(define (derive expr var)
(cond ((number? expr) 0)
((variable? expr) (if (same-variable? expr var) 1 0))
((sum? expr) (derive-sum expr var))
((product? expr) (derive-product expr var))
((exp? expr) (derive-exp expr var))
(else 'Error)))
To implement the system, we will use the following data abstraction. Sums and products are lists, and they are simplified on construction:
; Variables are represented as symbols
(define (variable? x) (symbol? x))
(define (same-variable? v1 v2)
(and (variable? v1) (variable? v2) (eq? v1 v2)))
; Numbers are compared with =
(define (=number? expr num)
(and (number? expr) (= expr num)))
; Sums are represented as lists that start with +.
(define (make-sum a1 a2)
(cond ((=number? a1 0) a2)
((=number? a2 0) a1)
((and (number? a1) (number? a2)) (+ a1 a2))
(else (list '+ a1 a2))))
(define (sum? x)
(and (list? x) (eq? (car x) '+)))
(define (addend s) (cadr s))
(define (augend s) (caddr s))
; Products are represented as lists that start with *.
(define (make-product m1 m2)
(cond ((or (=number? m1 0) (=number? m2 0)) 0)
((=number? m1 1) m2)
((=number? m2 1) m1)
((and (number? m1) (number? m2)) (* m1 m2))
(else (list '* m1 m2))))
(define (product? x)
(and (list? x) (eq? (car x) '*)))
(define (multiplier p) (cadr p))
(define (multiplicand p) (caddr p))
Question 9: Derive Sum
Implement derive-sum
, a procedure that differentiates a sum by
summing the derivatives of the addend
and augend
. Use data abstraction
for a sum:
(define (derive-sum expr var)
'YOUR-CODE-HERE
)
Use OK to test your code:
python3 ok -q derive-sum
Question 10: Derive Product
Implement derive-product
, which applies the product
rule to differentiate
products:
(define (derive-product expr var)
'YOUR-CODE-HERE
)
Use OK to test your code:
python3 ok -q derive-product
Question 11: Make Exp
Implement a data abstraction for exponentiation: a base
raised to the
power of an exponent
. The base
can be any expression, but assume that the
exponent
is a non-negative integer. You can simplify the cases when
exponent
is 0
or 1
, or when base
is a number, by returning numbers from
the constructor make-exp
. In other cases, you can represent the exp as a
triple (^ base exponent)
.
Hint: The built-in procedure expt
takes two number arguments and raises
the first to the power of the second.
; Exponentiations are represented as lists that start with ^.
(define (make-exp base exponent)
'YOUR-CODE-HERE
)
(define (base exp)
'YOUR-CODE-HERE
)
(define (exponent exp)
'YOUR-CODE-HERE
)
(define (exp? exp)
'YOUR-CODE-HERE
)
(define x^2 (make-exp 'x 2))
(define x^3 (make-exp 'x 3))
Use OK to test your code:
python3 ok -q make-exp
Question 12: Derive Exp
Implement derive-exp
, which uses the power
rule to derive
exps:
(define (derive-exp exp var)
'YOUR-CODE-HERE
)
Use OK to test your code:
python3 ok -q derive-exp