Distributed: Wednesday, 22 November 2006
Due: 9:00 a.m., Friday, 1 December 2006
No extensions.
This page may be found online at http://www.cs.grinnell.edu/~rebelsky/Courses/CS151/2006F/Exams/exam.03.html.
This exam is also available in PDF format.
Contents
There are five problems on the exam. Some problems have subproblems.
Each problem is worth twenty (20) points. The point value associated
with a problem does not necessarily correspond to the complexity
of the problem or the time required to solve the problem.
This examination is open book, open notes, open mind, open computer,
open Web. However, it is closed person. That means you should not
talk to other people about the exam. Other than as restricted by that
limitation, you should feel free to use all reasonable resources available
to you. As always, you are expected to turn in your own work. If you
find ideas in a book or on the Web, be sure to cite them appropriately.
Although you may use the Web for this exam, you may not post your answers
to this examination on the Web (at least not until after I return exams
to you). And, in case it's not clear, you may not ask others (in person,
via email, via IM, by posting a please help
message, or in any
other way) to put answers on the Web.
This is a take-home examination. You may use any time or times you deem
appropriate to complete the exam, provided you return it to me by the
due date.
I expect that someone who has mastered the material and works at
a moderate rate should have little trouble completing the exam in a
reasonable amount of time. In particular, this exam is likely to take
you about four to six hours, depending on how well you've learned topics
and how fast you work. You should not work more than eight hours
on this exam. Stop at eight hours and write There's more to life
than CS
and you will earn at least 75 points on this exam.
I would also appreciate it if you would write down the amount of time
each problem takes. Each person who does so will earn two points of
extra credit. Since I worry about the amount of time my exams take,
I will give two points of extra credit to the first two people who
honestly report that they've spent at least five hours on the exam or
completed the exam. (At that point, I may then change the exam.)
You must include both of the following statements on the cover sheet of the
examination. Please sign and date each statement. Note that the
statements must be true; if you are unable to sign either statement,
please talk to me at your earliest convenience. You need not reveal
the particulars of the dishonesty, simply that it happened. Note also that
inappropriate assistance
is assistance from (or to) anyone
other than Professor Rebelsky (that's me) or Professor Davis.
1. I have neither received nor given inappropriate assistance on this
examination.
2. I am not aware of any other students who have given
or received inappropriate assistance on this examination.
Because different students may be taking the exam at different times,
you are not permitted to discuss the exam with anyone until after I
have returned it. If you must say something about the exam, you are
allowed to say This is among the hardest exams I have ever taken.
If you don't start it early, you will have no chance of finishing the
exam.
You may also summarize these policies. You may not tell
other students which problems you've finished. You may not tell other
students how long you've spent on the exam.
You must present your exam to me in two forms: both physically and
electronically. That is, you must write all of your answers using the
computer, print them out, number the pages, put your name on the top
of every page, and hand me the printed copy. You must also email me
a copy of your exam. You should create the emailed version by copying
the various parts of your exam and pasting them into an email message.
In both cases, you should put your answers in the same order as the
problems. Failure to name and number the printed pages will lead to a
penalty of two points. Failure to turn in both versions may lead to
a much worse penalty.
In many problems, I ask you to write code. Unless I specify otherwise
in a problem, you should write working code and include examples that
show that you've tested the code.
Just as you should be careful and precise when you write code and
documentation, so should you be careful and precise when you write prose.
Please check your spelling and grammar. Since I should be equally
careful, the whole class will receive one point of extra credit for each
error in spelling or grammar you identify on this exam. I will limit
that form of extra credit to five points.
I will give partial credit for partially correct answers. You ensure
the best possible grade for yourself by emphasizing your answer and
including a clear set of work that you used to derive the
answer.
I may not be available at the time you take the exam. If you feel that
a question is badly worded or impossible to answer, note the problem you
have observed and attempt to reword the question in such a way that it
is answerable. If it's a reasonable hour (before 10 p.m. and after 8
a.m.), feel free to try to call me in the office (269-4410) or at home
(236-7445).
I will also reserve time at the start of classes next week
to discuss any general questions you have on the exam.
Topics: Sorting, Orderings, Higher-order procedures, Objects
As you've probably noticed, there are two key postconditions of a
procedure that sorts lists: The result is a permutation of the original
list and the result is sorted. We're fortunate that the unit
test framework lets us test permutations (with test-permutation!).
Hence, if we wanted to test merge sort in the unit test framework, we
might write
(define some-list ...)
(test-permutation! (merge-sort some-list pred?) some-list)
However, we still need a way to make sure that the result is sorted,
particularly if the result is very long.
a. [5 points]
Document a procedure, (sorted? lst may-precede?) that checks
whether or not lst is sorted by may-precede?.
b. [5 points]
Write that procedure.
For example,
> (sorted? (list 1 3 5 7 9) <)
#t
> (sorted? (list 1 3 5 4 7 9) <)
#f
> (sorted? (list "alpha" "beta" "gamma") string<?)
#t
Note that we can use that procedure in a test suite for any
sorting routine with
(test! (sorted? (sort some-list may-precede?) may-precede?) #t)
c. [10 points] Suppose that we are representing students in this class
with objects, as in the file
student-object.scm.
Suppose also that we have stored all of the student records in a
list, csc151.
i. Write an expression that checks whether csc151
is sorted by student id (from alphabetically first to alphabetically
last).
ii. Write an expression that checks whether csc151 is
sorted by participation grade, from largest to smallest.
iii. Write an expression that checks whether csc151 is
sorted by name, from alphabetically first to alphabetically last.
(Students may have the same last name; two students with the same last
name should be ordered by first name.)
iv. Write an expression that checks whether csc151 is
sorted by average exam grade, from largest to smallest.
v. Write an expression that checks whether csc151 is
sorted by graduation year (class) and, within graduation year, is sorted by
last name. (You need not worry about first names.)
Topics: Vectors, Higher-order procedures, map, Objects
As you may recall, the map procedure typically takes two
parameters, a unary procedure and a list, and then builds a new list by
applying the procedure to each element of the list. One definition of
this simple version follows. (Advanced Schemers know that the
built-in map
can take more than two parameters; but that's irrelevant to this problem.)
(define map
(lambda (proc lst)
(if (null? lst)
null
(cons (proc (car lst)) (map proc (cdr lst))))))
As you've seen many times this semester, we often find it useful to
write vector procedures that mimic (or at least are similar to) list
procedures. For the case of map, we might write a
map! that applies a procedure to each element of a vector,
modifying the value in place.
For example,
> (define grades (vector 80 70 85 90))
> grades
#4(80 70 85 90)
> (map! (l-s + 5) grades) ; give everyone five points
> grades
#4(85 75 90 95)
> (map! (compose exact->inexact (r-s / 25)) grades) ; Divide grades by 25 for 4-point scale
> grades
#4(3.4 3.0 3.6 3.8)
a. [5 points] Document the map! procedure.
b. [10 points] Implement the map! procedure.
c. [5 points] In
student-object.scm,
you will find an implementation of a simple student object. Build
a vector of five randomly-generated student objects. Then, use map!
to give each of them one additional homework grade, a plus.
Topics: Higher-order procedures
As you may recall, we began our investigation of recursion by considering
how we might step through a list of numbers, adding them (or multiplying
them or ...). Here are variants of the two versions we wrote:
One uses simple recursion:
(define sumr
(lambda (numbers)
(if (null? (cdr numbers))
(car numbers)
(+ (car numbers) (sumr (cdr numbers))))))
The other uses a helper that accumulates the results.
(define suml
(letrec ((kernel
(lambda (remaining so-far)
(if (null? remaining)
so-far
(kernel (cdr remaining) (+ so-far (car remaining)))))))
(lambda (numbers)
(kernel (cdr numbers) (car numbers)))))
If you think carefully about it, the two processes compute results
slightly differently. Given the list (n0 n1 n2 n3 n4),
the first computes
(+ n0 (+ n1 (+ n2 (+ n3 n4))))
while the second computes
(+ (+ (+ (+ n0 n1) n2) n3) n4)
For addition, this doesn't make a difference. If the operation were
subtraction, it would certainly make a difference.
Now, the process of inserting a binary operation into a list of values
is quite common. We've seen it for addition, subtraction, and multiplication.
We might also use it for appending strings and many other operations.
Computer scientists have various names for such a process. We'll call
it fold, but others use insert (as in insert this
procedure
) or reduce.
a. [5 points] Write a procedure, (foldr proc lst) that mimics
sumr, but with proc in place of
+. That is,
(foldr proc (list v0 v1 v2 ... vn-1 vn)) should compute
(proc v0 (proc v1 (proc v3 (... (proc vn-1 vn) ... a))))
For example,
> (foldr + (list 1 2 3 4))
10
> (foldr * (list 1 2 3 4))
24
> (foldr - (list 1 2 3 4))
-2
> (foldr (lambda (s1 s2) (string-append s1 ", " s2))
(list "sentient" "malicious" "powerful" "stupid"))
"sentient, malicious, powerful, stupid"
> (foldr list (list 'a 'b 'c 'd 'e))
(a (b (c (d e))))
b. [5 points] Write a procedure, (foldl proc lst) that mimics
suml, but with proc in place of +.
That
is,
(foldl proc (list v0 v1 v2 ... vn-1 vn)) should compute
(proc ... (proc (proc (proc v0 v1) v2) v3) ... vn)
For example,
> (foldl + (list 1 2 3 4))
10
> (foldl * (list 1 2 3 4))
24
> (foldl - (list 1 2 3 4))
-8
> (foldl (lambda (s1 s2) (string-append s1 ", " s2))
(list "sentient" "malicious" "powerful" "stupid"))
"sentient, malicious, powerful, stupid"
> (foldl list (list 'a 'b 'c 'd 'e))
((((a b) c) d) e)
c. [10 points] Some computer scientists prefer an alternate version of
fold, which we'll call folder. Rather than taking
two parameters, their folder
takes only one, proc, a binary procedure. The
folder procedure then returns a new, variable-arity, procedure,
that inserts proc between its arguments. For example,
> (define comma-splice (lambda (s1 s2) (string-append s1 ", " s2)))
> (comma-splice "Fred" "Barney")
"Fred, Barney"
> (comma-splice "George" "Jane" "Elroy" "Judy")
procedure comma-splice: expects 2 arguments, given 4: "George" "Jane" "Elroy" "Judy"
> (comma-splice "Scooby-Doo")
procedure comma-splice: expects 2 arguments, given 1: "Scooby-Doo"
> (define comma-splicer (folderr (lambda (s1 s2) (string-append s1 ", " s2))))
> (comma-splicer "Fred" "Barney")
"Fred, Barney"
> (comma-splicer "George" "Jane" "Elroy" "Judy")
"George, Jane, Elroy, Judy"
> (comma-splicer "Scooby-Doo")
"Scooby-Doo"
Implement folderr or folderl.
Topics: Binary search, Randomness, Counting Steps
As you may recall, the binary search procedure quickly searches through
sorted vectors by keeping track of a region of the vector in which
a value should fall. At each step
, it identifies the midpoint
of the region, grabs the value at that region, compares it to the
desired value, and narrows the region.
Some have criticized this procedure for being too predictable.
Rather than choosing the middle of the region, they think we should choose
a random
dividing point.
Let's see what happens with this technique for vectors of integers
(so that you don't have to worry about what comparison routine to use).
In particular, we'll define a procedure (rbsi val vals) that
searches for val in the sorted vector of integers,
vals, and returns an index of val if it
appears or #f if it does not appear. (Note that
rbsi stands for random binary search for integers
.)
a. [5 points] Document this alternate version of binary search.
b. [5 points] Implement this alternate version of binary search.
c. [10 points] Traditional binary search takes approximately
log2n recursive calls for a vector of length
n. Experimentally determine whether this version takes
approximately the same number of steps.
You may find the following procedure useful for building sorted
vectors:
(define random-sorted-vector
(letrec ((kernel
(lambda (vec pos len seed)
(if (= pos len)
vec
(begin
(vector-set! vec pos seed)
(kernel vec (+ pos 1) len (+ seed (random 5))))))))
(lambda (len)
(kernel (make-vector len 0) 0 len (random 5)))))
For example,
> (random-sorted-vector 15)
#15(3 3 7 9 10 12 15 17 21 25 27 28 30 34 36)
> (random-sorted-vector 15)
#15(3 6 7 10 11 12 13 17 20 22 24 28 31 31 35)
> (random-sorted-vector 15)
#15(0 3 4 6 7 8 11 14 16 19 23 26 30 33)
You may find the following procedure useful for figuring out the value
of log2n
(define log2 (lambda (x) (/ (log x) (log 2))))
For example,
> (log2 16)
4.0
> (log2 1024)
10.0
> (log2 1000)
9.965784284662087
> (log2 (* 1024 1024))
20.0
Topics: Higher-order procedures, map
Although we've used the map procedure with two parameters,
a procedure and a list, it can take an arbitrary number of parameters.
For example,
> (map list (list 1 2 3) (list 'a 'b 'c) (list "he" "she" "it"))
((1 a "he") (2 b "she") (3 c "it"))
> (map + (list 1 2 3) (list 4 4 4) (list 2 3 5) (list 100 200 300))
(107 209 312)
Scheme also provides an apply procedure, which acts
similarly to the streme-apply we've written previously.
> (apply + (list 1 2 3))
6
> (apply sqrt (list 4))
2
a. [10 points]: By conducting some experiments (that is, by entering
a variety of expressions and seeing their results) and by reading
documentation, figure out what
map does when given more than two parameters. Explain
it in your own words. (Make sure to note any restrictions on the parameters.)
Include at least three illustrative examples.
b. [10 points]: By conducting some experiments (that is, by entering
a variety of expressions and seeing their results) and by reading
documentation, figure out what
apply does. Explain
it in your own words. (Make sure to note any restrictions on the parameters.)
Include at least three illustrative examples.
These are some of the questions students have asked about the
exam and my answers to those questions.
- What are the elements of
csc151?
- It's a list of students. Here's a straightforward way to make a
three element list of random students.
(define csc151 (list (random-student) (random-student) (random-student)))
- How can we generate a longer list?
- Write a longer definition. Write a procedure that generates an
arbitrary length list of random students. Whatever you find most
natural.
- How should we handle situations in which the two values are equal and
may-precede? is <?
- Arguably,
may-precede? should be <=
rather than < (which is more accurately
precedes?). However, you can do whatever you see
fit, provided you document what you've done.
- What should the expressions for part c look like?
(sorted? csc151 (lambda (s1 s2) ...))- In what order should graduation year and last name be sorted?
- Graduation year should be in increasing order. Within each year,
students should be in order by last name, from alphabetically first
to alphabetically last.
- How do I get access to the code from
student-object.scm?
- You could copy and paste the code into your exam or you could load that code with
(load "/home/rebelsky/Web/Courses/CS151/2006F/Examples/student-object.scm")
- What types should the
map! procedure support?
- The
map! procedure should accept any vector as its
second argument. The first argument should be a procedure. It
is up to you to document any restrictions on the procedure argument.
- Does
map! need to support empty vectors?
- Certainly.
- How do I get access to the code from
student-object.scm?
- See the answer to the previous question.
- Do you realize that
(student ':add-homework! 'plus)
returns void?
- Yes.
- So, how do we use
map! to modify the vector?
- Write an anonymous procedure that both adds the grade and returns
the modified student.
- May I use
vector->list?
- No.
- Can you give me a hint as to the structure for
folderr?
- Sure. You're writing a procedure of one parameter, and it needs to
return a variable-arity procedure, so the form should be something like the
following:
(define folderr
(lambda (proc)
(lambda params
...)))
- I've found that in implementing variable arity procedures, I often need
a kernel. What if I want to write a recursive kernel, too?
- Well, if you insist upon writing a recursive kernel, you probably
want something like the following:
(define folderr
(letrec ((kernel
(lambda (proc params)
...)))
(lambda (proc)
(lambda params
...))))
- The tone of your previous answer suggests that you don't think that
a kernel is necessary.
- I don't. However, if the most natural way for you solve this is to
use a recursive kernel, then use a recursive kernel.
Whatever
floats your boat
, as they say.
- Doesn't the comma belong inside the question mark in the previous
answer?
- I use British style for quotations, rather than American style,
because I feel that British style is more logical.
- Do we get extra credit for making both
folderl and
folderr?
- No.
- Do
foldr or foldl have to handle empty
lists?
- No.
- I've found it easier to count only recursive calls, and not the outermost
call. This means that when it guesses right on the first call, it seems
to take 0 steps. Is it okay if we don't count the outermost call to our
procedure?
- Yes, that's okay.
- Why does Dr. Davis have fewer questions on part 1.c?
- Perhaps she's nicer than I am.
Here you will find errors of spelling, grammar, and design that
students have noted. Remember, each error found corresponds to a
point of extra credit for everyone. I usually limit such extra credit to
five points. However, if I make an astoundingly large number of errors,
then I will provide more extra credit.
- Random
a
in the middle of Scheme code. [SR, 1 point]
- In the list of questions and answers,
Sam wrote
list or random students
instead of
list of random students
. [DB, 1 point]
- Sam is repetitious:
a region of of the vector
.
[JP, 1 point]
- Sam continues to be repetitious; he used two open quotes instead
of one. [AB, 1 point]
- Sam wrote
add-grade! when he meant to write
add-homework!. [PR, 1 point]
- In the Q&A section about 1c, Sam got the parameters to
sorted? in the wrong order;
(sorted? pred lst) rather than the correct
(sorted? lst pred).
[JB]
Thursday, 16 November 2006 [Samuel A. Rebelsky]
- First draft version, based on preliminary discussions with Janet Davis.
Saturday, 18 November 2006 [Samuel A. Rebelsky]
- Updated most of the problems after further discussion with Janet Davis.
- Added two new potential problems.
Tuesday, 21 November 2006 [Samuel A. Rebelsky]
- Added objects to the
sorted? problem.
- Added objects to the
map! problem.
- Removed a build-your-own-object problem (rectangles).
- Other, minor, cleanup.
Wednesday, 22 November 2006 [Samuel A. Rebelsky]
Monday, 27 November 2006 [Samuel A. Rebelsky]
- Added some questions and answers. The questions and answers on problem
3 were inspired by a discussion with a student. The questions and
answers on problems 1 and 2 were inspired by a similar set of questions
and answers on Dr. Davis's version of this exam.
Tuesday, 28 November 2006 [Samuel A. Rebelsky]
- More questions and answers.
- More errors.
;
