CSC151.02 2010S Functional Problem Solving : Exams
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References: [A-Z] [By Topic] - [Scheme Report (R5RS)] [R6RS] [TSPL4]
Related Courses: [CSC151.01 2010S (Weinman)] [CSC151 2009F (Rebelsky)]
Misc: [SamR] [MediaScript] [GIMP]
Assigned: Friday 30 April 2010
Due: Beginning of class, Friday 7 May 2010
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.
There are ten problems on this exam. Correct or mostly-correct answers to nine or ten problems will earn you an A. Correct or mostly-correct answers to seven or eight problems will earn you a B. Correct or mostly-correct answers to five or six problems will earn you a C. Correct or mostly-correct answers to three or four problems will earn you a D. Correct or mostly-correct answers to fewer than three problems will earn you an F.
Read the entire exam before you begin.
We 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 two to three hours, depending on how well you've learned the topics and how fast you work. You should not work more than four hours on this exam. Stop at four hours, show your work on at least four problems, and write “There is more to life than CS” and you will earn at least a C+ 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 have completed the exam in three hours or less or have spent at least three hours on the exam. In the latter case, they should also report on what work they've completed in the three hours. After receiving such notices, I may change the exam.
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. If you use code that you wrote for a previous lab or homework, cite that lab or homework and the other members of your group. If you use code that you found on the course Web site, be sure to cite that code. You need not cite the code provided in the body of the examination.
Although you may use the Web for this exam, you may not post your answers to this examination on the Web. 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.
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 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 Weinman.
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. Do not include images; I should be able to regenerate those.
Unless I state otherwise, you should document any procedures you write. Your documentation should use the six-P documentation style.
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. I am best able to give such partial credit if you include a clear set of work that shows how you derived your answer. You ensure the best possible grade for yourself by clearly indicating what part of your answer is work and what part is your final 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 each class before the exam is due to discuss any general questions you have on the exam.
assoc
for Repeated Keys
Topics: Sequential search, assoc
,
List recursion
As you may have noted, one potential deficiency of assoc
is that when multiple entries contain the same key,
assoc
returns only the first entry with a particular
key.
Write, but do not document, a procedure, (
, that
makes a list of all the values in
assoc-all
key
lst
)lst
whose car is key
.
You need not check the preconditions for this procedure.
Note that assoc-all
should return the empty list if
no entry contains the desired key.
For example,
(define people (list (list "Adams" "Amy" "Architecture") (list "Jones" "Jack" "Geography") (list "Jones" "Jane" "Geology") (list "Smith" "Sarah" "Science") (list "Jones" "Jenna" "Genetics")))
>
(assoc-all "Jones" people)
(("Jones" "Jack" "Geography") ("Jones" "Jane" "Geology") ("Jones" "Jenna" "Genetics"))
>
(assoc-all "Smith" people)
(("Smith" "Sarah" "Science"))
>
(assoc-all "Davis" people)
()
assoc-all
Topics: Documentation, Preconditions
Write the six-P documentation for assoc-all
.
Topics: Binary search, vectors, divide-and-conquer, numeric recursion
As you've observed, when a key is repeated, binary search picks
one value with that key, but not necessarily the first value with
that key. Write, but do not document, a variant,
binary-search-first
, that
uses the ideas of binary search to find the first
element in the vector that contains the given key.
You should strive to make your procedure as efficient as possible.
While one strategy for implementing binary-search-first
is to find some value with the given key (which binary search already
does) and then to step left in the vector until you find the first
value with that key, stepping through elements one-by-one is inefficient.
Write your version so that it continues to
“divide and conquer” in its attempt to find the first
element with that key.
Topics: Lists, pairs, trees, tree recursion
Finding the average of a list of numbers can be a fairly
straightforward process. For example, if we have
the sum
procedure from the list recursion
reading, we could use something like the following.
(define list-sum (lambda (numbers) (if (null? numbers) 0 (+ (car numbers) (list-sum (cdr numbers)))))) (define list-length (lambda (lst) (if (null? lst) 0 (+ 1 (list-length (cdr lst)))))) (define average-v1 (lambda (numbers) (/ (list-sum numbers) (list-length numbers))))
However, this implicitly makes two passes over the list, once to do
the sum and once to calculate the length. If we
have very long lists, we might wish to compact
this into a single pass by keeping track of both a sum of the
elements in the list (so-far) as well as the count of the number of
elements in the list (so-far). Essentially, performing the
operations of both code
and length
at the same time. For example:
(define average-v2 (lambda (numbers) (let kernel ((sum-so-far (car numbers)) (count-so-far 1) (remaining (cdr numbers))) (if (null? remaining) (/ sum-so-far count-so-far) (kernel (+ sum-so-far (car remaining)) (+ count-so-far 1) (cdr remaining))))))
Alternately, we might try something like the following.
; Given a list of numbers, simultaneously compute the sum and ; length of the list. (define numbers-info (lambda (numbers) (if (null? numbers) (list 0 0) ; 0 sum, 0 length (let ((results (numbers-info (cdr numbers)))) (list (+ (car numbers) (car results)) (+ 1 (cadr results))))))) (define average-v3 (lambda (numbers) (let ((info (numbers-info numbers))) (/ (car info) (cadr info)))))
Show the steps used in computing the average of the list
(2 5 7)
using average-v2
and
average-v3
. That is,
show the key steps the Scheme interpreter goes through in figuring out
this average. For example, if we were showing the steps for
average-v1
, we might write:
(average-v1 (2 5 6)) => (/ (sum (2 5 6)) (length (2 5 6))) => (/ (+ 2 (sum 5 6)) (length (2 5 6))) => (/ (+ 2 (+ 5 (sum 6))) (length (2 5 6))) => (/ (+ 2 (+ 5 (+ 6 (sum ())))) (length (2 5 6))) => (/ (+ 2 (+ 5 (+ 6 0))) (length (2 5 6))) => (/ (+ 2 (+ 5 6)) (length (2 5 6))) => (/ (+ 2 11) (length (2 5 6))) => (/ 13 (length (2 5 6))) => (/ 13 (+ 1 (length (5 6)))) => (/ 13 (+ 1 (+ 1 (length (6))))) => (/ 13 (+ 1 (+ 1 (+ 1 (length ()))))) => (/ 13 (+ 1 (+ 1 (+ 1 0)))) => (/ 13 (+ 1 (+ 1 1))) => (/ 13 (+ 1 2)) => (/ 13 3) => 13/3
Consider the problem of averaging the values in a number tree.
In one sense, it's very easy: We might use sum-of-number-tree
to sum the values and leaves-in-number-tree
(or a variant
thereof) to count the values. But that requires us to traverse
the tree twice.
If our only criterion is conciseness of code, we might even flatten
the number tree and then use one of the procedures from the previous
problem. However, flattening a tree is a relatively expensive procedure
due to the repeated appending within tree-flatten
.
Is it possible to do both the summing and counting in trees like we
did with lists? Of course! However, we must take a slightly
different approach than the tail
recursive average-v2
. Instead of passing along
results-so-far, we must use basic recursion (as
in numbers-info
) to compute the results from
the bottom upward.
Write, but do not document, a procedure
(
that returns a list whose first item is the sum of all the numbers in
the tree and whose second item is the total count of numbers in the
tree.
number-tree-info
ntree
)
>
(number-tree-info (cons 1 (cons 2 (cons (cons 3 4) (cons 5 6)))))
(21 6)
Hint: Think carefully first about your base case. Then, how will you combine the results of the recursive calls?
Note that once we've written this procedure, computing the average is quite
easy, remarkably like the average-v3
procedure
above.
(define number-tree-average (lambda (number-tree) (apply / (number-tree-info number-tree))))
Topics: Trees, higher-order procedures
Document number-tree-info
.
Topics: Characters, strings, higher-order procedures
A substitution cipher is a code that simply replaces one character with another in a one-to-one fashion. This is the kind of code that must be broken when solving the typical newspaper puzzle called a "cryptogram." A special case of this code is called a Caesar cipher, named for the Roman emperor Julius Caesar, who used it to communicate with his generals. A Caesar cipher simply shifts each character to another a certain number of places down the alphabet. In a Caesar cipher with a shift of 3, the letter A (character 0) would become D (character 3), P (character 15) would become S (character 18), and so on. Typically, Y (character 24) would wrap-around to the front of the alphabet to become B (character 27, or, actually character 1).
Write and document a procedure, (
that
shifts the given character the number of places, ignoring any wrap-around.
char-shift
character
amount
)
Of course, Caesar ciphers are much more interesting when applied to strings, rather than to individual characters. It is also natural to think of a cipher as a function from strings to strings, rather than from numbers to strings. Hence, we'd like a procedure that builds ciphers.
Write, but do not document, a procedure
(
that returns a procedure
that takes a string as input and produces a string where each character is
replaced by another whose collating sequence number differs from
the original by shift-encoder
amount
)amount
.
You should strive to make your procedure as concise as possible.
>
(define encode-1 (shift-encoder 5))
>
(encode-1 "HELLO")
"MJQQT"
>
(define encode-2 (shift-encoder 2))
>
(encode-2 "MJQQT")
"OLSSV"
>
(define decode-12 (shift-encoder -7))
>
(decode-12 "OLSSV")
"HELLO"
>
((o decode-12 encode-2 encode-1) "WORLD")
"WORLD"
Topics: Higher-order procedures
Document and write a procedure, (
that returns
the function c*x^{n}.
polynomial-term
c
n
)
For example,
>
(define two-x-cubed (polynomial-term 2 3))
>
(two-x-cubed 1)
2 ; 2 * 1 * 1 * 1
>
(two-x-cubed 3)
54 ; 2 * 3 * 3 * 3
>
(two-x-cubed 5)
250 ; 2 * 5 * 5 * 5
>
(define three-x-squared (polynomial-term 3 2))
>
(three-x-squared 1)
3 ; 3 * 1 * 1
>
(three-x-squared 3)
27 ; 3 * 3 * 3
>
(three-x-squared 5)
75 ; 3 * 5 * 5
>
((polynomial-term 5 4) 2)
80 ; 5 * 2^4
Write, but do not document, a procedure
(
that takes a list of coefficients for the terms x^{0},
x^{1}, x^{2}, ... of a
polynomial and produces a
function that takes a single value, evaluating the polynomial given
those coefficients at that value.
polynomial
coefs
)
>
(define line (polynomial (list 1 4)))
; Create the polynomial f(x) = 1*x^0 + 4*x^1 = 1 + 4*x>
(line 5)
; Evaluate f(5) = 1 + 4*521
>
(define cubic (polynomial (list 1 4 3 -2)))
; Create the polynomial g(x) = 1 + 4*x + 3*x^2 - 2*x^3>
(cubic 5)
; Evaluate g(5) = 1 + 4*5 + 3*5^2 - 2*5^3-154
Note: You should try to make your solution as concise as possible (even at the potential cost of some efficiency).
Note: Since a polynomial normally has at least one term, you may assume that the list has at least one term. You need not check this precondition (or any precondition, for that matter).
Here we will post answers to questions of general interest. Please check here before emailing your questions!
binary-search-first
take?
(binary-search-first
vec
get-key
may-precede?
key
)
(expt x y)
in Problems 9 and 10?
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. We usually limit such extra credit to five points. However, if we make an astoundingly large number of errors, then we will provide more extra credit.
(three-x-squared 3)
is 27, not 18. [MS & RN, 1 point]"Genetics"
was missing a close quote in the example. [RN, 1 point]average-v1
does not use list-length
and list-sum
. [IY, 1 point]null?
[RN & JW, 1 point](2 5 7)
but the example uses (2 5 6)
[JW, 1 point]list-length
should have used lst
throughout. [BK & AW]Primary: [Front Door] [Schedule] - [Academic Honesty] [Instructions]
Current: [Outline] [EBoard] [Reading] [Lab] - [Assignment] [Quiz]
Groupings: [Assignments] [EBoards] [Examples] [Exams] [Handouts] [Labs] [Outlines] [Projects] [Quizzes] [Readings]
References: [A-Z] [By Topic] - [Scheme Report (R5RS)] [R6RS] [TSPL4]
Related Courses: [CSC151.01 2010S (Weinman)] [CSC151 2009F (Rebelsky)]
Misc: [SamR] [MediaScript] [GIMP]
Copyright (c) 2007-10 Janet Davis, Matthew Kluber, Samuel A. Rebelsky, and Jerod Weinman. (Selected materials copyright by John David Stone and Henry Walker and used by permission.)
This material is based upon work partially supported by the National Science Foundation under Grant No. CCLI-0633090. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
This work is licensed under a Creative Commons
Attribution-NonCommercial 2.5 License. To view a copy of this
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