This section provides a basic description of the discipline of computer science and outlines this lab-based, two-day introductory workshop.
Computers are useful, because they can help us solve problems. The discipline of computer science studies all elements of the process of using computers to solve these problems.
Unfortunately, popular culture sometimes associates characteristics to computing that are misleading. Here are two examples:
Computer scienceis the traditional name given to the discipline (at least in the United States), but even the name is somewhat misleading. In fact, in our studies, we rely on the tools and techniques from a number of other disciplines -- the scientific method is only part of the picture. From mathematics, we take proof techniques, formal language for describing problems and solutions, and even core ideas. From science, we take experimental techniques. From engineering, we take techniques for designing and constructing things. From psychology and the social sciences, we take techniques for understanding the relationship of our work to human endeavors. Overall, the discipline of computer science involves a much wider range of activities than the name suggests. However, we're stuck with the name and popular views of what it entails.
The primary goal of this two-day laboratory experience is to give you a
sense of what it means to do
computer science. We'll carefully
specify some problems, design some algorithms to solve them, implement
them in the programming language Scheme, and evaluate them experimentally.
Today we'll focus on some background ideas. We'll start by writing an algorithm for an everyday problem. We'll continue by looking at a traditional and well-studied problem in computer science, that of putting a list of things in order. We'll spend a little time working with the basics of Scheme. We'll conclude by reflecting on what we've done today.
Tonight you'll have the opportunity to read some more formal descriptions of some common sorting algorithms in preparation for tomorrow's lab. You'll also reflect on what you've done today.
Tomorrow, we'll consider these sorting algorithms and some related algorithms in more depth. We'll be doing a good deal of experimental analysis and also look at some related design and theory principles.
Write a set of instructions for making a peanut butter and jelly
sandwich. Assume that the person you're writing instructions for
is fairly clueless (e.g., an ivory tower
professor, a computer
programmer, a politician, or whatever group to whom you assign little
common sense).
Given your experience of dealing with a simulated clueless person, write a set of instructions for making N peanut butter and jelly sandwiches, where N is a non-negative integer. That is, the person should be able to follow these instructions to make 1, 10, 5, or even 0 sandwiches. You can assume that the person has enough supplies to make N sandwiches and that the person can count.
As you may have noted, there are some common aspects to algorithms. That is, there are techniques that we use in many of the algorithms we write. It is worthwhile to think about these algorithm parts because we can rely on them when we write new algorithms. Common parts of algorithms include
As we write algorithms, we like to name things. Sometimes we use
long names, such as the piece of bread in your left hand
.
Sometimes, we use shorter names, such as bread-left
. As
we start to write more formal algorithms, we may explicitly say
that we're using names.
Many algorithms work on data that are presented to the algorithm.
For example, our PBJ algorithm should work no matter what kind of
bread or peanut butter we give to the algorithm. That algorithm
also requires that we give it bread and peanut butter. Similarly,
an algorithm to find a name in the phone book might take the name
and the phone book as input values. We call such input values
parameters
.
At times, our algorithms have to account for different conditions, doing different things depending on those conditions. In our PBJ algorithm, we might check whether the jar of peanut butter is open or what kind of lid is on the jelly jar. We call such operations conditionals. They typically take either the form
if some condition holds
then do something
of the more complex form
if some condition holds
then do something
otherwise do something else
At times, we need to decide between more than two possibilities.
At times, our algorithms require us to do something again and again.
In our PBJ algorithm, we may have had to turn the twisty-tie again
and again until it was untwisted. In our many-sandwiches algorithm,
we made one sandwich again and again. We call this technique
repetition
. Repetition takes many forms. We
might do work until we've reached a desired state.
repeat
some action
until some condition holds
We might continue work as long as we're in some state.
repeat
some action
while some condition holds
We might repeat an action a fixed number of times.
repeat
some action
N times
You can probably think of other forms of repetition.
Many algorithms require common actions for their operation. For
example, to make N sandwiches, you benefit from knowing how to make
one sandwich. To make a peanut butter and jelly sandwich, it helps
to know how to spread something on bread. We can write additional
algorithms for these common actions and use them as part of our
broader algorithm. We can also use them in other algorithms.
We call these helper algorithms subroutines
.
As strange as it may seem, we sometimes find it useful to define an algorithm in terms of itself. In particular, we can have an algorithm deal with a large or complex input by using itself as a helper with a smaller or less complex input.
Consider the following example
To make N peanut butter and jelly sandwiches If N is 0 then conditional Celebrate, you're done! Otherwise Make one PBJ sandwich subroutine Make N-1 PBJ sandwiches recursion
We call this technique recursion. Mastering recursion is one of the key steps on your path to becoming a computer scientist.
A helpful way to think about recursion: Every time you use a subroutine, you hand both the input and the instructions for the algorithm to a friend. That friend executes the algorithm and hands the result back to you. It shouldn't matter whether the algorithm you give your friend is a different one than you're using or the same one.
Write an algorithm that tells someone how to put a collection of CDs in alphabetical order by title. You may find it useful to design additional algorithms to solve parts of the problem.
Write an algorithm that tells someone how to put a list of numbers in
order from smallest to largest. Your algorithm should use a different
strategy than your sort CDs
algorithm. Once again, you may find
it useful to design additional algorithms to solve parts of the problem.
Please don't be intimidated! Although this lab contains many details which may seem overwhelming at first, these mechanics will become familiar rather quickly. To help the process along, you might want to spend some time on your own before the next class, working through the lab a few times, exploring various buttons, and experimenting with options. Feel free to talk to the instructor or with a MathLAN User Consultant if you have questions or want additional help!
Short Version:
To use any of the computers in the Mathematics Local-Area Network, one must log in, identifying oneself by giving a user name and a password. You will have received a MathLAN user name and password from the instructor if you did not already have one. If you have not received a MathLAN user name and password, or if you have forgotten either one, please tell your instructor.
When it is not in use, a MathLAN workstation displays a login screen with a space into which one can type one's user name and, later, one's password. (If the workstation's monitor is dark, move the mouse a bit and the login screen will appear.) To begin, move the mouse onto any part of the box containing the login box. Type in your user name, in lower-case letters, and press the <Enter> key. The login screen will be redrawn to acknowledge your user name and to ask for your password; type it into the space provided and press <Enter>. (Because no one else should see your password, it is not displayed on screen as you type it in.)
At this point, a computer program that is running on the workstation consults a table of valid user names and passwords. If it does not find the particular combination that you have supplied, it prints a brief message saying that the attempt to log in was unsuccessful and then returns to the login screen -- inviting you to try again. Consult the instructor or the system administrator if your attempts to log in are still unsuccessful.
Short Version:
Once you have logged in, a control panel will appear at the bottom of the screen. Some other windows also may be visible in other parts of your screen. All of these areas are managed by a special program, called a windowing system. On MathLAN, login chores and other administrivia are handled by a program or operating system, called Linux, and the windowing system is called Gnome. (While the Linux operating system supports many languages and environments, you may be interested to know that the Linux/Gnome windowing system itself is written in Scheme.)
Short Version:
Many of the fundamental ideas of computer science are best learned by reading, writing, and executing small computer programs that illustrate them. One of our most important tools, therefore, is a computer program designed specifically to make it easier to read, write, and execute other computer programs. In our introductory courses, we will often use a programming environment named DrScheme.
Short Version:
To start DrScheme, click the mouse on the large, round, red icon containing
the Greek letter lambda in the middle of the control panel. If you don't have
that icon, click on the icon of a computer screen with a foot on it, Soon
you'll see a gray window on the screen, type drscheme &
in that window.
Shortly, DrScheme will start up and appear as a new window, with two white rectangular text areas against a dark gray background.
At the top of the window, just below the frame, is a menu bar, providing ways of activating various operations that DrScheme can perform. Eventually we'll explore a number of these, but for the moment let's just look at one that you're certain to need: the operation of shutting itself down.
Move the mouse pointer onto the word File at the left end of the menu bar and click the left mouse button. A small menu appears.
Move the mouse pointer onto the word Quit at the bottom of this menu and click the left mouse button. DrScheme responds by popping up a confirmation box of its own, the purpose of which is to make sure that you don't shut down DrScheme by mistake.
Move the mouse pointer onto the word Quit in the confirmation box and click the left mouse button. Both the confirmation box and the main DrScheme window disappear.
Short Version:
DrScheme can deal with computer programs written in a number of dialects of the Scheme programming language. When you start DrScheme for the first time, it expects you to use Beginning Student Scheme, which is a kind of training-wheels dialect that enables DrScheme to catch and diagnose some common mistakes of novice programmers.
Although you are novice programmers,
we're going to proceed directly to Full Scheme,
which is more
nearly standard. To inform DrScheme of this decision, we'll need another
of the operations accessed through the menu bar. If you exited from
DrScheme, restart it now. When the window appears, move the mouse pointer
onto the word Language on the menu bar and click the
left mouse button to bring up the Language menu. Move the mouse pointer
onto the phrase Choose Language... and click the left
mouse button again to select that operation. Another window appears.
Move the mouse pointer over the triangle to the left of PLT and click on it until it points downward. You should see a few more words indented. Click on Graphical (MrEd, includes MzScheme).
Move the mouse pointer onto the word OK and click the left mouse
button. The extra window disappears, leaving the DrScheme window. Next,
move the mouse pointer onto the word Execute (next to a green
arrow, just below the menu bar) and press the left mouse button. DrScheme
now expects programs in the dialect it calls Graphical Full Scheme
.
DrScheme retains the information that you prefer to use full Scheme, so that when you log in again tomorrow and start DrScheme again, it will automatically expect programs in that dialect. You won't need to use the Choose Language operation again unless you decide that you'd like to experiment with Beginning Student Scheme or some other dialect.
Make sure that all members of your group login and choose the appropriate dialect of Scheme before you go on to do any other work in Scheme.
Short Version:
(sqrt 137641)
Now you're ready to look at the parts of DrScheme.
In the interactions pane -- the lower of the two large text areas -- DrScheme displays a one-line greeting, a reminder of which dialect of Scheme it expects to see, and a prompt (in this case, a greater-than sign), indicating that DrScheme is ready to deal with any command that you type in.
To enter a Scheme program, move the mouse pointer to the right of the prompt, click the left mouse button, and type in the program. (If you prefer, you can select the program from another window by moving the mouse pointer to the beginning of the program, pressing and holding the left mouse button, dragging the mouse pointer to the end of the program, and releasing the left mouse button. The background color against which the text that you have selected changes during this process, so that you can see the boundaries of the selection clearly. You can then paste the selected text into the interactions pane by moving the mouse pointer to the right of the prompt and clicking the middle mouse button.)
To get DrScheme to execute your program, press the Enter key after the right parenthesis. At this point, DrScheme examines your program, translates it into a sequence of instructions to the computer's central processor (the electronic circuit that directs the movement and transformation of data inside the computer), executes it, and prints out the result of its computation. Because this particular program is extremely simple, the result is printed immediately.
You may notice that DrScheme prints out another prompt after
executing your program. This is because DrScheme cannot be sure that it
has seen all the steps in the program. A program written in Scheme has a
particularly simple structure: it is a sequence of definitions and commands
-- any number of them, in any order. DrScheme reacts to each definition
that you type into the interactions pane by memorizing it and to each
command by carrying out the command. (The expression (sqrt
137641)
is a command -- Compute the square root of 137641!
)
Because a program might contain several commands rather than just one,
DrScheme has to be prepared to receive another after carrying out the
first.
Short Version:
The upper text area in the DrScheme window, which is called the
definitions pane, is used when you want to prepare a program
off-line,
that is, without immediately executing each step. Instead of
processing what you type line by line, DrScheme waits for you to click on
the button labeled Execute (the second button from the right, in
the row just below the menu bar) before starting to execute the program in
the definitions pane. If you never click on that button, fine -- your
program is never executed.
As its name implies, the definitions pane usually contains definitions rather than commands, although either kind of expression can be written in either pane. The difference is simply that we generally want an immediate response to a command, whereas definitions are usually processed in bulk.
Warning: When you click on the Execute button, the contents of the interactions pane are erased. The idea is that executing the program in the definitions pane may invalidate the results of previous interactions. Erasing the results that may now be inconsistent with the new definitions ensures that all visible interactions use the same vocabulary. This is actually a helpful feature of DrScheme, but it can take you by surprise the first time you see it happen. Just make sure that you have everything you need from the interactions pane before clicking on Execute.
In addition to unstructured
data types such as symbols and numbers,
Scheme supports lists, which are structures that contain other
values as elements. There is one list, the empty list, that
contains no elements at all. Any other list is constructed by attaching
some value, called the car of the new list, to a previously
constructed list, which is called the cdr of the new list.
Scheme's name for the empty list is a pair of parentheses with nothing
between them: ()
. When we refer to the empty list in a Scheme
program, we have to put an apostrophe before the left parenthesis, so that
Scheme won't mistake the parentheses for a procedure call:
> () ()
Since this conventional name for the empty list is not very readable, our
implementation of Scheme also provides a built-in name, null
,
for the empty list. We follow this usage and recommend it.
> null ()
The simplest way to build a list is with the list
procedure.
You tell Scheme to build a list by writing an open parenthesis, the
procedure name list
, the values you want in the list, and
a close parenthesis.
> (list 1 2 3) (1 2 3) > (list 3 1 2 5 6) (3 1 2 5 6)
You can name a list you've created with the define
procedure.
> (define lst (list 1 2 3 4 5)) > lst (1 2 3 4 5)
You can also add an element to the front of a list with cons
.
It takes two arguments and returns a list that is just like the second
argument, except that the first argument has been added at the beginning,
as a new first element. Once again, this change does not affect the
original list.
> (define lst (list 1 2 3 4 5)) > lst (1 2 3 4 5) > (cons 6 lst) (6 1 2 3 4 5) > (define newlst (cons 0 lst)) > newlst (0 1 2 3 4 5) > lst (1 2 3 4 5)
Task: Try this example!
You can even use cons
to build up a list from nothing
(or at least from null
).
> (define singleton (cons 5 null)) > singleton (5) > (define doubleton (cons 8 singleton)) > doubleton (8 5) > (define tripleton (cons 0 doubleton)) > tripleton (0 8 5) > (cons 1 (cons 2 (cons 3 (cons 4 null)))) (1 2 3 4)
Task: Try this example!
Once you've built a list, you can extract the first element with
car
and the rest of the list (all but the first element)
with cdr
. Neither operation affects the original list;
they simply create a new
value or list.
> (define lst (list 1 2 3 4 5)) > lst (1 2 3 4 5) > (car lst) 1 > (cdr lst) (2 3 4 5) > lst (1 2 3 4 5)
Task: Try this example!
Task: Figure out how to get the second value in a list.
Scheme provides several built-in procedures that operate on lists. Here are a few that are very frequently used:
The length
procedure takes one argument, which must be a
list, and computes the number of elements in the list.
Task: Play with length
to make sure you
understand it.
The reverse
procedure takes a list and returns a new list
containing the same elements, but in the opposite order.
> (reverse (list 1 2 3)) (3 2 1)
The append
procedure combines elements on shorter lists
to create a longer list.
> (append (list 1 2 3 4) (list 15 16 17)) (1 2 3 4 15 16 17)
As the example suggests, when applied to two lists, append
creates a new list which contains all elements of the first list followed
by all elements of the second list.
Scheme even lets you define your own procedures. Here's a simple procedure that gives you the second value in a list. You might want to enter it in the top half of the DrScheme window (the definitions pane).
(define (second lst) (car (cdr lst)))
Task:
(4 5 6)
.
In most cases, you'll fail.
(4 5 6)
.
Task:
Write a procedure,
(swap-first-two lst)
that swaps the first two elements of a list.
For example,
> (swap-first-two (list 1 2 3 4 5)) (2 1 3 4 5) > (swap-first-two (list 10 100)) (100 10)
Task:
Write a procedure,
(last lst)
that returns the last element on a list.
For example,
> (last (list 1 2 3 4)) 4
Scheme lets you write conditionals with the if
procedure.
It takes the form
(if test expression-to-use-if-test-holds expression-to-use-if-test-fails)
Scheme tests usually have the form
(test-operation val1 ... valn)
You can use the null?
operation to determine if a list
is empty.
> (null? (list 1 2 3)) #f > (null? null) #t > (null? (cdr (list 1))) #t
Task:
Write a procedure, only-one?
that determines if a list has
only one element. (Try not to use length
.)
For example,
> (only-one? (list 1)) #t > (only-one? null) #f > (only-one? (list 1 2 3 4 5)) #f
Scheme also provides five numeric tests,
<
,
<=
,
>
,
>=
, and
=
.
For example, to check if x is less than y, I might use
> (< 4 6) #t > (< 6 -2) #f > (define x 10) > (< x 4) #f
Here's a simple use of if
to find the smaller of two
numbers.
(define (smaller num1 num2) (if (< num1 num2) num1 num2))
Task: Test this procedure.
Task:
Update swap-first-two
so that it does not crash when
you try to swap the first two elements of a list with fewer than two
elements.
1. Review the work you've done so far and make a list of any questions or comments you may have.
2. Read the following document that discusses common sorting routines.
The problem of putting a collection of values in order appears so frequently that it's one of the most commonly studied problems in computer science. As with many problems, it has many different potential solutions. In this document, we consider a four of the standard algorithms that computer scientists have developed over the years.
As you read through these algorithms, you might want to try running them by hand to see how (and whether) they work.
Selection sort relies on your ability to easily identify the smallest value in the collection. To sort using selection sort, you repeatedly (1) identify the smallest value and (2) put it at the front of the values not yet identified. We can naturally phrase this algorithm recursively.
To sort a list of values using selection sort If the list is empty then You're done Otherwise Identify small, the smallest value in the list Remove small from the list Sort the modified list Shove small back on the front of the sorted list
Of course, for this algorithm to work, we need a way to find the smallest value in a list and to remove an element from a list. We'll consider finding the smallest value and leave removal as a thought problem.
It's fairly easy to find the smallest value in a list. Here are two techniques, one using repetition and one using recursion.
To find the smallest value in a list (version 1: repetition) Let guess be the first value in the list For each value, val, in the list If val < guess then Let guess be val guess is now the smallest value in the list
To find the smallest value in a list (version 2: recursion) If the list contains only one element then That element is the smallest value in the list Otherwise Let first be the first element in the list Let guess by the smallest value in the remainder of the list The smaller of guess and first is the smallest value in the list
Insertion sort works by building the sorted list from the bottom up. We repeatedly take the first element from the list to be sorted and put it in the correct place in the sorted list.
To sort a list, lst, using insertion sort Create an empty list sorted Insert each value in lst into the correct place in sorted To insert each value in lst into the correct place in sorted For each value, val in lst Insert val into the correct place in sorted
Note that we might also want to phrase a key step in insertion sort recursively.
To insert each value in lst into the correct place in sorted If lst is empty then You're done Otherwise Insert the first element in lst into sorted Insert the remaining elements in lst into the new sorted
In either case, we rely on a helper procedure. This time, the goal of the procedure is to insert a value into a sorted list.
To insert one value, val, at the proper position in a sorted list, slist Possibility One: slist is empty Make a list from val Possibility Two: val is less than or equal to the first value in slst Add val to the front of slst Possibility Three: val is greater than the first value in slst Keep the first value in slst and insert val into the rest of slst
Both selection sort and insertion sort build the sorted list one
element at a time. As you get more experience in computer science,
you'll quickly learn that there are sometimes better ways to break
up your work. The divide-and-conquer technique suggests
breaking your work in half (or other large parts). We can use that
technique along with recursion to come up with a sorting algorithm
called Merge Sort
.
To sort a list using merge sort Split the list into two equal halves (or as equal as possible) Sort each half Merge the two halves together
Once again, we rely on some helper algorithms. We need a way to split the list in half and a way to merge two sorted lists into one sorted list. We'll leave splitting as an exercise to the reader. Merging is somewhat more interesting.
To merge two sorted lists, slst1 and slst2 Possibility 1: slst1 is empty Just use slst2 Possibility 2: slst2 is empty Just use slst1 Possibility 3: The first element of slst1 is less than element of slst2 Use the first element of slst1 as the first element of the merged list. Build the rest of the merged list by merging the rest of slst1 with slst2. Possibility 4: The first element of slst1 is not less than the first element of slst2 Use the first element of slst2 as the first element of the merged list. Build the rest of the merged list by merging the slst1 with the rest of slst2.
Permutation Sortfirst forms a list containing all permutations of the initial list elements. It then methodically examines each of these permutations until it finds a permutation that also is sorted. Of course, generating all permutations can take a long time (even relatively small data sets may have many permutations), so this is not a particularly good sorting algorithm. However, for some problems, variations of this approach may be the only known solutions.
To sort a list using permutation sort Generate a list of all permutations of the original list Examine the list of permutations until a sorted permutation is found.
This time, we need several helper algorithms; various steps may be conceptually straightforward, but helper procedures allow us to organize various details. Here are two helper steps for finding a sorted permutation within a list of permutations.
To find a sorted permutation within a list of permutations If 1st permutation on the list is sorted then You're done Otherwise Find a sorted permutation among the remaining permutations
To determine if a list of numbers is sorted If list is null or contains 1 element then list is sorted Otherwise compare the first two items on the list If the first item is not less than the second then the list is not sorted Otherwise check that the elements after the first form a sorted list
(Note: Technically, the less-than test <
can be used
to check as many elements as you wish; the less-than test <
can be used to compare more than 2 elements. The above algorithm suggests
how this less-than test works.)
Generating permutations requires more work. An outline of one approach follows:
To generate all permutations of a listlst
If the list is null then the only permutation is the null list, so the list of all permutations is just the list containing the null list Otherwise generate a list of all permutations of(cdr lst)
insert(car lst)
successively into every position of every permutation of(cdr lst)
We've seen that it's possible to write different algorithms to solve the same problem. Given more than one algorithm that solves the same problem, what criteria might you use to select one algorithm rather than another?
Here is the selection sort algorithm as implemented in Scheme. (Note that I've eliminated many of the comments that would typically accompany the program code.)
(define (selection-sort lst) (if (null? lst) null (selection-sort-helper (smallest lst) lst))) ; The helper procedure handles the deletion of the smallest ; element and the repetition (define (selection-sort-helper smallest-value lst) (cons smallest-value (selection-sort (remove lst smallest-value)))) (define (smallest lst) (if (null? (cdr lst)) (car lst) (smaller (car lst) (smallest (cdr lst))))) (define (smaller val1 val2) (if (< val1 val2) val1 val2)) (define (remove lst val) (if (eq? val (car lst)) (cdr lst) (cons (car lst) (remove (cdr lst) val))))
You can use these procedures by adding the following line to your definitions pane and then clicking execute.
(load "/home/walker/public_html/sorting-workshop/Scheme/simple-selection-sort.ss")
1. Test the three key helper procedures, smallest
,
smaller
, and remove
. Here are some
examples to get you started.
> (smaller 1 2) > (smaller 2 1) > (smaller -5 2) > (smallest (list 1 2 3 4 5)) > (remove 4 (list 1 2 3 4 5))
2. Test selection-sort
on some simple lists you create yourself.
3. For more interesting testing, you need a way to create large lists. Here's a procedure I like to use.
(define (random-list n) (if (= n 0) null (cons (random 1000) (random-list (- n 1)))))
You can add this definition in your definitions window. I've also put a copy of this procedure in a file, which you can load with
(load "/home/walker/public_html/sorting-workshop/Scheme/random-list.ss")
Try creating random
lists of 5, 10, and 100 elements.
4. You can now combine pieces to build and sort larger lists as a way of testing selection sort. Here are some examples:
> (define sample (random-list 500)) > (length sample) 500 > sample (243 590 ... 866) > (selection-sort sample) (0 4 5 ... 999 999) > (selection-sort (reverse (selection-sort sample))) (0 4 5 ... 999 999)
a. Try some of your own examples.
b. Why do you think I included that last test?
5. As we discussed, it is often useful to be able to predict the running time of an algorithm. We'll use an experimental approach. In your CS courses, you will also see a more formal approach. We need to begin by gathering information on the running time of selection sort on various sizes of inputs. Here's a technique I've found useful. Enter the following in your definitions window
(load "/home/walker/public_html/sorting-workshop/Scheme/simple-selection-sort.ss") (load "/home/walker/public_html/sorting-workshop/Scheme/random-list.ss") (define sample (random-list 100)) (define result null) (time (set! result (selection-sort sample)))
When you click the Execute button, DrScheme will tell you how much time it spent sorting the list (use cpu time, it's the most accurate). We use the odd syntax because the default behavior of time is to print the result, and we really don't want the result printed.
Gather between ten and twenty data points for running time using different size lists.
6. Most of us can't see patterns in numeric data and instead rely on
visualization techniques. We'll use a program called Maple
to visualize these data.
a. You may have a Maple icon in your control panel. If so, click it. If not,
click on the picture of the terminal with the foot and then type
xmaple
.
b. In the untitled window, enter the following, which provides some sample data
with(plots): spoints := pointplot({[100,250],[200,1000]}, symbolsize=10, color=blue, view=[0..300,0..1100]): sfun := plot(x*x/1000, x=1..1000, color=green): display({spoints});
If all goes right, you should see points plotted at (100,250) and (200,4000). Replace those two values with the values you recorded earlier and re-plot.
c. Past experience tells us that the values for selection sort will be reasonable close to a quadratic curve so we want to draw a parabola that matches. Update the display line to read
display({spoints,sfun});
You should see a parabola that does not match your data, which is a good starting point.
d. Update the coefficient of x*x in the plot so that the curve matches your data fairly closely. You may have to try a variety of values before you find a good match.
The selection sort algorithm requires us to repeatedly remove a value from a list. Without worrying about how to remove an element from a list, carefully describe what it means to remove a value from a list. Think about it in the sense of writing a contract for the programmer: What guarantees do you require for the result of the algorithm?
Using the techniques you've developed for evaluating the running time of selection sort, compare the running times of selection sort, insertion sort, and merge sort. Which is fastest on large problems? On small? How much faster?
Note that you may have difficulty finding a quadratic function that matches the running time of merge sort. You might want to try other functions to see if you can get a better fit.
You can load an implementation of selection sort with
(load "/home/walker/public_html/sorting-workshop/Scheme/simple-selection-sort.ss")
That Scheme file provides the procedures
(selection-sort lst)
(smallest lst)
(remove val lst)
You can load an implementation of insertion sort with
(load "/home/walker/public_html/sorting-workshop/Scheme/simple-insertion-sort.ss")
That Scheme file provides the procedures
(insertion-sort list)
(insert val lst)
You can load an implementation of merge sort with
(load "/home/walker/public_html/sorting-workshop/Scheme/simple-merge-sort.ss")
That Scheme file provides the procedures
(merge-sort lst)
(merge slst1 slst2)
(split lst)
The permutation sort first generates all permutations of a list. If the list has N items, the number of permutations is N factorial or N * (N-1) * (N-2) * ... * 2 * 1. These permutations then must be tested in sequence until a sorted permutation is found. To test one permutation, successive elements must be checked to determine if they are in ascending sequence. This requires about N tests. Altogether, in the worst case, all permutations must be tested, and this requires about N * (N factorial) tests, so we might expect the performance of the permutation sort to be roughly proportional to N * (N factorial).
1. Write out the product of terms needed to computer N * (N factorial) for N being 4, 5, 6, 7, 8, and 9. Can you find a rule that allows you to compute one of these values from the previous?
2. You can load an implementation of permutation sort with
(load "/home/walker/public_html/sorting-workshop/Scheme/simple-permutation-sort.ss")
That Scheme file provides several procedures, including
permutation-sort
generate-permutations
find-sorted
3. Check that generate-permutations
does indeed produce
all permutations of a list. For example, you might try:
> (generate-permutations (list 3 1 2)
Similarly, you might check that permutation-sort
does
select the correct, sorted permutation:
> (permutation-sort (list 3 1 2))
4. Time the permutation sort with lists of 6, 7, and 8 elements. For example, you might try
> (time (permutation-sort (list 6 5 4 3 2 1))) > (time (permutation-sort (list 7 6 5 4 3 2 1))) > (time (permutation-sort (list 8 7 6 5 4 3 2 1)))
How well does the time from one of these tests allow you to predict the time required for the next?
5. Use your experience with the past experiments to estimate how long the permutation sort might take for a list of nine elements. Then compare your result with the actual time required for
> (time (permutation-sort (list 9 8 7 6 5 4 3 2 1)))
Here's a Scheme procedure that finds the smallest value in a list of numbers.
(define (smallest lst) ; If the list has only one element (if (null? (cdr lst)) ; That one element is the smallest (car lst) ; Otherwise, use the smaller of the first element and ; the smallest remaining element (smaller (car lst) (smallest (cdr lst))))) (define (smaller val1 val2) (if (< val1 val2) val1 val2))
1. Test those procedures to make sure that they work as you'd expect.
2. Write and test a Scheme procedure that finds the largest value in a list of procedures.
As you may have observed, smallest
and largest
are surprisingly similar, differing only in their use of a procedure to
select between two values and the procedure used on the rest of the list.
We can perhaps phrase the procedure more generally as
(define (best lst better) ; If the list has only one element (if (null (cdr? lst)) ; Then that value is the best (car lst) ; Otherwise, use the better of the first element and the ; best remaining element (better (car lst) (best (cdr lst) better))))
3. Test best
using a list of numbers and smaller as parameters.
4. Test best
using a list of numbers and larger as parameters.
5. Write a closer-to-100
procedure that, given two values,
returns the one that is closer to 100.
6. Using best
and closer-to-100
, find the
value in a list of numbers that is closest to 100.
Tuesday, 20 August 2002 by Samuel A. Rebelsky
Wednesday, 21 August 2002 by Samuel A. Rebelsky
Short Versionintroductions to the longer sections.
Monday, 18 August 2003 by Samuel A. Rebelsky
Saturday-Sunday, 14-15 August 2004 by Henry M. Walker
Monday-Tuesday, 16-17 August 2004 by Henry M. Walker
http://www.cs.grinnell.edu/~walker/sorting-workshop/index.shtml
created by Samuel A. Rebelsky in late summer, 2002 revised by Samuel A. Rebelsky in August 2003 revised by Henry M. Walker for 2004 last revised August 17, 2004 | |
For more information, please contact Henry M. Walker at (walker@cs.grinnell.edu) |