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| author | elioat <elioat@tilde.institute> | 2023-08-23 07:52:19 -0400 |
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diff --git a/js/games/nluqo.github.io/~bh/ssch24/spread.html b/js/games/nluqo.github.io/~bh/ssch24/spread.html new file mode 100644 index 0000000..4587d70 --- /dev/null +++ b/js/games/nluqo.github.io/~bh/ssch24/spread.html @@ -0,0 +1,592 @@ +<P> + +<P> +<CENTER><IMG SRC="../ss-pics/spread.jpg" ALT="figure: spread"></CENTER><A NAME="excel"></A><P><CENTER>Spreadsheet display from Microsoft +Excel +</CENTER><P> + + +<HTML> +<HEAD> +<TITLE>Simply Scheme: Introducing Computer Science ch 24: Example: A Spreadsheet Program</TITLE> +</HEAD> +<BODY> +<HR> +<CITE>Simply Scheme:</CITE> +<CITE>Introducing Computer Science</CITE> 2/e Copyright (C) 1999 MIT +<H2>Chapter 24</H2> +<H1>Example: A Spreadsheet Program</H1> + +<TABLE width="100%"><TR><TD> +<IMG SRC="../simply.jpg" ALT="cover photo"> +<TD><TABLE> +<TR><TD align="right"><CITE><A HREF="http://www.cs.berkeley.edu/~bh/">Brian +Harvey</A><BR>University of California, Berkeley</CITE> +<TR><TD align="right"><CITE><A HREF="http://ccrma.stanford.edu/~matt">Matthew +Wright</A><BR>University of California, Santa Barbara</CITE> +<TR><TD align="right"><BR> +<TR><TD align="right"><A HREF="../pdf/ssch24.pdf">Download PDF version</A> +<TR><TD align="right"><A HREF="../ss-toc2.html">Back to Table of Contents</A> +<TR><TD align="right"><A HREF="../ssch23/vectors.html"><STRONG>BACK</STRONG></A> +chapter thread <A HREF="../ssch25/spread-implement.html"><STRONG>NEXT</STRONG></A> +<TR><TD align="right"><A HREF="http://mitpress.mit.edu/0262082810">MIT +Press web page for <CITE>Simply Scheme</CITE></A> +</TABLE></TABLE> + +<HR> + + +<P>Until now, you may have felt that the programs you've been writing in Scheme +don't act like other computer programs you've used. In this chapter and the +next, we're going to tie together almost everything you've learned so far to +write a <EM>spreadsheet</EM> program, just like the ones accountants use. + +<P>This chapter describes the operation of the spreadsheet program, as a user +manual would. The next chapter explains how the program is implemented in +Scheme. + +<P>You can load our program into Scheme by typing + +<P><PRE>(load "spread.scm") +</PRE> + +<P>To start the program, invoke the procedure <CODE>spreadsheet</CODE> with no +arguments; to quit the spreadsheet program, type <CODE>exit</CODE>. + +<P> +<P>A spreadsheet is a program that displays information in two dimensions on +the screen. It can also compute some of the information automatically. Below +is an example of a display from our spreadsheet program. The +display is a rectangle of information with six columns and 20 rows. The +intersection of a row with a column is called a <EM>cell;</EM> for +example, the cell <CODE>c4</CODE> contains the number 6500. The column letters +(<CODE>a</CODE> through <CODE>f</CODE>) and row numbers are provided by the spreadsheet +program, as is the information on the bottom few lines, which we'll talk +about later. (The <CODE>??</CODE> at the very bottom is the spreadsheet prompt; +you type commands on that line.) We typed most of the entries in the cells, +using commands such as + +<P><PRE>(put 6500 c4) +</PRE> + +<P> +<PRE> -----a---- -----b---- -----c---- -----d---- -----e---- -----f---- + 1 NAME NUMBER PRICE GROSS DISCOUNT NET + 2 Widget 40.00 1.27 50.80 0.00 50.80 + 3 Thingo 203.00 14.95 > 3034.85< 15.00 2579.62 + 4 Computer 1.00 6500.00 6500.00 8.00 5980.00 + 5 Yacht 300.00 200000.00 60000000.+ 0.00 60000000.+ + 6 + 7 + 8 + 9 TOTALS 60009585.+ 60008610.+ +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +d3: 3034.85 +(* b3 c3) +?? +</PRE> + +<P>What's most useful about a spreadsheet is its ability to compute some of the +cell values itself. For example, every number in column <CODE>d</CODE> is the +product of the numbers in columns <CODE>b</CODE> and <CODE>c</CODE> of the same row. +Instead of putting a particular number in a particular cell, we put a <EM>formula</EM> in all the cells of the column at once.<A NAME="text1" HREF="spread.html#ft1">[1]</A> This implies that when we change the value of one cell, other +cells will be updated automatically. For example, if we put the number 5 +into cell <CODE>b4</CODE>, the spreadsheet will look like this: + +<P> +<PRE> -----a---- -----b---- -----c---- -----d---- -----e---- -----f---- + 1 NAME NUMBER PRICE GROSS DISCOUNT NET + 2 Widget 40.00 1.27 50.80 0.00 50.80 + 3 Thingo 203.00 14.95 > 3034.85< 15.00 2579.62 + 4 Computer 5.00 6500.00 32500.00 8.00 29900.00 + 5 Yacht 300.00 200000.00 60000000.+ 0.00 60000000.+ + 6 + 7 + 8 + 9 TOTALS 60035585.+ 60032530.+ +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +d3: 3034.85 +(* b3 c3) +?? +</PRE> + +<P>In addition to cell <CODE>b4</CODE>, the spreadsheet program has changed +the values in <CODE>d4</CODE>, <CODE>f4</CODE>, <CODE>d9</CODE>, and <CODE>f9</CODE>. + +<P>One detail we haven't mentioned so far is that at any moment there is one +<EM>selected</EM> cell. Right now cell <CODE>d3</CODE> is selected. You can tell +that because of the arrowheads surrounding it in the display, like this: + +<P><PRE>> 3034.85< +</PRE> + +<P>Also, the lines + +<P><PRE>d3: 3034.85 +(* b3 c3) +</PRE> + +<P>at the bottom of the screen mean that cell <CODE>d3</CODE> is selected, +its value is <CODE>3034.85</CODE>, and its formula is <CODE>(* b3 c3)</CODE>. + +<P><H2>Limitations of Our Spreadsheet</H2> + +<P>In commercial spreadsheet programs, you generally select a cell with arrow +keys or by clicking on it with a mouse. The highlighted cell is typically +displayed in inverse video, and there might be thin lines drawn in a grid on +the screen to separate the cells. + +<P>Our program leaves all this out for two reasons. First, details like this +don't add very much to what you learn from studying the program, but they +take a disproportionate effort to get exactly right. Second, the facilities +needed in Scheme to control screen graphics are specific to each model of +computer. We couldn't write a single program that would work in all versions +of Scheme. (A program that works in all versions is called <EM>portable.</EM>) + +<P>Similarly, our program prints an entire new screenful of information after +every command. A better program would change only the parts of the screen +for which the corresponding values have changed. But there is no uniform +way to ask Scheme to print at a particular position on the screen; some +versions of Scheme can do that, but not using standard procedures. + +<P>Also, of course, if you spend $500 on a spreadsheet program, it will have +hundreds of bells and whistles, such as graphing, printing your spreadsheet +on paper, changing the widths of columns, and undoing the last command you +typed. But each of those is a straightforward extension. We didn't write +them all because there are only two of us and we're not getting paid +enough! You'll add some features as exercises. + +<P><H2>Spreadsheet Commands</H2> + +<P>When you begin the spreadsheet program, you will see an empty grid and a +prompt at the bottom of the screen. + +<P>Most spreadsheet programs are controlled using single-keystroke commands (or +equivalent mouse clicks). That's another thing we can't do entirely +portably in Scheme. We've tried to compromise by including one-letter +command names in our program, but you do have to type the <CODE>return</CODE> or +<CODE>enter</CODE> key after each command. The single-letter commands are for +simple operations, such as selecting a cell one position up, down, left, or +right from the previously selected cell. + +<P>For more complicated commands, such as entering values, a longer notation +is obviously required. In our program we use a notation that looks very +much like that of a Scheme program: A command consists of a name and some +arguments, all enclosed in parentheses. However, the spreadsheet commands +are <EM>not</EM> Scheme expressions. In particular, the arguments are not +evaluated as they would be in Scheme. For example, earlier we said + +<P><PRE>(put 6500 c4) +</PRE> + +<P>If this were a Scheme expression, we would have had to quote the +second argument: + +<P><PRE>(put 6500 'c4) ;; wrong! +</PRE> + +<P><H2>Moving the Selection</H2> + +<P>There are four one-letter commands to move to a new selected cell: + +<P> +<P><TABLE><TR><TH>Command Name <TH>Meaning +<TR><TR><TD><CODE>f</CODE><TD>move Forward (right) +<TR><TD><CODE>b</CODE><TD>move Back (left) +<TR><TD><CODE>n</CODE><TD>move to Next line (down) +<TR><TD><CODE>p</CODE><TD>move to Previous line (up) +</TABLE> + +(These command names are taken from EMACS, the industry +standard text editor.) + +<P>If you want to move one step, you can just type the letter <CODE>f</CODE>, <CODE>b</CODE>, +<CODE>n</CODE>, or <CODE>p</CODE> on a line by itself. If you want to move farther, you +can invoke the same commands in Scheme notation, with the distance to move +as an argument: + +<P><PRE>?? (f 4) +</PRE> + +<P>Another way to move the selection is to choose a particular cell by name. +The command for this is called <CODE>select</CODE>: + +<P><PRE>(select e12) +</PRE> + +<P>The spreadsheet grid includes columns <CODE>a</CODE> through <CODE>z</CODE> and rows <CODE>1</CODE> through <CODE>30</CODE>. Not all of it can fit on the screen at once. If you +select a cell that's not shown on the screen, then the program will shift +the entire screen display so that the rows and columns shown will include +the newly selected cell. + +<P><H2>Putting Values in Cells</H2> + +<P>As we've already seen, the <CODE>put</CODE> command is used to put a value in a +particular cell. It can be used with either one or two arguments. The +first (or only) argument is a value. If there is a second argument, it is +the (unquoted) name of the desired cell. If not, the currently selected +cell will be used. + +<P>A value can be a number or a quoted word. (As in Scheme programming, most +words can be quoted using the single-quote notation <CODE>'word</CODE>, but words +that include spaces, mixed-case letters, or some punctuation characters must +be quoted using the double-quote string notation <CODE>"Widget"</CODE>.) +However, non-numeric words are used only as labels; they can't provide +values for formulas that compute values in other cells. + +<P>The program displays numbers differently from labels. If the value in a +cell is a number, it is displayed at the right edge of its cell, and it is +shown with two digits following the decimal point. (Look again at the +screen samples earlier in this chapter.) If the value is a non-numeric word, +it is displayed at the left edge of its cell. + +<P>If the value is too wide to fit in the cell (that is, more than ten +characters wide), then the program prints the first nine characters followed +by a plus sign (<CODE>+</CODE>) to indicate that there is more information than is +visible. (If you want to see the full value in such a cell, select it and +look at the bottom of the screen.) + +<P>To erase the value from a cell, you can put an empty list <CODE>()</CODE> in it. +With this one exception, lists are not allowed as cell values. + +<P>It's possible to put a value in an entire row or column, instead of just one +cell. To do this, use the row number or the column letter as the second +argument to <CODE>put</CODE>. Here's an example: + +<P><PRE>(put 'peter d) +</PRE> + +<P>This command will put the word <CODE>peter</CODE> into all the cells in +column <CODE>d</CODE>. (Remember that not all the cells are visible at once, but +even the invisible ones are affected. Cells <CODE>d1</CODE> through <CODE>d30</CODE> are +given values by this command.) + +<P>What happens if you ask to fill an entire row or column at once, but some of +the cells already have values? In this case, only the vacant cells will be +affected. The only exception is that if the <EM>value</EM> you are using is +the empty list, indicating that you want to erase old values, then the +entire row or column is affected. (So if you put a formula in an entire row +or column and then change your mind, you must erase the old one before you +can install a new one.) + +<P><H2>Formulas</H2> + +<P>We mentioned earlier that the value of one cell can be made to depend on the +values of other cells. This, too, is done using the <CODE>put</CODE> command. The +difference is that instead of putting a constant value into a cell, you can +put a formula in the cell. Here's an example: + +<P><PRE>(put (+ b3 c5) d6) +</PRE> + +<P>This command says that the value in cell <CODE>d6</CODE> should be the +sum of the values in <CODE>b3</CODE> and <CODE>c5</CODE>. The command may or may not have +any immediately visible effect; it depends on whether those two cells +already have values. If so, a value will immediately appear in <CODE>d6</CODE>; +if not, nothing happens until you put values into <CODE>b3</CODE> and <CODE>c5</CODE>. + +<P>If you erase the value in a cell, then any cells that depend on it are +also erased. For example, if you erase the value in <CODE>b3</CODE>, then the +value in <CODE>d6</CODE> will disappear also. + +<P>So far we've seen only one example of a formula; it asks for the sum of +two cells. Formulas, like Scheme expressions, can include invocations of +functions with sub-formulas as the arguments. + +<P><PRE>(put (* d6 (+ b4 92)) a3) +</PRE> + +<P>The "atomic" formulas are constants (numbers or quoted words) +and cell references (such as <CODE>b4</CODE> in our example). + +<P>Not every Scheme function can be used in a formula. Most important, there +is no <CODE>lambda</CODE> in the spreadsheet language, so you can't invent your own +functions. Also, since formulas can be based only on numbers, only the +numeric functions can be used. + +<P>Although we've presented the idea of putting a formula in a cell separately +from the idea of putting a value in a cell, a value is really just a +particularly simple formula. The program makes no distinction internally +between these two cases. (Since a constant formula doesn't depend on any +other cells, its value is always displayed right away.) + +<P>We mentioned that a value can be <CODE>put</CODE> into an entire row or column at +once. The same is true of formulas in general. But this capability gives +rise to a slight complication. The typical situation is that each cell in +the row or column should be computed using the same algorithm, but based on +different values. For example, in the spreadsheet at the beginning of the +chapter, every cell in column <CODE>d</CODE> is the product of a cell in column +<CODE>b</CODE> and a cell in column <CODE>c</CODE>, but not the <EM>same</EM> cell for +every row. If we used a formula like + +<P><PRE>(put (* b2 c2) d) +</PRE> + +<P>then every cell in column <CODE>d</CODE> would have the value <CODE>50.80</CODE>. +Instead we want the equivalent of + +<P><PRE>(put (* b2 c2) d2) +(put (* b3 c3) d3) +(put (* b4 c4) d4) +</PRE> + +<P>and so on. The spreadsheet program meets this need by providing +a notation for cells that indicates position relative to the cell being +computed, rather than by name. In our case we could say + +<P><PRE>(put (* (cell b) (cell c)) d) +</PRE> + +<P><CODE>Cell</CODE> can take one or two arguments. In this example we've used the +one-argument version. The argument must be either a letter, to indicate a +column, or a number between 1 and 30, to indicate a row. Whichever +dimension (row or column) is <EM>not</EM> specified by the argument will be +the same as that of the cell being computed. So, for example, if we are +computing a value for cell <CODE>d5</CODE>, then <CODE>(cell b)</CODE> refers to cell <CODE>b5</CODE>, but <CODE>(cell 12)</CODE> would refer to cell <CODE>d12</CODE>. + +<P>The one-argument form of <CODE>cell</CODE> is adequate for many situations, but not +if you want a cell to depend on one that's both in a different row and in a +different column. For example, suppose you wanted to compute the change +of a number across various columns, like this: + +<P> +<P><PRE> -----a---- -----b---- -----c---- -----d---- -----e---- -----f---- + 1 MONTH January February March April May + 2 PRICE 70.00 74.00 79.00 76.50 81.00 + 3 CHANGE 4.00 5.00 -2.50 4.50 +</PRE> + +<P> +<P>The value of cell <CODE>d3</CODE>, for example, is a function of the +values of cells <CODE>d2</CODE> and <CODE>c2</CODE>. + +<P>To create this spreadsheet, we said + +<P><PRE>(put (- (cell 2) (cell <1 2)) 3) +</PRE> + +<P>The first appearance of <CODE>cell</CODE> asks for the value of the cell +immediately above the one being calculated. (That is, it asks for the cell +in row 2 of the same column.) But the one-argument notation doesn't allow +us to ask for the cell above and to the left. + +<P>In the two-argument version, the first argument determines the column and +the second determines the row. Each argument can take any of several +forms. It can be a letter (for the column) or number (for the row), to +indicate a specific column or row. It can be an asterisk (<CODE>*</CODE>) to +indicate the same column or row as the cell being calculated. Finally, +either argument can take the form <CODE><3</CODE> to indicate a cell three before +the one being calculated (above or to the left, depending on whether this is +the row or column argument) or <CODE>>5</CODE> to indicate a cell five after this +one (below or to the right). + +<P>So any of the following formulas would have let us calculate the change in +this example: + +<P><PRE>(put (- (cell 2) (cell <1 2)) 3) + +(put (- (cell 2) (cell <1 <1)) 3) + +(put (- (cell * 2) (cell <1 2)) 3) + +(put (- (cell * <1) (cell <1 2)) 3) + +(put (- (cell <0 <1) (cell <1 <1)) 3) +</PRE> + +<P>When a formula is put into every cell in a particular row or column, it may +not be immediately computable for all those cells. The value for each cell +will depend on the values of the cells to which the formula refers, and some +of those cells may not have values. (If a cell has a non-numeric value, +that's the same as not having a value at all for this purpose.) New values +are computed only for those cells for which the formula is computable. For +example, cell <CODE>b3</CODE> in the monthly change display has the formula <CODE>(- b2 a2)</CODE>, but the value of cell <CODE>a2</CODE> is the label <CODE>PRICE</CODE>, so no +value is computed for <CODE>b3</CODE>. + +<P><H2>Displaying Formula Values</H2> + +<P>Formulas can be used in two ways. We've seen that a formula can be +associated with a cell, so that changes to one cell can automatically +recompute the value of another. You can also type a formula directly +to the spreadsheet prompt, in which case the value of the formula will +be shown at the bottom of the screen. In a formula used in this way, +cell references relative to "the cell being computed" refer instead to the +selected cell. + +<P><H2>Loading Spreadsheet Commands from a File</H2> + +<P>Sometimes you use a series of several spreadsheet commands to set up some +computation. For example, we had to use several commands such as + +<P><PRE>(put "Thingo" a3) +</PRE> + +<P>to set up the sample spreadsheet displays at the beginning of this +chapter. + +<P>If you want to avoid retyping such a series of commands, you can put the +commands in a file using a text editor. Then, in the spreadsheet program, +use the command + +<P><PRE>(load "<EM>filename</EM>") +</PRE> + +<P>This looks just like Scheme's <CODE>load</CODE> (on purpose), but it's +not the same thing; the file that it loads must contain spreadsheet +commands, not Scheme expressions. It will list the commands from the file +as it carries them out. + +<P><H2>Application Programs and Abstraction</H2> + +<P>We've talked throughout this book about the importance of abstraction, +the act of giving a name to some process or structure. Writing an +application program can be seen as the ultimate in abstraction. The user of +the program is encouraged to think in a vocabulary that reflects the tasks +for which the program is used, rather than the steps by which the program +does its work. Some of these names are explicitly used to control the +program, either by typing commands or by selecting named choices from a +menu. Our spreadsheet program, for example, uses the name <CODE>put</CODE> for the +task of putting a formula into a cell. The algorithm used by the <CODE>put</CODE> +command is quite complicated, but the user's picture of the command is +simple. Other names are not explicitly used to control the program; +instead, they give the user a <EM>metaphor</EM> with which to think +about the work of the program. For example, in describing the operation of +our spreadsheet program, we've talked about rows, columns, cells, and +formulas. Introducing this vocabulary in our program <EM>documentation</EM> +is just as much an abstraction as introducing new procedures in the program +itself. + +<P>In the past we've used procedural abstraction to achieve <EM>generalization</EM> of an algorithm, moving from specific instances to a +more universal capability, especially when we implemented the higher-order +functions. If you've used application programs, though, you've probably +noticed that in a different sense the abstraction in the program <EM>loses</EM> generality. For example, a formula in our spreadsheet program can +operate only on data that are in cells. The same formula, expressed as a +Scheme procedure, can get its arguments from anywhere: from reading a file, +from the keyboard, from a global variable, or from the result of invoking +some other procedure. + +<P>An application program doesn't have to be less general than a programming +language. The best application programs are <EM>extensible.</EM> Broadly +speaking, this means that the programmer has made it possible for the user +to add capabilities to the program, or modify existing capabilities. This +broad idea of extensibility can take many forms in practice. Some +kinds of extensibility are more flexible than others; some are easier to use +than others. Here are a few examples: + +<P>Commercial spreadsheet programs have grown in extensibility. We mentioned +that our spreadsheet program allows the user to express a function only as a +formula attached to a cell. Modern commercial programs allow the user to +define a procedure, similar in spirit to a Scheme procedure, that can then +be used in formulas. + +<P>Our program, on the other hand, is extensible in a different sense: We +provide the Scheme programs that implement the spreadsheet in a form that +users can read and modify. In the next chapter, in fact, you'll be asked to +extend our spreadsheet. Most commercial programs are provided in a form +that computers can read, but people can't. The provision of human-readable +programs is an extremely flexible form of extensibility, but not necessarily +an easy one, since you have to know how to program to take advantage of it. + +<P>We have written much of this book using a home computer as a remote terminal +to a larger computer at work. The telecommunication program we're using, +called <EM>Zterm,</EM> has dozens of options. We can set frivolous things +like the screen color and the sound used to attract our attention; we can +set serious things like the file transfer protocol used to "download" a +chapter for printing at home. The program has a directory of telephone +numbers for different computers we use, and we can set the technical details +of the connection separately for each number. It's very easy to customize +the program in all of these ways, because it uses a mouse-driven graphical +interface. But the interface is inflexible. For example, although the +screen can display thousands of colors, only eight are available in Zterm. +More important, if we think of an entirely new feature that would require +a modification to the program, there's no way we can do it. This program's +extensibility is the opposite of that in our spreadsheet: It's very easy +to use, but limited in flexibility. + +<P>We started by saying that the abstraction in an application program runs the +risk of limiting the program's generality, but that this risk can be +countered by paying attention to the goal of <EM>extensibility.</EM> The ultimate form of extensibility is to +provide the full capabilities of a programming language to the user of the +application program. This can be done by inventing a special-purpose +language for one particular application, such as the <EM>Hypertalk</EM> +language that's used only in the <EM>Hypercard</EM> application program. Alternatively, the application +programmer can take advantage of an existing general-purpose language by +making it available within the program. You'll see an example soon, in the +database project, which the user controls by typing expressions at a Scheme +prompt. The EMACS text editor is a better-known example that includes a +Lisp interpreter. + +<P><H2>Exercises</H2> + +<P> + +<P><P>For each of the following exercises, the information to hand in is +the sequence of spreadsheet commands you used to carry out the assignment. +You will find the <CODE>load</CODE> command helpful in these exercises. + +<P><B>24.1</B> Set up a spreadsheet to keep track of the grades in a course. Each column +should be an assignment; each row should be a student. The last column +should add the grade points from the individual assignments. You can make +predictions about your grades on future assignments +and see what overall numeric grade each prediction gives you. + + +<P> +<B>24.2</B> Make a table of tax and tip amounts for a range of possible costs at a +restaurant. Column <CODE>a</CODE> should contain the pre-tax amounts, starting at +50 cents and increasing by 50 cents per row. (Do this without entering each +row separately!) Column <CODE>b</CODE> should compute the tax, based on your +state's tax rate. Column <CODE>c</CODE> should compute the 15% tip. Column <CODE>d</CODE> should add columns <CODE>a</CODE> through <CODE>c</CODE> to get the total cost of the +meal. Column <CODE>e</CODE> should contain the same total cost, rounded up to the +next whole dollar amount. + + +<P> +<B>24.3</B> Make a spreadsheet containing the values from Pascal's triangle: Each +element should be the sum of the number immediately above it and the +number immediately to its left, except that all of column <CODE>a</CODE> should have +the value 1, and all of row 1 should have the value 1. + + +<P> + +<HR> +<A NAME="ft1" HREF="spread.html#text1">[1]</A> We did it by +saying + +<P><PRE>(put (* (cell b) (cell c)) d) +</PRE> + +<P>but we aren't going to talk about the details of formulas for a +while longer.<P> +<P><A HREF="../ss-toc2.html">(back to Table of Contents)</A><P> +<A HREF="../ssch23/vectors.html"><STRONG>BACK</STRONG></A> +chapter thread <A HREF="../ssch25/spread-implement.html"><STRONG>NEXT</STRONG></A> + +<P> +<ADDRESS> +<A HREF="../index.html">Brian Harvey</A>, +<CODE>bh@cs.berkeley.edu</CODE> +</ADDRESS> +</BODY> +</HTML> |