% CHANGES TO VOLUME 1 OF THE ART OF COMPUTER PROGRAMMING % % Copyright (C) 2011, 2012, ..., 2018, 2019 by Donald E. Knuth % This file may be freely copied provided that no modifications are made. % All other rights are reserved. % % Three levels of changes to the books are distinguished here: % % "\bugonpage" introduces the correction of an error; % "\amendpage" introduces new material for future editions; % "\improvepage" introduces ameliorations of lesser importance. % % (Changes introduced by \improvepage do not appear in the hardcopy listing.) % % Also, "\planforpage" introduces some of the author's half-baked intentions. % % NOTE: TO PUT THE INDEX ON A SEPARATE PAGE, RUN THIS WITH THE COMMAND LINE % tex "\let\indexeject+ \input err1" \newif\ifall % \alltrue means show the trivial items too \relax % hook \def\vertical{|} \def\inref#1 #{\expandafter\def\csname\vertical#1\endcsname} \catcode`|=\active \let|\inref \input \jobname.ref \catcode`|=12 \input taocpmac % use the format for TAOCP, with modifications below \def\becomes{\ifmmode\ \hbox\fi{\manfnt y}\ } % wiggly arrow indicates a change \def\bugonpage#1.#2 #3 (#4) { \medbreak\defaultpointsize \line{\kern-5pt\llap{\manfnt x}% print a black triangle in left margin {\bf Page #2}\enspace #3 \leaders\hrule\hfill\ \eightrm\date#4.} \nobreak\smallskip\iftrue\noindent} \def\amendpage#1.#2 #3 (#4) { \medbreak\defaultpointsize \line{\kern-5pt{\bf Page #2}\enspace #3 \leaders\hrule\hfill\ \eightrm\date#4.} \nobreak\smallskip\iftrue\noindent} \def\improvepage#1.#2 #3 (#4) {\ifall \medbreak\ninepoint \line{\kern-6pt{\sl Page #2\enspace #3\/} \leaders\hrule\hfill\ \eightrm\date#4.} \nobreak\smallskip\noindent} \def\planforpage#1.#2 #3 (#4) { \medbreak\defaultpointsize \line{\kern-5pt{\bf Page #2}\enspace #3 \leaders\hbox to 5pt{\hss.\hss}\hfill\ \eightrm\date#4.} \nobreak\smallskip\begingroup\let\endchange=\endgroup\sl\noindent} \let\endchange=\fi \def\nl{\par\noindent} \def\nlh{\par\noindent\hangit} \def\hangit{\hangindent2em} \def\cutpar{{\parfillskip=0pt\par}} \def\date#1.#2.#3.{% convert "yy.mm.dd." to "dd Mon 19yy" #3 \ifcase#2\or Jan\or Feb\or Mar\or Apr\or May\or Jun\or Jul\or Aug\or Sep\or Oct\or Nov\or Dec\fi \ \ifnum #1<97 \hundred#1\else19#1\fi} \def\hundred{20} % the "century" for dates before '97 \def\ex #1. [#2]{\ninepoint \textindent{\bf#1.}[{\it#2\/}]\kern6pt} \def\EX #1. [#2]{\ninepoint \textindent{\llap{\manfnt x}\bf#1.}[{\it#2\/}]\kern6pt} \def\foottext#1{\medskip \hrule height\ruleht width5pc \kern-\ruleht \kern3pt \eightpoint \smallskip\textindent{#1}} \def\volheadline#1{\line{\cleaders\hbox{\raise3pt\hbox{\manfnt\char'36}}\hfill \titlefont\ #1\ \cleaders\hbox{\raise3pt\hbox{\manfnt\char'36}}\hfill}} \def\refin#1 {\let|\inref \input #1.ref \let|\crossref} \let\defaultpointsize=\tenpoint %%%%%%%%%%%%%% opening remarks %%%%%%%%%%%%%%%%%%%%%%%% \def\lhead{INTRODUCTION} \let\rhead=\lhead \titlepage \volheadline{THE ART OF COMPUTER PROGRAMMING} \bigskip \volheadline{ERRATA TO VOLUME 1 (after 2010)} \bigskip \noindent This document is a transcript of the notes that I have been making in my personal copy of {\sl The Art of Computer Programming}, Volume~1 (third edition, 27th printing), since it was first printed in 2011. Previous errata are recorded in another file `{\tt all1-pre.ps}'. \ifall Four levels of updates\dash---``errors,'' ``amendments,'' ``plans,'' and ``improvements''\dash---appear, indicated by four \else Three levels of updates\dash---``errors,'' ``amendments,'' and ``plans''\dash---appear, indicated by three \fi different typographic conventions: \begingroup\def\hundred{17} \bugonpage 0.666 line 1 (76.07.04) Technical or typographical errors (aka bugs) are the most critical items, so they are flagged with a `\thinspace{\manfnt x}\thinspace' preceding the page number. The date on which I first was told about the bug is shown; this is the effective date on which I paid the finder's fee. The necessary corrections are indicated in a straightforward way. If,~for example, the book says `$n$' where it should have said `$n+1$', the change is shown thus: \smallskip $n$ \becomes $n+1$ \endchange \amendpage 0.666 line 2 (89.07.14) Amendments to the text appear in the same format as bugs, but without the~`\thinspace{\manfnt x}\thinspace'. These are things I wish I had known about or thought of when I wrote the original text, so I added them later. The date is the date I drafted the new text. \endchange \def\hundred{19} \planforpage 0.666 line 3 (17.11.20) Plans for the future represent a third kind of item. In such notes I~sketched my intentions about things that I wasn't ready to flesh out further when I~wrote them down. You can identify these items because they're written in slanted type, and preceded by a bunch of dots `\hbox to 6em{\leaders\hbox to 5pt{\hss.\hss}\hfill}' leading to the date on which I recorded the plan in my files. \endchange \improvepage 0.666 line 4 (38.01.10) The fourth and final category\dash---indicated by page and line number in smaller, slanted type\dash---consists of minor corrections or improvements that most readers don't want to know about, because they are so trivial. You wouldn't even be seeing these items if you hadn't specifically chosen to print the complete errata list in all its gory details. Are you sure you wanted to do that? \endchange \endgroup \ifall\else\medskip\ninepoint My personal file of updates also includes a fourth category of items, not shown in this list. They are miscellaneous minor corrections or improvements that most readers don't want to know about, because they are so trivial. If you really want to see all of the gory details, you can download the full list from Internet webpage $$\.{http://www-cs-faculty.stanford.edu/\char`\~knuth/taocp.html}$$ by selecting the ``long form'' of the errata. \fi \medskip \tenpoint My shelves at home are bursting with preprints and reprints of significant research results that I want to digest and summarize, where appropriate, in the ultimate edition of Volume~1. I didn't do that in the third edition because I would surely have to do it over again later: New results continue to pour forth at a great rate, and I will have time to rewrite that volume only~once. Volumes 4 and~5 need to be finished first. So I've put most of my effort so far into writing up those parts of the total picture that seem to have converged to their near-final form. It follows, somewhat paradoxically, that the updates in this document are most current in the areas where there has been least activity. On the other hand I do believe that the changes listed here bring Volume~1 completely up to date in two respects: (1)~All of the research problems in the previous edition\dash---i.e., all exercises that were rated 46 and above\dash---have received new ratings of 45 or less whenever I learned of a solution; and in such cases, the answer now refers to that solution. (2)~All of the historical information about pioneering developments has been amended whenever new details have come to my attention. \beginconstruction The ultimate, glorious, 100\% perfect editions of Volumes 1--4A are works in progress. Please let me know of any improvements that you think I ought to make. Send your comments either by snail mail to D.~E. Knuth, Computer Science, Gates Building 4B, Stanford University, Stanford CA~94305-9045, or by email to {\tt taocp{\char`\@}cs.stanford.edu}. (Use email for book suggestions only, please\dash---all other correspondence is returned unread to the sender, or discarded, because I have no time to read ordinary email.) Although I'm working full time on Volume~4B these days, I~will try to reply to all such messages within a year of receipt. Current news about {\sl The Art of Computer Programming\/} is posted on $$\.{http://www-cs-faculty.stanford.edu/\char`~knuth/taocp.html}$$ and updated regularly. \par\endconstruction \rightline{\dash---Don Knuth, January 2011} \bigskip \bigskip {\quoteformat What happened? The subject took the bit in its teeth and ran away with it, that's what happened. I know now how Sir James Frazer felt when, after setting out to dash off a brief monograph on a single obscure rite, he found himself in the embarrassing possession of the 12 volumes of ``The Golden Bough.'' \author WAVERLEY ROOT (1974) % International Herald Tribune, 22 May 1974, p8 \bigskip No matter how many times you proofread your book, after publication there will be roughly one typo per page. \author JOSEPH H. SILVERMAN (2017) % "A tale of two books", Bull AMS 54 (2017), 593 \vfill\eject } \def\today{\number\day\space\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\year} %%%%%%%%%%%%%%% CHANGES FOR VOLUME 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\lhead{CHANGES TO VOLUME 1: FUNDAMENTAL ALGORITHMS} \let\rhead=\lhead \titlepage \volheadline{FUNDAMENTAL ALGORITHMS} \bigskip \rightline{Copyright \copyright\ 2011, 2012, \dots, 2018, 2019, 2020 Addison\with Wesley} \rightline{Last updated \today} \bigskip \rightline{\sl Most of these corrections have already been made in recent printings.} \smallskip \let\defaultpointsize=\tenpoint \bugonpage 1.0 (on the back page of the dustcover, line 23) (12.09.10) {\eightss Whatever your backgound, \becomes Whatever your background,} \endchange \bugonpage 1.v line 8 (18.12.13) \eightssi McCall's Cookbook \becomes McCall's Cook Book \endchange \bugonpage 1.x line 14 (16.03.24) American Mathematical Monthly \becomes The American Mathematical Monthly \endchange \amendpage 1.xi new paragraph to follow line 20 (13.11.25) My efforts to extend and enhance these volumes have been enormously enhanced since 1980 by the wise guidance of Addison--Wesley's editor Peter Gordon. He has become not only my ``publishing partner'' but also a close friend, while continually nudging me to move in fruitful directions. Indeed, my interactions with dozens of Addison--Wesley people during more than three decades have been much better than any author deserves. The tireless support of managing editor John Fuller, whose meticulous attention to detail has maintained the highest standards of production quality in spite of frequent updates, has been particularly praiseworthy. \endchange \improvepage 1.xi line 5 before the author's signature (15.09.23) nobody noticed \becomes nobody else noticed \endchange \amendpage 1.xiv date of the Herbert quotation (19.11.26) {\eightss (1640)} \becomes {\eightss (1651)} \endchange \amendpage 1.xiv replacement for the Vauvenargues quotation (19.12.21) \vskip0pt % this actually helps {\quoteformat Les meilleurs auteurs parlent trop. \author VAUVENARGUES, {\sl Paradoxes, m\^el\'es % de R\'eflexions et Maximes}, 115 (1746) % "the best authors say too much" } \endchange \amendpage 1.xvi line 16 (11.06.16) a {\it 45\/} rating \becomes a {\it 40\/} rating \endchange \amendpage 1.xvi line 20 (13.01.30) creativity.\ \becomes creativity. All exercises with ratings of {\it46\/} or more are open problems for future research, rated according to the number of different attacks that they've resisted so far. \endchange \amendpage 1.2 line $-7$ (19.09.13) also usually appears \becomes often appears also \endchange \improvepage 1.6 lines 1--3 (13.03.13) $\{m,n\}$ \becomes $m$ and $n$ [twice]\nl $\{n,r\}$ \becomes $n$ and $r$ [twice] \endchange \improvepage 1.8 lines $-5$ and $-4$ (13.03.13) Such a computational method \dots\ it is also \becomes Every step of such a computational method is clearly effective, and experience shows that pattern-matching rules of this kind are also \endchange \improvepage 1.8 line $-11$ (19.07.30) for strings $\alpha$ and $\omega$ \becomes for some strings $\alpha$ and $\omega$ in~$A^*$. \endchange \amendpage 1.9 the rating of exercise 7 (14.04.05) \ninepoint {\it M21\/} \becomes {\it\HM21\/} \endchange \improvepage 1.11 lines 3--4 (15.05.06) An expansion \dots\ can \becomes A more leisurely presentation of much of the following material, together with related concepts, can \endchange \improvepage 1.15 line $-13$ (19.04.20) consists of proving \becomes constructs a proof \endchange \bugonpage 1.17 line 26 (13.12.15) {\bf11} (1838) \becomes {\bf12} (1838) \endchange \amendpage 1.18 replacement for line 5 of exercise 2 (19.04.19) \ninepoint$$ a^{(n+1)-1}=a^n={a^{n-1}\times a^{n-1}\over b^{n-1}}={1\times 1\over 1}=1, \quad\hbox{where $b=a^{(n-2)/(n-1)}@$;} $$ \endchange \improvepage 1.18 clarification in exercise 4 (13.02.10) \ninepoint $\phi^{n-2}$. \becomes $\phi^{n-2}$ for all positive integers $n$. \endchange \improvepage 1.19 clarification in exercise 5 line 1 (13.02.05) exact divisors \becomes positive integer divisors \endchange \amendpage 1.19 in exercise 9 (19.09.13) $1-na$ \becomes $1-na$ for all integers $n$ \endchange \improvepage 1.22 lines 1 and 3 after {\eq(8)} (13.01.20) $10^x$ \becomes $b^x$ \qquad[two places] \endchange \amendpage 1.22 line 3 after {\eq(8)} (19.09.17) is 1.999\dots\ \becomes begins with 1.999\dots \endchange \amendpage 1.23 line 1 after {\eq(13)} (15.02.22) Edward M. Reingold. \becomes Ernesto Trucco [{\sl Bull.\ Math.\ Biophysics\/ \bf18} (1956), 130]. \endchange \improvepage 1.25 lines 1 and 2 of exercise 11 (19.11.22) \ninepoint and $x\approx\log_{10}2$, to how \dots~value of $x$ in order to determine \becomes and $x$ is approximately equal to $\log_{10}2$, how many decimal places of $x$'s value will guarantee that we can determine \endchange \improvepage 1.25 in exercise 18 (19.11.22) $\half\lg x$ \becomes $\half\lg x$, for all real $x>0$. \endchange \bugonpage 1.27 line 1 (11.07.22) \ninepoint $n>1$ \becomes $n>1$ and $n\ne e$. \endchange % actually I've also changed $n$ to $x$ in this exercise \amendpage 1.29 replacement for five lines after {\eq(9)} (19.12.22) \noindent where $S'(j)$ specifies summing over all $j$ such that ``there is an integer $i$ such that both $R(i)$ and $S(i,j)$ are true''; and $R^{@\prime}(i,j)$ specifies summing over all~$i$ such that ``both $R(i)$ and $S(i,j)$ are true.'' For example, if the summation is $\sum_{i=1}^n \sum_{j=1}^i a_{ij}$, then $S'(j)$ sums over $j$ where ``there is an integer $i$ such that $1\le i\le n$ and $1\le j\le i$,'' that is, $1\le j\le n$; and $R^{@\prime}(i,j)$ sums over~$i$ where ``$1\le i\le n$ and $1\le j\le i$,'' that is, $j\le i\le n$. Thus, \endchange \improvepage 1.30 last line of Example 1 (19.04.20) consists of simplifying \becomes simplifies \endchange \amendpage 1.35 replacement for line 2 of exercise 18 (19.12.22) \ninepoint\noindent $R(i)$ specifies summing over all~$i$ such that ``$n$ is a multiple of $i$'' and $S(i,j)$ specifies summing over all~$j$ such that ``$1\le jl$, because that could happen only if $(n+d_1/10+\cdots+d_l/10^l+1/10^l)^m\le u$. \endchange \amendpage 1.470 line 2 of answer 11 (19.09.17) $10^x=1.99999\ldots$ \becomes $10^x$ begins with $1.99999\ldots$ \endchange \improvepage 1.470 in step E1 of answer 28 (11.10.28) If $1-\epsilon$ \dots $k\gets1$. \becomes Set $x\gets1-\epsilon-x$, $y\gets y_0$, and $k\gets1$, where $1-\epsilon$ is the largest possible value of~$x$, and $y_0$ is the nearest approximation to $b^{1-\epsilon}$. \endchange \amendpage 1.471 new answer 18 (19.12.22) \ans18. $S'(j)$ sums over $j$ with ``$1\le jj$''. \endchange \amendpage 1.472 replacement for last line of answer 26 (20.06.12) The answer is $P_1=(P_1P_2)^{1/2}=\bigl(\prod_{i=0}^n a_i\bigr){}^{n+2}$, because $P_1=\bigl(\prod_{i=0}^n \prod_{j=i}^n a_ia_j\bigr)=P_2$. \endchange \amendpage 1.472 new copy for end of answer 31 (11.10.20) Consequently we have $\bigl(\sum_{j=1}^n u_j\bigr)\bigl(\sum_{j=1}^n v_j\bigr) \le n\sum_{j=1}^n u_jv_j$ when $u_1\le u_2\le\cdots\le u_n$ and $v_1\le v_2\le\cdots\le v_n$, a result known as {\it Chebyshev}'s monotonic inequality. [See {\sl Soobshch.\ mat.\ obshch.\ Khar'kovskom Univ.\ \bf4},\thinspace2 (1882), 93--98.] \endchange \amendpage 1.474 lines 3--5 of answer 43 (11.11.26) as in exercise 44 \dots\ $(x_i-1)\bigr)$. \becomes by setting $x=1$ in exercise~40 and obtaining ${\prod_{k\ne i}(x_k-1)/x_i\prod_{k\ne i}(x_k-x_i)}$. After multiplying numerator and denominator by $x_i-1$, we can sum on $i$ by applying exercise~33 with $r=0$ to the $n+2$ numbers $\{0,1,x_1,\ldots,x_n\}$. \endchange \amendpage 1.476 replacement for answer 4 (13.12.22) \ans4. By part (f), $x\le\lceil x\rceil0$. The $k$th term is $r/(r-tk)$ times $$\displaylines{\quad{1\over n!}\Bigl({n\atop k}\Bigr) \prod_{0\le jn-1$ \becomes $y\ge n-1$ \endchange \amendpage 1.491 replacement for last line of answer 67 (12.02.18) ${n\choose k}\ge\bigl({(n-k+1)e\over k} \bigr){}^k{1\over ek}$, which is less memorable (but often sharper) than ${n\choose k}\ge\bigl({n\over k}\bigr){}^k$. \endchange \amendpage 1.492 replacement for answer 16 (16.06.21) \ans16. $H_{2n}-\half H_n$. \endchange \amendpage 1.493 new answer for section 1.2.7 (13.02.11) \ans25. $H_n^{(0,v)}=\sum_{k=1}^nH_n^{(v)}$ and $H_n^{(u,0)}=H_n^{(u-1)}$; so the identity generalizes~\eq(8). [See L.~Euler, {\sl Novi Comment.\ Acad.\ Sci.\ Pet.\ \bf 20} (1775), 140--186, \S2.] \endchange \amendpage 1.495 last line of answer 34 (18.07.10) Generalizations \dots 7.1.3. \becomes For generalizations, see exercise 4.5.3--52, exercise 5.4.2--10, and Section 7.1.3. \endchange \bugonpage 1.503 last line of answer 3 (13.03.05) 1937 \becomes 1927 \endchange \improvepage 1.504 last line of answer 8 (12.02.18) ${cn^2\choose n}\big/c^n{n^2\choose n}$ \becomes ${cn^2\choose n}\big/\bigl(c^n{n^2\choose n}\bigr)$ \endchange \amendpage 1.509 new answer 23 (18.02.28) \ans23. First, by R. D. Dixon:\hskip5em Second, by George Wahid, if $\rA\ge0$: \smallskip \line{\vtop{\mixthree (3000)&ENT1&4\cr (3001)&LDA&200\cr &SRA&0,1\cr &SRAX&1\cr\endmix}\hfil\vtop{\mixthree &DEC1&1\cr &J1NN&3001\cr &SLAX&5\cr &HLT&0\quad\slug$_{\phantom j}$\cr\endmix} \hfil\hfil\hfil\vtop{\mixthree (3000)&ENT1&5:5\cr (3001)&SLA&1\cr &ST1&3003(4:4)\cr (3003)&ADD&200(*)\cr\endmix}\hfil\vtop{\mixthree &DEC1&1:1\cr &J1P &3001\cr &HLT&0\quad\slug\cr\endmix}} \endchange \improvepage 1.514 replacement for line 5 of the program (13.04.30) \nobreak\vskip-12pt\ansmixthree 3H \ \ \ &ENT2&9*8-8,1&Start at row 9.\cr \endmix \endchange \bugonpage 1.514 replacement for lines 19 and 20 of the program (13.03.10) \nobreak\vskip-12pt\ansmixthree PHASE2&ENT3&9*8 \ \ \ &At this point $\rA =\min_jC(j)$\cr 3H&ENT2&0,3&Prepare to search a row.\cr \endmix \endchange \amendpage 1.518 line 14 (14.08.27) Victorius of Aquitania \becomes Victorius of Aquitaine \endchange \bugonpage 1.545 line 8 of answer 3 (19.04.23) $\rA \gets\.{INFO($\rI1$)}$ \becomes $\rA \gets\.{INFO($\rI3$)}$ \endchange \amendpage 1.546 last lines of answer 9 (14.06.30) plagues many automatic programming systems.) \becomes plagued many automatic programming systems in the old days.) \endchange \amendpage 1.547 new answer 13 (18.03.15) \ans13. Sha and Kleitman [{\sl Discrete Math.\ \bf63} (1987), 271--278] \vadjust{\vskip-1pt}% proved that the number is at most $\prod_{k=0}^n{n\choose k}\losup{n\choose k}$; \vadjust{\vskip1pt}% Brightwell and~Tetali [{\sl Order\/ \bf20} (2003), 333--345] improved this upper bound to $\exp\bigl((2en2^n\!/r)^r\prod_{k=0}^n{n\choose k}!\bigr)$, where $r=(2^{n+1}-2)/(n+1)$. The obvious lower bound is $\prod_{k=0}^n{n\choose k}!= 2^{2^n(n+O(\log n))}$. For the initial values $\langle a_n\rangle= \langle1,1,2,48,1680384,\ldots{}\rangle$, up to $a_7\approx6.305\times10^{137}$, see OEIS sequence A046873. \endchange \bugonpage 1.552 in answer 12 (12.06.30) line 1: $29p$ \becomes $27p$\nl line 3: $3\over4$ \becomes 78\% \endchange \amendpage 1.570 new sentence to follow line 1 of answer 30 (13.10.08) Thus $\.{LOC(T)}=\.{HEAD}$, and $\.{HEAD}\symord/$ is the first node of the binary tree in symmetric order. \endchange \improvepage 1.570 line $-3$ of answer 30 (13.10.08) the algorithm of exercise 21 \becomes Algorithm~U in answer 21 \endchange \amendpage 1.576 new answer(s) (14.09.09) \anss 14, 15. See Martin Ward and Hussain~Zedan, ``Provably correct derivation of algorithms using FermaT,'' {\sl Formal Aspects of Computing\/ \bf26} (2014), 993--1031. \endchange \amendpage 1.587 replacement for top eight lines (17.08.12) \noindent types ($\alpha@a$, $\delta Na$, $\alpha@d$, $\delta Na$, $\alpha@a$, $\delta Nd$, $\alpha@d$) on the main diagonal have the form $$ \vbox{\offinterlineskip \def\bit{height 3pt&\omit&&&&&&&&&\cr} \def\midbit{height 3pt&\omit&&&\omit&&\omit&&&&\cr} \def\ctr#1{\hbox to 4em{\hfil #1\hfil }} \halign{ \vrule #&\vrule height 9pt depth 5pt width 0pt \ctr{$#$}&\ctr{$#$}&\ctr{$#$}&\vrule #&\ctr{$#$}&\vrule # &\ctr{$#$}&\ctr{$#$}&\ctr{$#$}&\vrule #\tabskip0pt\cr \noalign{\hrule} \bit &\alpha@a&\beta K\?Q&\alpha@b&&\beta@QP&&\alpha@a&\beta K&\alpha@b&\cr &\gamma J\?P&\delta Na&\gamma R&&\delta@K\?Q&&\gamma J\?L&\delta Nb&\gamma P&\cr &\alpha@c&\beta DS&\alpha@d&&\beta@Y\?\?QT&&\alpha@c&\beta S&\alpha@d&\cr &\multispan3\hrulefill&&&&\multispan3\hrulefill\cr \midbit &\gamma PQ&\delta J\?P&\gamma X\?P&\omit&\delta Na&\omit&\gamma RQ&\delta R&\gamma R&\cr \midbit &\multispan3\hrulefill&&&&\multispan3\hrulefill\cr \bit &\alpha@a&\beta KU&\alpha@b&&\beta DP&&\alpha@a&\beta K&\alpha@b&\cr &\gamma@JT&\delta Nc&\gamma S&&\delta SD&&\gamma JS &\delta Nd&\gamma@T&\cr &\alpha@c&\beta@QS&\alpha@d&&\beta DT&&\alpha@c&\beta S&\alpha@d&\cr \bit \noalign{\hrule}\cr}} $$ \endchange \amendpage 1.587 line 12 (19.06.22) For a similar \becomes (Six of the 92 types can actually be omitted; see exercise 7.2.2.1--121.) For a similar \endchange \amendpage 1.588 line 3 of answer 4 (20.01.05) at $z=\alpha$. \becomes at $z=\alpha$. (See exercise 7.2.2.2--348.) \endchange \amendpage 1.589 last line of answer 6 (18.02.25) 140.] \becomes 140. Incidentally, there are $g_{n-1}$ oriented trees with $n$ {\it leaves\/} and in-degree~2 at nonleaves.] \endchange \amendpage 1.589 in answer 7 (20.01.05) $a\approx2.483253536173$. \becomes and $a\approx2.483253536173$ satisfies $F(1/a)=0$. \endchange \amendpage 1.598 last line of answer 3 (17.12.28) two. \becomes two. See exercises 7.2.1.6--26 through 7.2.1.6--34. \endchange \bugonpage 1.607 lines 1--9 (16.06.24) [In some printings, the ``primes'' disappeared from step names; for example, step C5$^\prime$ was simply called step C5.] \endchange \let\defaultpointsize=\tenpoint % begin appendices \bugonpage 1.611 line 5 (14.07.25) inner loop A3* \becomes inner loop A2.5* \endchange \bugonpage 1.613 last line of answer 28 (14.04.14) $\.{AVAIL[$k$]}\gets \.L$. \becomes $\.{AVAILF[$k$]}\gets \.L$. \endchange \improvepage 1.614 line $-2$ of answer 33 (17.12.28) Cohen and Nicolau \becomes J. Cohen and A. Nicolau \endchange \amendpage 1.624 through page 626 (12.04.07) [replace the notation $(R\Relbar\joinrel\joinrel\joinrel\Rightarrow x;\;y)$ by $(R{?}\ x{:}\ y)$ in eleven places] \endchange \amendpage 1.628 new entries for Appendix C (11.04.17) \ninepoint\noindent Program 1.2.10M, 145, 186.\nl Program 1.4.3.1M, 204--211, 530.\nl Program 2.1A, 236.\nl Program 2.1B, 535.\nl Program 2.3.1S, 325. \endchange \bugonpage 1.629 in Appendix C (11.04.17) \ninepoint\noindent Algorithm 2.4B$'$, 606. \becomes Algorithm 2.4B$'$, 605. Algorithm 2.4B$''$, 606.\nl Algorithm 2.5G, 613. \becomes Algorithm 2.5G, 613--614. \endchange \expandafter\ifx\csname indexeject\endcsname\relax\else\vfill\eject\fi \amendpage 1.630 and following (11.01.01) Miscellaneous changes to the existing index of Volume~1 are collected here, including corrections and amendments to the old entries as well as new entries that are occasioned by the new material. Thus, the lines of the full index that have changed serve also as an index to the present document. However, when a correction or amendment has caused an old index entry to be deleted, the deletion is usually not indicated. \input exotic \begindoublecolumns \indexformat \gdef\Uni1.08:{\bitmap24:1.08:} \hangindent 2em $\pi$ (circle ratio), as ``random'' example, 397, 591. % 42nd Bernstein, Sergei Natanovich ({\rus Bernshtei0n, Sergei0 Natanovich}), 104. %30th Brightwell, Graham Richard, 547. % 39th Carroll, Lewis (= Dodgson, Charles Lutwidge), 465. % 32nd Chebyshev (= Tschebyscheff), Pafnutii Lvovich ({\rus Chebyshevp2, Pafnut{\char12}i0 Lp1vovichp2} = {\rus Chebyshev, Pafnutii0 Lp1vovich}), inequalities, 98, 104, 472. % 31st Chia, Hsien (\JC48:48:0:40% J7643 <00000000001800000000003cfffffffffffe7fffffffffff0000f00f00000000f00f0000% 0c00f00f00600e00f00f00f00ffffffffff80ffffffffffc0f00f00f00f80f00f00f00f0% 0f00f00f00f00f00f00f00f00ffffffffff00ffffffffff00f00000000f0060000000060% 00000000000000000000000000ffffffff0000ffffffff0000f000000f0000f000000f00% 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Comfort, Webb Tyler, 461. % 34th Contour integration, 49, 92. % 37th Depth of node in a tree, \see Level. % 30th Diagrams of structural information, tree structures, 309--315, 337, 346, 349, 460, 465. % 32nd Dodgson, Charles Lutwidge, \see Carroll. % 32nd Doolittle, Myrick Hascall, 303. % 29th $e$ (base of natural logarithms), 23, 619--620, 626. % 30th Empty list, 244--245, 247, 258, 260--261, 273--275, 278, 280, 540, 546. % 30th Euler, Leonhard ({\rus Ei0lerp2, Leonardp2} = {\rus E1i0ler, Leonard}), 49, 50, 52, 57, 75, 76, 87, 111, 374, 407, 472, 493, 496, 536, 600. % 31st Family trees, 310--311, 317, 406, 465. % 32nd Fletcher, William Thomas, 527. % 39th Floating point arithmetic, 131, 230, 306, 411. % 42nd Fraenkel, Aviezri Siegmund ({\heb\Hll/\Hqq/\Hnn/\Hrr/\Hpp/ \Hyy/\Hrr/\Hzz/\Hai/\Hyy/\Hbb/\Haa/}), 251. % 39th Fuller, John Edward, xi. % 32nd Geek art, xx. % 34th GO button of \MIX, 126, 139, 143--144, 211. % 29th Gordon, Peter Stuart, xi. % 32nd Greedy algorithm, 489. % 40th Hamel, Georg Karl Wilhelm, 480. % 39th Huskey, Harry Douglas, 229. % 34th Invariants, 2, 17. % 41st Kinkelin, Georg David Hermann, 504. % 39th Knuth, Nancy Jill Carter (\GC75:75:-3:62% G2463 <0000001f0000000000000000001ff8000000000000000007ff000000000000000003ff80% 00000000000000007f8000000000000000007f8000000000000000003f8000003c000000% 00003f8000007e00000000001f800000ff00000000000f800001ff00ffffffffffffffff% ff80ffffffffffffffffffc07fffffffffffffffffe03800000000000000000000000000% 00000000000000000000000070000000000078000000f800000000007f000001fc000000% 00007ffffffffe00000000007fffffffff00000000003ffffffffe00000000003f000000% fc00000000003f000000fc00000000003f000000fc00000000003f000000fc0000000000% 3f000000fc00000000003f000000fc00000000003f000000fc00000000003f000000fc00% 000000003ffffffffc00000000003ffffffffc00000000003ffffffffc00000000003f00% 0000fc00000000003f000000fc00000000003f000000fc00000001c03e0000000003c000% 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1f00f80007c00007c0001e00f80007c00007c0003c00f80007ffffffc0007000f80007ff% ffffc0006000f80007ffffffc000c000f80007c00007c0008000f80007c00007c0000000% f80007c00007c0000000f80007c00007c0000000f80007c00007c0000000f80007c00007% c0000000f80007c00007c0000000f80007c00007c0000000f80007c03fffc0000000f800% 07c01fffc0000000fc0007c007ffc0000000fc000fc001ffc0000000fc000fc0007fc000% 0001fc000fc0003f80000001f8000fc0001f80000001f8000000001e0000>%% Unicode char "7cbe %\JC47:48:-1:40% J4586 %<0001801800000001e01e00000001f01f00180001e01e003cfffffffffffefffffffffffe% % 0001e01e00000000c00c00001800306000c01e00787801e01ffff87ffff01ffff87ffff0% % 1e00787801e01e00787801e01ffff87fffe01ffff87fffe01e00787801e01e00787801e0% % 1ffff87fffe01ffff87fffe01e00787801e01e00363001e01e00078001e01e0007c031e0% % 1e00078079e01efffffffde01e7ffffffde01e00078001e01e0c078181e01e0fffffe1e0% % 1e0fffffe1e01e0f0783c1e01e0f0783c1e01e0fffffc1e01e0fffffc1e01e0f0783c1e0% % 1e0f0783c1e01e0fffffc1e01e0fffffc1e01e0f3fc3c1e01e067ff981e01e007fbe01e0% % 1e00f79f01e01e01e78f81e01e03c78fffe01e078787cfe01e1e0783c7c00c7803018380% % >%% Unicode char "862d \GC76:72:-2:62% G3228 <000000000000380000000000700000003c00000000003c0000007e00000000003e000000% 7f80000000001f8000007fc0000000001fc000007fe0000000000fe000007f8000000000% 07f00000ff800000000007f80000ff000000000003fc0000fe000000000003fc0000f000% 0000000001fe0001f0000000000001fe0001e0000000000000fe0003e0000000000000fe% 0007c00000000000007e0007c00000000000007e000f800000000000007e001f00000000% 0000003e003f000780000000003c003e000fc00000000038007c000fe000000000000078% 001fe0000000000000f0003ff00003fffffffffffffff80001fffffffffffffffc00007f% fffffffffffffc0000180000000000000000000000000000000000000000000000000000% 000000000000000000000000000000000000000000000000000000000000000000000000% 000000000000000000000000000000000000000000000000000000000000000000000000% 000000000000000000000000000000000000000000000000000000000000000000000000% 00380000000000000000007c000000000000000000fe000000000000000001fe00000000% 0000000003ff0000000fffffffffffff8000000fffffffffffffc0000003ffffffffffff% c0000000c000000000000000000000000000000000000000000000000000000000000000% 000000000000000000000000000000000000000000000000000000000000000000000000% 000000000000000000000000000000000000000000000000000000000000000000000000% 000000000000000000000000000000000000000000000000000000000000000000000000% 00000000000000000000000000000000380000000000000000007c000000000000000000% fe000000000000000000ff000000000000000001ff800000000000000003ffc0ffffffff% ffffffffffe0fffffffffffffffffff03ffffffffffffffffff008000000000000000000% >%% Unicode char "5170 ), x, xx. % 33rd K\"onigsberg bridges, 374. % 36th Lagrange, Joseph Louis (= Giuseppe Lodovico Lagrangia = Giuseppe Ludovico (= Luigi) De la Grange Tournier), inversion formula, 392, 594. % 39th %Lam\'e, Gabriel L\'eon Jean Baptiste, 407. % 34th Lam\'e, Gabriel, 407. % 37th Leaf of tree, 308, \see Terminal node of a tree. % 39th L\'evy, Paul Pierre, 363. % 39th Lilius, Aloysius (= Aloigi Giglio, Luigi Lilio), 159. % 33rd Lineal chart, 310--311, 465. % 32nd log (without subscript), 110. % 33rd Markov, Andrei Andreevich ({\rus Markov, Andrei0 Andreevich}), the younger, 9. % 31st McCall's Cook Book, v. % 41st Mealy, George Henry, II, 462. % 36th Mirsky, Leon ({\rus Mirskii0, Leonid}), 587. % 36th Neumann, John von (= Neumann J\'anos Lajos = Margittai Neumann J\'anos), 18, 229, 231, 457. % 34th Nonnegative coefficients, 396, 501. % 30th Number system, decimal, 21, 619. % 32nd OEIS\regtm: The On-Line Encyclopedia of Integer Sequences\regtm\ ({\tt oeis\period org}). % 43rd Oldenburg, Henry (= Heinrich), 57. % 36th Overflow toggle of \MIX, 126, 131, 134, 142, 144, 208, 214, 228. % 29th Pak, Igor Markovich ({\rus Pak, Igorp1 Markovich}), 407. % 37th Pattern matching, 8. % 31st Pedigree, 310--312, 465. % 32nd Pi ($\pi$), as ``random'' example, 397, 591. % 42nd Reingold, Edward Martin ({\heb \Hdd/\Hll/\Hvv/\Hgg/\Hnn/\Hyy/\Hyy/\Hrr/},\indexbreak{\heb \Hfm/\Hyy/\Hyy/\Hkh/ \Hfn/\Hbb/ \Hhh/\Hsh/\Hmm/ \Hqq/\Hkh/\Hts/\Hyy/}), 518. % 33rd Residue theorem, 472--473. % 37th Reversion storage, 229, 240. % 34th Root of a tree, 308, 309, 317, 465. % 32nd Schwenk, Allen John Carl, 495. % 42nd Staver, Tor B{\o}hm, 582. % 28th Stirling numbers, 66--69, 71--74, 78, 99--100, 506, 582, 591, 625. % 34th sup: Supremum, least upper bound, 37. % 40th Swift, Jonathan, 630. % 39th Szpilrajn (later Marczewski), Edward, 268. % 36th Terminal node of a tree, 308, 318, 352, 397, 589, 597. % 39th Tetali, Venkata Sitarama Vara Prasad \indexbreak({\tl\|0142|\C86\\2\|0128|\C86\\2\C222\ \|0141|\C107\\2\C9\|1128|\C71\\2\C81\ \|2132|\C110\\2\|0129|\C86\\5\|0129|\C103\\6\|0128|\C101\ \|0128|\C107\\2\|0128|\C103\ \|1128|\C97\\{-7}\C175\\3\|0129|\C110\\5\|1148|\C88}), 547. % 39th Trees, diagrams of, 309--315, 337, 346, 349, 460, 465. % 32nd Trucco, Ernesto, 23. % 33rd Victorius of Aquitaine, 518. % 33rd von Neumann, John (= Neumann J\'anos Lajos = Margittai Neumann J\'anos), 18, 229, 231, 457. % 34th Wahid Aziz Abdel-Malek, George \indexbreak ({\arab\Afc/\Aml/\Amm/\Ail/\Aa/ \Afd/\Amb/\Aiai/ \Afz/\Aiy/\Afz/\Aiai/ \Afd/\Amy/\Aihh/\Aw/ \Atch/\Ar/\Afw/\Aitch/}), 509. % 39th Ward, Martin Paul, 576. % 32nd Windley, Peter Francis, 523. % 34th Zeckendorf, \smash{\'E}douard, 495. % 36th Zedan, Hussein Saleh Mamoud\indexbreak ({\arab\An/\Aa/\Afd/\Aiy/\Az/ \Ad/\Afw/\Amm/\Amhh/\Aim/ \Afhh/\Ail/\Afa/\Aiss/ \Afn/\Amy/\Ams/\Aihh/}), 576. %32nd \vfill \enddoublecolumns \endchange \bye [The next printing will be the 44th.] Not mentioned above: page 4, line -16: one half -> one-half "Hindu mathematics" and "Islamic mathematics" leave the index ARTICLES "TO APPEAR" THAT ARE STILL PENDING: