331092

cl THE STATE OF THE ART OF COMPUTER PROGRAMMING

~by

D.E. Knuth

STAN-CS-76-551JUNE 1976

COMPUTER SCIENCE DEPARTMENTSchool of Humanities and Sciences

STANFORD UNIVERSITY

" -J"Tt"N STATMENT A

Approved for public release;Distribution Unlimitod

DDC

A

:1F1

The State of The Art of' Computer 1Frogramming

Donald M. Knuth(: !nj)u t(e' ""C rWOf ri-1 1,01*1110f1 I

8trt l'ord lII i IV'r'::i Iy

.t.anU'ord, (aiJ.i l'oruia qI05

This report lists all corrections and changes to volumes 1 and 5 ;

The Art of Computer Progranuning, as of May 14, 1976. The changes apply

to the most recent printings of both volumes (February and March, 1975):

if you have an earlier printing there have been many other changes not

indicated here. Volume 2 has been completely rewritten and its second

edition will be published early in 1977. For a summary of the changes

made to volume 2, see SIGSA4 Bulletin 9, h (November 1975), p. iOf --

the changes are too numerous to list except in the forthcoming book itself.

On any given day the author likes to feel that the last bug has

finally disappeared, yet it appears likely that further amendment.:r will be

made as time goes by. Therefore a family of computer programs has been

written to maintain a collection of errata, in the form printed here, but

encoded as an ad-hoc sequence of ASCII characters. The author wishes to

thank Juan Ludlow-Saldivar for the enormous aemount of help he provided

in order to get this system rolling. (Some readers who have access to

the Stanford A.I.-Lab computer may wish to consult the change file before

they report a "new" error; the file name is ACP.MAS [ART,DEK]. Entries

for page nnn of volume k begin ith j3kOlnnn (but change the 01 to O

if nnn is the Arabic equivalent of a Roman numeral); since B is the

control character " tC", you may rather search for simply the string

" kOlnnn". The text of the correction usually include- special coder

following the symbol " ", for things like font changez, etc.)

The author thanks all the bounty hunters who have rejorted dlf'-",t_ e

they spotted. 'Te reward to fir.t finder of each error is still ^l for

the first edition and t2 for the second, gratefully paid. Volume . res ins

rather far from completion, so there is tlenty of time to work all the

exercises in volumes 1 - : and to catch all the remaining errors therein.

This research was supported in part by Nationa.1 Science Foundation grant14CS 72-05752 A03 and by the Office o2 :ava . Resa.rch contract NXOlh-7. -C-055J.Reproduction in whole or in part is permdttd for any purpose of .he UnitedStates Government.

1

The Tirt of Computer Programming

2, 4 ;LU line 5

for 'is hitimcIf I% hing encoiaragrd

t U.N[K line 10 2

antiw'r anwerr

new quote for bottom of page

We can race our prob'tam.We can arrange such facts as wi have

with order and method.-- HERCULE POIROT, in Murder on the Orient Express (1934)

211. lina 23 4

VO. -. EO. (boldface)

2i,2. line 3

prove /16 '\-4 prov that /6

2L2W line -1 16

3n0 ^. 3n

ILI t LO lines -3 and -2-7-

T < 3nO, where 13 or the original vai:r n n, T I., ', . ,

~.. .... .....

t !Ch

-~

LIiu' ex 25 8

deeIe ItCp 1.5 and Move the I to the P1d or Step 14

ex 25, change step L3 to: 9

L3. [Shift.] If x-- < 1, set z r- hifted right 1, k +- k.l, and repeat this .t.p.

.i line 15, new sentence 10

harIwarr. -v hardware. The idea gn, hark in PrM;rnre to Henry Brij:g., who uscd it (indeecinal rather than binary form) t) compute logarithm tablcs, published in 1624.

L,'' line 23 1

example, N,4 example

L exercise 40 12

a periil (.) shoulI appear after the displayed equation

2L, line 2 two changes 13

(i) the (/p) and (p/q) don't iatch each other. (i) th, ,,:Ft two liner of p44 shotull be

mnv , bark tn p43, otherw;ie the reader will think cxercire 47 =r complete without turningfhe P.aLe.

k,.A'. line 20 14

1/12n lff/(2n)

ex 15

put FpareF in the firMt matrix, i.e.abc "%' f h e

de.f ' def

3

I line 7 after Table 1 16

Shikh-ehich ^.- Shih-Chieh

--. left side of eq. (17) 17

move the k a little left, to renter it

, line 7 after (26) 1

Shiih-el,,h \,* Shih-Chich

2Ji line 8 after (26) 19

tie hnlolfare 3 apprarr to he in wrong font (too i-niall)

22'/ 14 places 20

chanpe It to i (Homan type) in the notation for Ieta ftinctinn, namely in line i, line 2, line3 (twice), line 4 (thrice), line 5 (twirc), line 7, line 10 (twice), Itne 12, linc 15.

2,'2 exercise 47 21

in thpl.ayr, iornmala: chanj:e tipper jn,lirc" from n, n.1/2, 241, 2n.l-k to r, r-1/2, 2r, 2r-k rcj wrti,.Jyline 3:n -1. IN- r'-/

2-,'/t lines -3 and -2 22

ranrr the Renaisrance. I'v* during the Mildle Agem.

2L- line 2 23

1)63-) A- 19613-),

2J St i between (23) and (24) 24

series ^. o4 , ri. s (cf. (17))

Ali

2l!,1!J insert new sentence just after (26): 23

S,-e I). A. Znve, Inf. Prof. Lettera 5 (1976), to aprear, for a further generalization.

.iSL! replace (25) by new equation (25): 26

( ) (I/(i-z)) " Zkxo(I~k-iI) (%*k ) k, m > O.

2,,1S' lines 4-8 27

move ihe copy for each step to the left next to the step numbers (standard format Se e.g.Algorithm V on p2 )

-,,I! line -4 28

[ lines 3 and 4 after Fig. ]1 29

).; that "%'e X -halvalues, 'we -\ vjlte -we

, line 5 30

,dulrilirtinn, the i* dsht r oin, we can unre r-gnificantly on Chcby.hev'sincquuhaity: '|'he

-A line after (13) 3)

P),{2.() ten4l A-, )42k41)(r) an j(2k4)(r) tend

J.1 line 11 32

C ". C (Roainn, not italies)

r r line 20 33

, line 5 (two places) 34

record V block

row 5 column 4 of the table 35

I+T -% IT

2L '. Fig. 14 in both steps P7 and P6 36

Pr iME [K) PRIME[K]

2,'L'. line 4 37

fix hrokr, type in the ( of PRIME[iH]

21.2±' line 9 38

felelrt the exclamalion point (0)

LI "1" ex 3, first line of program 39

X+I I-",- X+I (0)

2, '/ last !ine of ex 18 40

aturnr' acmltne that

f EX; line 5 41

incert iinre pare after ihe permiol, thit lane*c to narrow

, line no. 21 of the program 42

PERM I ... ERM I,....

t.2-L!I line 8 43

4itF.elf " "N-,' :tel.

2-1 L- top of page 44

lthe "I" ir hrakrn i "1.3.3"

, '"'L- line 14 45

the 0 ir broken

L,?( line 16 46

0. J., 0A.-i.

2,Y v line -10 47

print) print),

L,-i Fig. 3(a) 48

deleie ihe ftniiy little box whirh appcars betwern "th'il fram top" and "fourth fro top"

a. L: just after (1) 49

remove black ,zprrk

.' A lines -3 and -2 50

tIcile the gentenee "!A there - obtainable?"

,i, bottom line 3S

TOP ' TOP (twire)

7

,L,=. line 3 52

<L < <L

&;;LL1 after step names GI and G2 53

broken type r for

LL' line -1 54

BASE, BASE+1, BASE+2, A.-. BASE+i, BASE+2, BASE+3,

move the heavy bar to the right so that it ir aligntl vertically with the heavy bar in (11)

l ,a-e " comment for line 18 of the program 56

Ti3 "%,*"14

new paragraph before the exercises 57

Its ciste of Ihe fart that Algorilhm T ir so efficient, we will Are an L'ven haeteralgorithm for olinlogical Anrtung in Section 7.4.

changes to Program A 58

lint- (1: SH - 1Hline 05: lirrnmec line 06

gli 06: li,'hrnrc line 07line 07: brorne line 05, ,nd lelr't ciae "IH" and rhanize x l'mline 12: heronle a the following Iwo liies

12 L02 1:3(L INK) q O.-LINK(O1.13 JP 28 q Repeat.

lirans 13-35 heroinre liner 14-36charge 6B "'.* 18 in what war line 17 (now line 18)

4

. Z line -1 3s9

line -4 60

exceril Isi rxN- rei h-

L, .-' ) line 12 61

29 " 27 (twire)

l MiL Table 1, left column 62

the line for time 0200 is out of plare, it belongg junt before the line for lime 0256

2- E Fig. 12 63

the lmlhh: ,n fil,; figure uytterintidy , ippcarc-l frenn the 3rd column of noilr,, n the,'cronl cstion. (FIrs- rlitinn it OK!)

, 'jY line 7 64

2419200 A 2,119,4200

, two lines before 11) 63

si the l)w gt value .,% pomnt' to the bottom-moct value

L. ! LL exercise 20 line 3 66

All.J) -N--. A[I.J]

TA

ILL new exercise 67

21. ( 0] S igr'l a ctorage allnation ftncuin for nXni matrirr. %here if if. variabr.- Thercleirnt. A [ ,JI for I IJ .S n ihoull ocrcupy n2 con~ccutivc locatirn, regardless of thevalut, of I.

, tree illustration near bottom of page 68

the nii:nher "(9)" muht b iterted at the right of thirt diaram

2,Z line - 13 69

p* , p*

- between (2) and (35 70

tilt the od.arra. , anti w- Sta 't 'tilt the ehanr.i an berI i:,.gntly, obtaining

4. Fig. ;"P 71

I-., phnlnrl -,,'.r: hat la.e, :p the two p rt of ihic figuirr inprowirly in Ih.t elition; theleft-han,: -.alf of the illu'tmation d nulh ,e lowered " that the trcet are fluith at the

lint!. m -- :hc nrianc tl at cnrrepaiaing ]-ttert will te an the ame lc in bath left and

r.,ghl prt.t of the illustratinn

1 line -9 72

of (7) N.. of the Ieft-hand tree. in (7)

. line 16 73

nole io 'n nore with

:' E line 9 74

only unwar,! link, are ituffarurnt O ,opward linktz ire 4ufficwrnt by thrmdrlAw

;.- .-- .. _ ._;-: _ ... . .. . . .-. . . . .... .. .*

l4* 'S in(G17) 7,

delete "." outride the boxeq (for consistency in style)

UIL!J exercise 11 76

change script I to italic I in five places (lines 5,6,6,23,25)

, Theorem A part (a) 77

line 4 78

remove hairline between "fin" and "("

, line -3 79

or it "/ or

L $ , '' line -2 80

Exercise -,V exercise

. .J.L exercise 12 81

Suppose [ (20) Suppose

itL'% line -14 82

partic lar v particular

L. ' line -4 83

3 -. )

21AL' Fig. 40 84

tile shave of tile box eontaiiii B6. slimild Iave touiiileal silles (like tbat, of 112); on tileoilier imii'l, Ole box Iith a "Error" sliotilil lbe rcie.ingiolar

~ line -5

t Ii ik ilhlayrd lne A hoiglil he raigell lialf a Apace An uiat it is Arparaicil fromt line -4 by Oihesanie ainount aF it ig Aerarated fromn line -6

LIL lines 4,6,7 86

aiud itlolln . conitionenireriie+ V einergicirr.henlce *N Ii'nce

L L7U line -7 87

two level - two-level

~~ exercise 39 line 3 8

2.i~/ line 18 89

t.TL " first line of quote 90

me that me . .. that

--- - - - - - - - -

,,Jl exercise 3 91

line ^5: let r he 'N$ let m lieline 4: lfr 0 , V Ifm - 0 ,

line 5: f/r % n/mr ad let in he ^%,4 m aid let it be

lines 6 anti 7 (steps F4 anti F5) dele, tedline 8: F6. -"" F4.

, t. better answer to exercise 3 92

3. -1/27, but the text hasn't defined it.

. exercise 13 93

first s--tare.e shoulI become:Add " < 3(n-d).k" to aprertionc /3. /4, /5. /16, where k takes the re'pective values2,3,3,1.

LlIV line 16 94

Plermenrt a and h ./%4 elements, a < h,

LS SLLZ2, ! exercise 3 95

the valise 3 is .. Iwo n. v* n2 • 3 occurs for no it, ani in the second place ,j2 4occurs for tuwo n.

1Y3T~ line 10 96

388. 'A4* 388; V. S. L.inkii ZA. Vych. Aft. i Afat. Fiz. 2 (1P), 90-119.

new answer replacing answer 10 97

9,10. No, the applhations of rule (d) asume that n 0. (1Thie result is correct for it " -1but the derivation isn't.)

15

L,'i/ " exercise 41 line 4 98

1/4 1/8 (twice)

exercise 31 99

We have N [Thr mln war firnt obainel in LOti form by J. F. Pfaff, Notio aelaarad. itriflI. Tht'r. 1 1 (1797), 38-57.) We have

.,LL and extending to page 487 100

change iI in R (Roman type) in the solitions to Iexer,l w~c 40, 41 (twice), 42, 48 (twic,-).

..LL exercise 14 101

ff4/' 'e" ?1

L, . exercise 10 line 2 102

Ms) -^s,- (17)

21 ,L LO qJ exercise 15 103

line 3 (.Gn .2i(playil forniIa): lGclect e period, then add a new line:when z -1/4; G (-1/4) (n.l)/2" for n >0.

2 AF bottom of page, a new answer to exercise 1.2.11.2-3:104

3. IR2k, < 1II'k!/(2k)!: fIi 1120)(,)dr. [C.. H. Hetn, ah nhirvr. that R2k f l"(l'2k.2 -

I?.,2({ -})) A 2kL2 ))d 42kU2)!, aiI that '12k.2 - 112k-2({') alhay' he litween 0 and (2-

2-2k-i)lI'/Ik. Thlerefore if (2ki(x) it [lot ,(2k#3)(x) terd mmintonirally to zero, (13)

.mill hiolk for conie i with 0 ( < 2 - 2 2k-i.

,E!L exercise 6 105

O(n"3 ) "v* O(n'3))

1},

r- -. ._ =- -= , > ) - _ ..:i-. __ _ = - -= --- . .. - -.-- -, - ...-=-- --- -- - . --

[ exercise 14 106

line 3: MOVE - MOVEline 4: JSJ*+1 AV. JSJ *+1

1, i exercise 17(b) 107

(Ilsfing awmbly ... section.)(A slightly farter, but quite prepostrro, s, program umsi, 993 STZ's: JMP 3995; STZ1,2: STZ 2,2; ...; STZ 993,2; J2N 3399; OEC2 993; J2NN 3001; ENN10,2: JMP 3000,1.)

, exercise 18 add new sentence: 0os

(Ilrleq the program itself apprarr in lor.ation; 0000-0015.)

4 ".a' exercise 20 109

Fukuoka) . Fukuoka.)

L, I V exercise 16 line 1 1)0

(49): A (49);

Ll EI T new line just before answer no. 23:

For small hy:o size, the entries ±613 would not appear.

exercise 6 line 3 112

" line -13 113

e.g. the -v4 e.g., the

15

L,2W exercise 22(d) 114

Sine theo W are irlepcently choww, the hie

2L iJ, 5"" U exercise 23 113

line 1: ol... (In ). -Ne f0Oexp(-1-/;())d1, where bi(")line 4: In n! e7y ^\, P7 II. i

line 6: 8310...; A 83100 83724 .;1796* [lth. Conp. 2 (1968), 411-415);

t,!i'L line 6 116

,Irv -/T"-/i). Ov* ,ev /7"-), when is> 2m.

*I " line 5 117

pro e woull loop imlefinitely;* ~algoriahni nrepukt. down (porsihly referr to buffer whiale 1/0 Is in progrcs-);

,j (~~i exercise 9 116

in revers.e, we can :et the inverqrharkwarok, we ran get the reverse of the inverim of he reverse

2i, TI exercise 12 119

1, EE1 ,, line 12 120

ro(:) "%V rq(z):r

2, h~. exercise 4() should have the following answer instead:

121

(ii) LDA X,7:7(0:2).

4 A

new answer 1221 3 . I. J . K le itm a n h a g sh o w n th a t lim n .too 2 "n In g jf(n ) li n. o V" o '0k ~ n (p .

['('o appe'ar.]

Ll line -5 123

COUNT " COUNT

1 ' and also page 544, answer to exercise 24 124

replac-e lite 8.-87 of tlre J11 X pro'rain bySTG X,1(QLINK) QLINK[r1 )- k.

Thrn rentmbpr lir;ex R,-118 in 86-116.Finally delete "Note: Wlhcn Ihe ... a; ihe loop." on p. 544.

6,N lines 1 i-i2 change to (with same indentation): 123

T"o. If P A A, r , QLINK(SUC W)] - k, P .-NEXT (P), and re,rat (hir step.

i. exercise 16 126

li 2: 29Z '\e 2 Z (twice)lie 8:6 - 4

F*L line -4 insert new sentence (no new paragraph) 127

[See exere 5 .2.3-29 for a fatrer algorithn.)

, exercise 1 line 4 12S

AVAIL v Y -

, t lin e 2 12 9

COL (P) oN.,. COL (PO)

17=

114:i change answer 18 (saving space for new answer 21):130

the firrt part up to "after I"" cart lie shortenel ar follow.;.18. The three pritt ; ,Mr. --. tive columnF 3,1,2, yield respectively

( !, ( ' ,(ime the rame n 'tr. , 1) 1 "quccze onto one line)

L- exercise 20 131

.;-6 k: new answer 132

Z1. For example, i - inax(l,J), LOC(AI ,JJ) * LOC(A[1,1]) * M1(1-I) I ] - J.(Such foritiflat have been propn-rl indepenmrkly Ivy many people. A. 1. H crnberg and H.R. Strong have ripgetrol the ,llowihg k-'hm:suonal gzcneralization: LOC (A [I, ..., I])

Lk Aheri- L 1 I LOC(A t, I..,I) I -, L, L. (r1)r r - - (M

-i), where Ms. . max(I 18.,Jt). JIBAI Terh. Fbsetlonure Bull. 14 (1972), 3026-3028.J)

2j,-kW exercise 15 i33

remove Lritche. in fira a-d ,;cronl lincr

2I.,1IL exercise 12 line 2 134

Am d A[m].

~~~L. new answer 135

13. (S.niitnn by S. Araujo.) Let miep, TI thros}i. *r4 k- isnclanjgrl, except that a newvari. ble 0 tF mtaiz;,i to A in cirp Ti; 0 will point to Ilse ia t no-le victirl, if an). StepTS' hcrcomr to mr'pc: T5. (Right branch loue?] If SIR NK (F') 2 or RLINK(P) = 0,

ro on to T6; oliherwire iet A e P, P - RUINK (P) an! return to T2. T6. (Virt P.] "Viit"NODE (P), x.et 0 P, anti return to T4' A mmilar proof appliem

L- iE 1 136

LOC (T). - V L!C(T).

I

r. L exercise I line 1 137

conFit 4" conrirtr

j,4 ... exercise 12 line 2 138

INFO(P2)-! ^N4 TREE(INFO(P2)-I)

Y'Z// exercise '8 line B 139

plenr,|er "=' pn~tnrdcr

/ exercise 7 140

the diagrams for Ci.ae I have two arron.hial in the wrong direction ... the arrows should

lead away from a and iown'.'s I both Before and After

~~~j line -8 141

32 I. 322

e> cise 12 iine 5 142

n(i) F (i) - ri,j) ., h(i) - j;a(j) ret n(j) - li,j) and h(j) i;

* V exercise 16

line 2: tit 'te ne o Iraring ouat

Imne. *Ii: wa .,ave an oriented rujkubtree ^v* the tated digraph it an oriented treline 5: confiriuration I *hgraphline 6: ,ubtree tree

2,'Zi last line 144

|), K Krith, H, H. i)awen and I. J. Good, lnn. Moth. Siat. 28 (!57).: 946-OS6; 1). E.Knuth,

Jq

~ exercise 24 l'ne 2 145

t. L I, Is line of exercise 23, add: 146

(Frtt- 2ti rv'alt w. -lite to C. Hlye Saifac-Marie, VI'Arint'dinire des Afattritnmtiriesix1 (1891), 107-110.)

2 2.1L exercise 3 line 3 14 7

2.iZ'3 e-^ercise 10 14S

t I!) second-last line before exercise 6 149

thip; line irn't rght-jimfiiril, aild xpac after Ilit rrinicaln

'L bottom line 130

eich.aiil. VNe rihmaat:.l. ftr Giuy 1. Sirele Jr., C/101i 18 (1975), 405-508, anti P.W&.tIkr, Ca(:AI 19 (1075), in .u;'gwar, for htheilr infnrrinaln.]/ Nte lisa' therit no rointno hierit~ Steele and Jr. in hsis nann./

Slines 19-21 replace by iS)

Several br'awuifi I.,,a-cappyin Ar~nrithin hhutril make subatially weaker.1-usnvitic abonmit i.ct rrre-nttiann have lurnsa ,1cviwid. See 1). W. Clark, CACAI 19(1976), to aprar, antl JI Xf Robcon, C/)CA 19 (1970), to appear.Ii' line 4 132minisu,,r N4mnz~u

- -~ -_ --

2. ,L! line before exercise 34 153

165.] I 165. Ser alro L. Wegbreit, Camp. J. 15 (1972), 204-208; I). A. Zavc, Jnf. Proc.IALeerz 3 (1975), 167-169.)

2,,LIitL line -7 14

det. (A)1 'e ,et(.A)

;UL- IV in several p!aces 155

Change • to .. . in the dcfinitionr of r upper k, x !owcr k, it factorial, and Stirlingnumlerc of both kmdni

L 2 I bottom line 156

give rectien reference 1.2.5 in right-hand column

ii.1U1[i definition of Beta function 157

!J -. B

L T line -20 (the entry for 1 degree of arc) 138

1154 e" I Iis

insert new paragraph after line 7: 159

See the Nnxwc to exercire 1.3.3-23 for the 40-digit value of another fundamentalrnnctanl.

, last line 160

9 -"2 9.

t /LW 161

Araujo, 'Satln, 560.

163

Bigr, I 14inry, 20.

LI.-15L Boizano entry 164

163

Carlyle, IhnmOF; 1vi.

L2.I LI L! 166

CIorsiic Mallawar., D).mr Agatha Mary Ctarima (Mmiler), xix.16

CI s ih-Clid, 52, 58.

Ll t A IJ263

Clark, l)ntjggI. We'lls, 594.

L2.WChebyshev's inequality entry 169

2111l p. 102

S170 9

lDa,nn Rerel, 57/8.

• 171

Donyl, Sir Arthur C nan, 46 .3. ;

["vrn, Shimon, 239. f

• 173 :

Flye Samnte-Marie, Camille, .580.

lti Z1 174

elvhir Ficlrr, David Allen

Llq

Hamlet, P'rince of D)enmark, 228.

• '' entry for Good. Irving John 176

adld p. 578

- ~line -8 F77

l': .r, "v exerwe

__ 1' 178

Kleitman, Iarth J.. S 4I .

_173

&~Et Knapp entry 179

~E ~ Krogdahl entry /sofix bsrokcri tyr

~~ ~line 18

"0 N 20.

ZLJILI182= 1incko,~ V. S., 470.

Pathalnia 399-40or.

fl'fI, Joan.ra, Fri-rtir, OLS.

L 183Phuicn 1:Z000, 120.

-"L* L186

Pnirnt, Ifrreodr, i

HCA 601, 120.

'1

t 61- ft 188

Renherg, Arnold leonard, 556.

,.tA Robson entry 189

add p. 594

Z1 190

Shakespeare, William, 228, 465.

I )-I L191

dele Shih-ehih, Chu entry

ii 192Steel(- Jr., Guy I,ewir (-Quux), 594.

193

Strong, Hovey Raymond, Jr., 556.

Tat'jan, Robert Endre, 239.

Wadler, Philip lee, 594

, last line 196

delete "theorem," (raver one line)

25

197

Werbrett, EhtIengr, 603.

Ww, I)avid Stephen, 434, 595.

L199

Zave, i)errk Alan, 90, 603.

L (namely the endpapers of the book) 200

dlh'te "Table 1"alo make the change Fprclfied for page 136

/ line 4 of the Preface 201

ryltVin IN.s ryct.In1A

1, L line 4 202

forcing himelf V4 being encouraged

,bK line 10 203

onrwer anu werK

~ ~(LU204

raige ti, 'lu~tration about 3/8 inch

F

A! making the quotation format more consistent 205

line 5: The Prince -% The Prince iline 10: MASON (The Case ... 1951) -VMASON, in The Case of the Angry Mourner (1951)

,"_ t new exercises 206

21. (AI25] (G. I). Knott.) ,'how that the permutation n,... n is oblitable with a staci:, in

the sniste of exerrire 2.2.1-5 or 2.3.1-6, if and only if . Cj 1+i for I < i < n -n thenotation of exrrerie 7.ZZ. (M28] (C. Meyer.) When m is relatively prime lo ii, we know that the sequence (mimol i) (2ri moil it) ... ((n-l)m inil it) is a periutation of (1,2...,--1). Show that thenumber of inversions of this permutation can be expresse in terms of ])vdleki sums (cf.Sertion 3.3.3).

, line -9 207

4.588A5 v 45855

1, T lines 5-8 after (38) 208

Curiously ... s aituation to the d\,4 An interf'tuni one-to-one correcponlence between suchpermutation, anl loinary t ree, more ihreet than lhe romiidabout method via Algorithm Ithat we havei usel here, has been found by i). Rotemn (Inf. Proc. Ietters 4 (1975), 58-61];similarly there is a

Z,1 insert new sentence after (53): 209

Actoially the 0 Irni here should have an extra 91 in the exponent, but our manipulationsmake it clear that thi, O would dis-appear if we had carried further accuracy.

Xz exercise 28, three changes 210

the average is \, the iverage lI I,

sort ll1! " for Fouic ohsire reason, g sorting,"2,171; ... "/.) Ns, 2./T. I. A. .ierp and II. F. lo;an have lurovril that limi infii.0o

ln/ViTi > 2 (to appear).)

'7

- ~ ~ ~ ~ , -2---- - - -

,2Z' - figure 9 step D3 211

COUNT [Kj] -%- COUNT [K1 I

' AI!1'l/ addition to step B2 212(if BOUND 1, this means go directly to R4.)

1,2-1 line -5 213

th ie ,rline F.hlmldn't. be b~roken,

1 L .' comments for lines 14 and 15 of the program 214

BOUND -^" BOUND

,.i line 9 21.

(-i) ,nbhrr, 197/4).) 1%-' (19704, 287-289.)

, line 8 216

In, -N Ig:

, Z2..., exercise 15 line 2 217

sulsrrtpts an raperpciriptr are in wrong font

,2."LL2t line 1 218

L LL. line 3 219

rIS V rli

=4

L2i. line -18 220

Slast line of Th4ble 2 221

179 -Nv4 170

LII line -2 222wise, oracle ^V4 flangerous, adveritary

LtM! lines - 13, -12, -9, -8 223

prn~rnncrg~,,~ N~outconmrs (four C11211gc"

AV- lines -7 thru -3 224

oracles".- advecarics.oracle ad~versary (five changrit)

4 -7

z",L~lb lines 7-23 mnust be replaced by new copy: 225

Conistructing lower bounds. Taicorczn %I &showF that lihe "infoartiation thieoretic'l ow( r linil (2) ran lie art it rarity fa r f romn life trial lower liniml; tis lii Ite Irtecu ii ii e ma -rto pirove' Ihenoni M gi-4 uc another way to ilicrover lower liniik. a. ; proofteclinitili' is often vieweil as ft, creation of an admorsary, a rcrniciats iraiig who tnics tomake .alvormcrnic rim dowly. When anl algorithin for amesrging i-vmii're it) coiiparm Ai : 1tile ailva','ary ilete'rinite'; tie fate of the ronmisarimi a. ar to force life algorithm olowni themore dliffir'tt patth. If we canl invent a sitiable aihverary, as int the proof of Theorem M,we Carl eiiire ft.-it every vatiil mierging algorillhm will have to make a rallier large mnsmberof cninplarion. (sonme propte have iieril Life wore onracek' or 'iemon' instead of 'adlver.sary';li it is prefferable to avouil Fsch terms in thmic context, since 'oratcle%' ha~c Clifits, a

oliffer'nt m'onnot at ion in t he Itheory of rcmirive fumitious, amid 'elenmii' appear itt, st ill ailiffa'rent p'iic-s within languaage for artificial intetlijgcnee.)

We s-hall make time of coanstramued adversaries, %lance power ic limited with rvgardl to

tilie oitcnies of certaini conipart-nio. A mrigismg nictlmodl whih a, itier tile influence of acotiutraiii". ail'.ercat m n otim know a.ubout tisp eoiu't raiinte to it mmmiti make tile nrcme-arycrampa1risn even thonugli their Oltomies hiave bem'r: jarril"'.me .. For example, mn our proof

of Trori c i'oiit rainedi A alos"r lay comhitsmn (5), yet tie mincai:ng *Agoriti han, wasuammab1te to make ims of flitfil fu't in ric.r to avoidl any of thle comnparimc'The comnt raumntc that we shall tire fit tile fnllowinig dliscusgsi apply to tife left andright cns of tife fils. Left constraints, are s-ymbolizel bay

~L ~ lines 7 and 16 226

quti.ctionc -v4 coampar .nn;

lie aniwermil -,' recult in

S,11 lines 9, 10, 18 227

oracle dV adversary (four changer.)

S,&.1" line 12 226

lhen we di-rfle litle,

t L,- L line 15 229

, line 18 230the orache * lie

, Z lines 2, 11, 16, 20, -9,-6 231orac:e \,4 adversary (IWx changes)

, LI line 1 232

ORACIE ^-. AI)VRSARY

, line 4 233

its I- hir,

.'LL[Z exercise 10, line 2 234

oracle ^v* adversary

...Ll exercise 23, line 6 295

oracle ^v4 a'vcrxary

S32 J! line 2 236

oracle i; aske!fI" aIIvercary is about to ,lcrile

line 3 237The oracle N lie.

I.AL lines 5, 11, 20, 24, 27, 30 23S

Say ". Dec -de (six chaniger)

51]

,,.2Z line -2 239

"nracle", -^v4 "a.lversary" ar .n . I"!3I 5.3.2,

240

line 13: fending an oracle e conkiructing an adversaryline 15: oracle tirlare A adversary cause

lirA 17, 20, 23: orarle ".' adversary

L,. replace the eight lines preceding Table 1 by: 241

may '!p xsuliet in luriber jimrnvrmnr. The art tlhat V4I(7) 10 -hows that (11) is alreadyoff hy :2 whe.n it 7.

A fiwrly gnni lower bnduI !nr the celection prni.kn h~a, lenr obtained by Davidl C.Kirkpatriek [PI,.). threcc, II. of Tnronio, 197], %ho rnnarutrtle a.. adverary which provesthatV'(:a) > ' - O(j<t-2 rlg((n,2-:)/tt-j))l, it > 2t-1. (12)

Kirkptrick hac ako ciahlrhrI the ex.t lchatmr Orn t-3 by Fhowing that V3(n) it

r1g((ia-1)/2.5,)1 rl~(nj,-1)/4)1 for all n > 50 (cf. -terlc(e 22).

242

line 17: A. Sclh'inh ace *M. Paterson, N. Pippenger, and A. Srhiihagcline 18: la; iiaveline -1: (12) (13)

1! L-V 243

line -7: (13) * (14)

line -5: V (n) -e' Vt(n)

b, 244

line -21: a houinogrn.'us -N . n ohlirinasline -2 and -1: a hnon;ngenrorm .an nblhvioa

any linmntrnim I, , any nblivinas

,tL! lines 5-6 245

a suitable oracle.] ^.- an adversary.)

"'NI substitute for exercise 22 246

22. [24] (Iavid (. Kirkpatrick.) Show that when 4.2k ( l-I ( 5 "2k, the upper bound (11)

for V.i(n) can he rtluced by 1 ac follnwx: (i) Form four "knockout treer" of ize 2k (ii) ind

the minsinin of the four ina ima, ani ,hrcard all 2k el,,menL of it., tree. (im) Using the

knowno iflnrmatinn, build a inigle knockouit tree or. size i--2k. (iv) Continue as in theprnof of (11).

W,, , caption 247

A inovingennur -"./* An olivinus,

, line 3 248

1972), Chapter IS] I, 1973), 163-172]

QM- upper left corner of Fig. 51 249

ther::' a clot inising on the reco,,i line of the diagram for n-6

I line 3 new sentence 230

A. ,:. Yao ani F. F. Yao have pr.oe that A(2,n) C(tn) c r4.,i andi that .fM(mn,) .4nig(riv1) for in ( n (JACH, to ap.'.ar].

SL;S' line 12 251

16 is in the wrong hol-fact font

, , line 13 232

RECORD (0) V RECORD(0)

__- -

~i..g line.1 253

delf-Ii,a: ... 4.i."%nce the proof of that throrrm iF bring rhange'l in thle r-econdroiion of vol. 2

tiLa u' line 6 234

other 1. . M"h:er P.]

~ i line 15 255

t C.N' to C~jif M> .

~~ lines -16 and -1526

sonTlO -N'SORT10SORT01 -N.- SORT01

~. "i bottoam line 237

~L ~ ij~.258

SoJ""I A' onicminrary formn of the "Soum;'lrx"

E&:L 7.239

I'mne 17: formmaslatell -Ne roirtzc-,7r.httvw6 M-20: smercely -. He.ahionr -^" .pproximr.-Iriy ~drioprmnm.a to I/11. (Tilt Psymo-Ih0ir' of lImnese O1-1n, .:Hniighlto, Ni1ffian, 19M!); Ilumrin Behavior wed thmePrinigdipe of ILost F,*fora Ha't~

~~iiU~ extra annotation on line 08 of Program826

LrA/2. I- LrA/2J. (rX cIhangrg [on)

:4 1U line -7 261

only all ",,- only if all

,4,L I L line 13 262betwern '" betwen antl nutmide the extreme values nf the

= L (6) 263

1 <j ^,4 2<j

line -10 and also line -18 264

80o \ 500

LLI2./ line 18 265

memory. It -N. memory. 'he ,hfferenre betw,-ea Ig Ig N and k, N is not sub-tantialijnkmc N ix quite large, ant, typical files aren't ;ufficeutsy randrm rn:her. lnterpolat:on

new paragraph after iine 14: 266

hterinlatinn arc s .t'ymptnatrally rupermr I hinary qrarch: anr step of binaryrearrh r-enthally rrpL|.rr- the amomnt f earc|i, n, by bm, %hle one step of interpolatinn-rarrh rcsieully rrplarr- n hy / if the key- it. the table are rannlnmly tJstr butel.Ur'nro it ran he oFhIfnl that ,nmtrmpation ieah (ake ahnist Ig Ig N xtep; on the average.( " r rr, 22.)

'Lt '/ replace lines -5 ihru -I I by: 26701 13 0) 3 29 0 08 53 20 49 12 2701 13 1431 52 30 49 09 0" 1201 13 43 10 48 48 49 41 1501 13 48 10 30 48 1622 59 25 'TS 55 33 20Oh 14 0 26 40 4836

.'LL bottom line 268

war '. erim to have been

269

line 11: 1 * 1,2line 12: "rh'riiary, February

-L2 line 2 270

1lhe lam p.rl ir is nearly prfret .illibaheltir order! "VIhr alplialie, nordler in f1Ie Lict part ic rtuh.lnlzally lbet ir.

LL t replace exercise 22 271

22. (A.4131 (A. C. Yan awl F. F. Yan.) Show that an appropriate formulation ofinihrplJton ceareli rrqisirec a-yi . iotically lg Ig N .onparinn% on [ic average, whcnapplil to N ,ilepenlrnt ,niform ranlome keys that have liren sartel. Furthermore, illSearh algorthilin on sulch iablet must make asymptotically Ig Ig N onmparisons, or' the

,'LtiJ line -8 272

A . - A), zince the necerary op.rationc are trivial when ROOT " A.

1,.",t[ line 17 273

Algorithm I. I Algoriulta T

S,.-L lines 7 and 8 274

Clearly eoki'l, ilell 1i41i l iffer'cl ,lrlion'; I"coiii-tris,'ltl n* oliffrent ,!c.letion, ae for each j;

~ 4 Li~~:273

line 6: A fairiy An emrn moreline -7: tome. \ ttne. In far. M. F rnin ha lhiown that 0( ) mi,: -,f tnime .ifficr, ifthe right Ilaia ktrirtiorec are imres (ICI Symp. Teory of Comap. 7 (1975), 240-2441

[ JL1.) and following pages 276

in lte recotil coliton of val. 31 1 intimt revile lte fiibectioii aboutt the Hsi-Tuckrr algorithmto take ietint of the ne~w Garsa-Wachr algorithsm. Meanwhile I conlil have improved myv-iatnirnt of h1it-Tuicker by leaving the external noiler oust of lte priority queues (cf. (23)

on p. 4t44, an unnreescrarily ctimber-nine approach).

'A replace lines 3-5 by: 277

t hat [lie rcrijliing 'unxiinuar rtiltrr weight, :I1ax(uiO,k-l ),trn(k,,s)), is; I% fiall ac osbeThik approach can also lie fairly poor3 lscauau' it may chioom a nolc with very -;m.;l Pk tohe tie root; liowrv.er, Paul J. Bayer hars proved that lte resulting tree will always have aweighitril path length near the optimum (ree ciercife 36).

SLJ exercise 30 278

A146 A14 41

t).IEN ' new version of exercise 36 279

36. (Rf4] (Paul J. Bayer.) Genr-lizing lte tipper bound o, *r:,.orem G, prove that thecoct or any optimm binary scarch tree with ,innnezative weights muslt be at most thetotal Wrig~ht .1 Z q , l*~ 7~i~i time, 11 - 2, where

II fc,[i5o-oA pocrhr hc rert-I cho'rcorats that nminnize the

Sdiagrams (2)20

p-1t extra little vertical lines above the topmn-t nodex (11 and X, rtspectively), forconrirtency with (1)

z Y"7 line 2 281

K~~N K

37

t'.'L( 1J replace lines -4 and -3 by: 282

iuiltrale that tile aver.ge imnibr of rampiartorn, ncilel to incrrt tile NIh item irapproximately 1.01 Ig N * 0.1 excrpt when N ir snall.

,iLL bottom line of Table 1 283

2.8 IN- 2.78

1,Y Eq. (14) 284

.L'ito, 'ine -12 285

k - I i jll-pl. . k ir 1i0-p).

I- 286

lnw 1: I tV* 0.25I 1.01 Ig N 0.1Imc2: 11.17 Ig N. 4.8 11.31 Ig N.3line 6: 6.5 I. N * 4.1 V 6.6 Ig N 3

IL- Figure 24 287

below thie tlird noite from tle left, the I hac a bar acrns, it, inaking it look like a 4 by

t line -9 288

R(P) " R FANK(P)

,'LL line 16 289

RANK (R). V RANK(R). Go to CIO.

Sline -4 290

' vt 1 irld) -\,I [#-r' Atio, Iloperoft, au'1 I Uman, TIhe 'e ;jz nn wd Aliimysis of'4 L ~~~.mlirpit I t, Ill gorithm; C (I~val i, AI * i~. 'i on- W eIcy, 1974 t, Cl1 d . ''r 4)

S, ~ lnes 1112 should be replaced by:2)

re's, IJsi le a,ru' %lir-I 'f. allow~ Owe livigt ilif'ereu,"i- of tinlit re"r I. wi at 'to'- 9% 'such'~t runt n. my Ili, ro;le'i 1il Ire,~cc (invarii -oe;'i tl.Iae:, m~ 01flllar

IMat- jre. , r'pt'mIl I lle ;pr('r ' . ,P II [ ). l.iiiliirica: test': oi H Jkr V~ e bee0%,,!~ ~.. by - I.. K arlton al., CI]C! 1 9 t076), 23-28.

71 Iv.'w exercise 292"

1 ~31. , ) (N. L.. Vredin.) I tivemi a, repre~iittah ori of Iiicar hsts with the lror ty I hat

I tivuv. I tn of e. I ;w item I *.cwCCII posit Ioii nt-I ani in, 1gvii in, Likes O(krq it) ;:m t'. of

t I tle

Y, ew paragraph before the exercises 293

,N-Iiw Yno Ili& lirnviI that tlei .irvi nuimbe'r of iolr's ifler rari'lom ,,.;ertif-tvwli lI hf' ome ilow feaoi'rr ,, ill lir 1V/,ft In 2) 4 OCN/n.), for lo,, v N ind. hit, sn tilez~a di:'c 1Iiii .. m'i-n %T lie appi osiiiam ely In 69.3 perreiit jIm ' l'nIifvrr iro, io

~IA line 1294

100 g long, but a~lways a 0111111 ple of 5 cli racter,

, -L, line -1029

~L~1~ line -4 296

HIOUSE A. HOUSE (twice)

V7 -7,7" MT7m- X, 7--. -- -

-i'L ' Iline 5 297

the ic ow of the' Irfi* I the tree if; tioraurpty) awl that it,; imiide

1iS.2 line 19 298

~~~~follow,.'. o~W:

,L!.," exercise 4 299

there will lip a nepw illijst riti, %illh pritalnniz iiiaiilerl finni I to 49 instvald or I to 55.I'Iie re. ti v , eti'lltri '. will If(:

(20) WAS THAT t(P,) OFBE THE Ills WiI CH WITH TiTlS(4) ON I HE A Ol (19)(3) TO HAD --- (14) BUT (1)(17) FOR BY IN FROM AND NOT(1) HER ARE IS IT AS AT(7) --- HAVE (3) --- YOU ---line 2: 55, -Nv 19liiv, after new illtitration: 20,1,14,...,. withl n 20,19,3,14,1,1 1,1,7.', 0,18,, within

, L. line 2 300

that t thal, if t 2 ,

,EL.LL exercise 39 301

A147 M43

' line -5 add new sentence after "of M." 302

(A prerl.;e formiila iq workcI out it exrercqe 34.)

program line 11? 303

einly ionenmpty

31

LLI

4 I, -16-3041delh'h' Iin' 15-18 (11,1 not really revilcid after all)

, line -1 305

anttally A t,, l1nally

,2 '/ line 13 306

', ,milar 'v weaker

,€tt three lines after (34) 307

prpnit z. 'N,4 rigrpo . lI fact, lero mtah atil Enilre SIrtier,i have mitCeedel inprovtin the dlfi'f (l theoren tlhat doti Ide haqhl, IC: afyinplothcally equivalent to iuniformprohini,, in the hinit a Al - rx. [To appear.)

Z, .I just after (37), insert new sentence: 308

IBy convention iv', alo ;t f(O,O) - 1.

new formula for (58) 309

CN - + . ("-Ihah)12h!) (2 * (&-1)b * (*24(a-) 2 (h-1))R(a,,)) * 01/A).

, line -2 310

until Nlrr,'.; ... 1968, "i untmil the late 1960's,

£"b- 311

line 1: The only ... .11nonjg I. The first pthlidrI iappar .re nf li wort] eem. to havebeen in II. Ilellernitan', Ink Digitnl Comlpiter Systen Prinrilet (New York: M|cGraw-Hll, 1967), p. 152; the only previous n,'cirren'e amongline 6: 1968 .. 4 1907

1 .i

exercise 5 lines 3 and 4 312

ten or Ire, I at most tenl

, exercise 10 313

AII 8 43

, 'L line -4 314

At'> Al t'. > H

,i/ exercise 45 315

-iY exercise 66 316

66. \4 ,66.

E, new exercise 317

67. [A125] (Andrew Yao.) Prove that all . e,-jie,,ittatio iigle-ha!;m/ihg chaemps in the

NeIM,. of exerie 62 -a t, tIy the inpiality C, > i 1/(l-a)). [Ilinu: Show that an

minr'cr-fqi .. rrh iaiker exactly k proh:, with prolZha)lity 1)k < (M-N)/AIJ

, lines -10, -8, -6 318

LONGITUDE ^,4 LONGITUDE (three places)

4.~, L~L~319

m- the ere m,,l ditihn I will lie revocing Scrlimnt 6.5 ag.im, -lelim: Ile matrte,.fl on pn-t-offire ireru. |.Ivii,' more a ti ieIion lt) l ,rnI ley)" k-d irec'', .nd "hil-lr ii ll tle searchprest-,l ir,, of lisik;la.i rd nd W; ,naly,-v by D hi ) t* l n r (cf. Stanfort' CS repoit ofSept. 1975)

S line 13, add: 320

[CA1 18 0975)S,50,9O-516f.1

$ hgs line 8 321

3 (In appear) -"'4 4 (1971), 1-10

, the numbers in (5) should be respectively: 322

.07048358; .00701159; .00067094; .00000786; .00000728; .00000082.

S, ' quotation 323

Alice's Adventures in Wonderland A A7ice's Adventures in Wonder7and

I E'J " iines 1-3 324

So we 11.1y ... (p-I )/2. "

In i'etrril if f i. any olivikor of p-I ansI d a.y livmor of rt'l(fMA), %e canT Fainilarly

dIet.rnir (,i/d) moil f by Inok ing tp lir v~aloa of h"-1)') .,. a tattle of Irngi l d. If I)-)

ha. the prime fairinr q, < q2 <q, ni-I if 7j i% small, we can therefore: romptite it

rapislly Iby fintig the dlgits frot right (o left in it. mxzeil-rulix representation, forralicer q1, .... q1. ('rhi i ea ir ,Jie to R. i.. Silvet.)

S,' exercise 6 323

the 'F in the eipnnints ride too high (twice)

,.t[L exercise 13 326

i "', Iinl, hm+l,

" exercise 20 327

.- innw~ky (in apprar". IN,' Zonnky, Discrete Mloth. 9 (1974), 293-298.

-N- ~ - - ~-~ -----

F V

Z, L 328,g new answer

22. Lin/nJ .. m i - Ln(j-i)/ni * 0 or 1; cirgl it 1, 0 ff Rij 1mI 1 > mi Irol i ince

lh nittirlicr of iitvrri.i-nI 1 (i .nj/.i - Lati/nJ) - LiLj-i)/inJ) O~r~nLmr/i h (r

. (k-r) (k-r-1)) which ran I t ra.,,orm"'I i , .-1)(,,-2) - 1,cr(nyi,0). (I. fUr diereinr und angce,. Aath. 198 (1957), 162-166.)

A- exercise 19 329

!e'rI hitr. 3-7 of utm an~wer.

line" : The. a.rwe.r N,4* (TiF flirtnila.ow ,ld a .ie% paragraph:

Note: A gen'er.I fori,;::la for ihe niiinlJber of wayc to placte inh Ieter {,2,...,u} in Lo anarray whi lh th ip "lifferenr&" of two tableau shapes (11, ..., 111n) \ (11, ... , 1n), where 0 <

1i < ni amil it - In i - 11i, hat lwen' funi by W. Vieit [Proc. Amer. Afah. or'. 4 (1953),

740-714). '11.m nu.mbwr is it! ,le (I /l(j-j) - (il-i))!)

,Z / line -4 330

4..N- * 2.5N - 6. I, (4.5N 2 * 2.5N - 6O,.

E,.L* LO) addendum tc. exercise 15 331

It it intrrreti g to note that Clw,t) - h(-w ,,)/P*(-tl,:), where /(q) ZO O z q' " /

fl< <l(I-q k) Is the grer.lu ml: l l function for partilone il mt" *p) into it partq, wherePj ,1 2 for 1 < j ( n and 1)n > 0 (cf. exercise 5.1-11,.

Z, kLZ1, exercise 31 line 03 332

INPUT-N.4 ^v- -INPUT-N.4

Z, L i addition to answer 2 333

(AIlorithm 5.2.3S lore exactly %rlir) xch'angr% .e rrre 5.2..3-4.)

L-- l" , line 12 334

0,o appearl. 1 (1975), 29-351

|.12.

. bottom two (clobbered) lines should sta-t respectively thus:335

43. At n -4 04,r())- In - r .) ,

1,UX"t' line -4 336

r if in wrong fnnt (,e line -2 for corrcct r)

,1. exercise .3 337

.197-404 .,-. 263-269

. lines -8, -7, -6, -3 333

oracle "4 aiIkrrsary

L Lb lines 2, 17 339

oracle -" ailver~ary

L, i, exercise 9 340

ennOpar=nr.) v- cnmparinis, yet tih proncolurr is not optnial.)

-,:'E' exercise 14 341

hn. 1: fn,.i In , foIu in (,I (n) <

ak ;a,,1 r,.w rntciire: (Krkpatrirk'r advkrsary actually prnove that (12) i, a lower boundfor 11,(114I) - 1.)

LI, T' line 2 342

oracle I aovi-ruary

new answer 343

2;2,. it penrrl when 2r k < 1142-C < (2 r+)o 2k al,1I I ( 2 r < 21t, this prrocdure mtartirig with

1.1 knockout frres of .iyc 2k will yield L(-)/21J fewer comparicons tihat (i), since at leastthis; many of the comparicons u-cdI to find the miimum in (it) can lie "rermed" in (iii).

1,L'.I, exercise 36 last line 344

=,ILLL1 insert new paragraph before line -2: 345

G. RItitI .and I). Stemirm,'oi¢an e t lar , d that rxercl r 3" anoi 31 comine to yield a41tpie snrin. invilid with (it Ig O)/k # 0(n) conp.arimon cycre on k prorr-or,: First cortk lifile- of l,, < rye/., tien inerge them in k pase ,:rng the "adhll-even transpositionmergr" of order k. (Tr appear.)

L exercise 2 line 4 346

)O -,- C8

new answer .347

10. See Pror. ,1GAI Symp. Theory of Com1 uting 6 (1974, 216-229.

L k exercise 3 for section 5.5, last line 348

= variahlr. ^.4 variablrr, without tranoforming the rerorisi in any way.

,:i ./ line -6 349

St ran' ~ Strnur

L exercise 7 line 3 350

80]. ^4 80; Fee alo 1. Giuhar, Arma Inforrmnati 4 (1975), 293-298.J

FI

351

hi.,: -9 (dih~playrl i.ohlr):r;Ns r 0

r2,) r1

line -8:r A r0

Al *'' A0

k>l ., k>0

iner -6 andl -5: the right ,itreec of ... ansi the re.,,lt I the refut

1, L7Ti new answer 352

30. Thi.i hia. been prov,', by R.mirIl Wcn hcnr [to appear].

S.67l" replace answer to 36 by: 353

36. Sce A/C Trrh. Arino. 69 (M.I.T., Noveinber 19T), 41 pp.

B_, L-' ' * exercise 19 334

the foorth rectangle in [ih left-han'l figiure ic tn diort -- it should he extendri ro thattr boltomn line ir at the rame level ar lite bottom of the first a,,I th,;r, rectaiglcr;I, 1 answer 20, tho line following the tree should become:

It may be ,lffireilt to imrrt a nrw nodue at the extreme left of thi-s tre.

answer 30 line 4 356

lefu ,ubtree of that " .e -ubtree rooteId at that

iV. ',VU, new answer 337

29. Partial r-l,,tion by A. Ya: With, N ) 6 krys tie lnwrgt level will contain an average of

'N-I) ,o-key notles, .(NI) two-key ,oirc. The average total number of na, l es ebetwren 0.0N and 0.79N, for large N. [/l(an iforinnticn, to -,ppear.]

,47

4

1,L7 - new answer 338

4 ~31. l ice a nearly baclatirrd Irce, with itlelict inal ticpar ;crci cks for lite Ideeci Part. Plas a1Clarwk of pa. I lintivl bealanc-e factor adpiminlnts alnng tis pathc. lEach inceni con lnes .1kocarcel ci accaccubr of the ljatc'rs)

L LL1 exercise 4 line 3 359

IONiC \-"" TRASH..- v n I\- fir

ccacri". ceew 4centricr All 1.1%1l inie: in:*:;; reaa le 4')-plare pirkcicig Is litie to J. SentFicle h :Ii, w..Io s.icnure th at 411 piaerc don aint '.aafrcre.]

L new answer t o exercise I1I (ext ends t o p. 633) 360

11. Nn; rhinaact eca a no-l % ella ncely ace - itpt% feclt rve wall "forget"C nice ccct tlce km~sof Ih ic anicpl y cibi nc. 'Ii uncle a ccneie, it c.slecl ie rt-placed! iy ane of its terminial

zj . .Iecceacclaic11c1 e.g~., lby sraclccug to lice righ lizciaever pa..cible.

i ~~exercise 12 361

1lc11 31: Ai-irclcin (..I\- th gn grcitm Aij!j!c%I4'e an the pre~ oaS icj- r

lact line: SV

: ,L LL exercise 34 line 1 362

1k2)A) A 1k2(k-1)

.L exercise 34, new answer to part (b) 363

(11) lie Icv 1 /fer'. ) part, of cce(fa'rc to rnticcir vaier c f j itilli 2 Ira i. For I < r < 2

Ir a e.i-'e~k xl e' r t, -frlit). For r %r~c have ZO(knI

S.LdL line -9 364

W 01

17.L

new answer 363

39. Sce Miyakawa, Ytiba, X1,g1ta, awul linoii,, SIAM.~ i. Conipuling, to apprar.

~.'~ line 12 366

ani V Witj, 0/0 1 whis k N A-I, anti

its it-lti . 1,'iljii t~t , I wilt r'.lc a r ei eraiI of i jae- nwrrc, IIoiIig MIike I'.~vsrsAa F-inii'Iifst'.flew .ippraurli to td

ZL5LL exercise 39 368

line 6 (foiurthl. ine oftiplarde formuitr): (1) N U) (t o eac)b

S~ ~ new answer 369

45. Y.Set, I-. ta n a ppet ~ar).

~S~ 'new answer 370

67. Let I'k 4 IkI thenf C F* -k>t -7k an"l )k niaz (,l(-iA-)A)

ii .M line 1 37'

it ~ Il)it I t 111C tlhr proviiity tha.i t a a.itla~mIr tso-udt v: i triie Amrp",1%urt (N-V..) / 'i hreV.

r-q krJ

~ ~. line -20 last column 372

Ik

, fi after Nne 7, a new paragraph: 373

A frw inrererming enu ,i.IemIn with-ii. t C irnlnnm riner have ari',, in cnrIer t e inn with theanalyiR of Inr lltg and crarchmi,: algnrith,.ws; .I-,igt alucr of thire, eons.tant. appcar inlthe atiqwer in xercriFer 5.2.3-27, 5.2.4-13, andi 6.3-27.

7/1iL:, left column 374

ilet (/11 J'V C eI(/)

E ,,1 IUL line -2 375

Z.':LI / definition of factorial 376

1 "2 "- n " , 1 2 ".. n

, / definitions of x lower k and Stirling numbers of both kinds37

line -12 378

.5.1.3. I\0 5.1.3

Atlhrr.ari.e, 20-201, 209, 211.-29 220.

.7'.I" L Aho entry 3.90

adh1 p. 468

___--

Ria.eil, Gerard, 0.40.

71 .2.L 382

Bayer, Paul Joep, , 0, 150.

I.Vi LL 383

Fell, Wah , 52.

FIihrn. J.1011 Scot, 680.

10 2.L Fredman entry 3S7

•stIll pp. 12 ), I'l

33

(;rac--rh 'Nl Gracrilh

, 2.LL Guibas entry 389

alt pp..52". h!2, 696

M E i390H i r a Ierbert 541 .'

!i f1. ir..nrii, 687. '

drieg. Iac eliry for "llninInq-nri- , cnanpaa..nss"

,2kTL Hyafii entry .93

d',, p. 215

, , ' L two new entries 394

if| [Ik treeri, 468. tfr, lit-IjhnnrI trre, 468, iee Ralne-,i Irre.

*' 24. Knockout tournament entry 395

a,d pp. 214, 220

',4 '2ItL Linear list representation entry ,96

468. N 471.

,S.' LL two new entries .97

Karle111, Ph1dap |x'-ic' 168K rkpatrock, I)D,.il (;;rr, 215, 220. 636.

l.,oran, lRenpini Franklin, Jr., S913

Meyer, Curt, 22.

400

N ,yk awa, M aali ro, 687.

5,_ t L 401

Olivioiss algorthim, 220-221.

W )- ."' L402

Orach.A, 200, xre, A1vvrsarim

7,,"/ 2L Parallel computation entry 403

add p. 640

404

llafvrnn, Mihael Stewart, 217.

L F L 405

Piplwi,'ier, Nirhola. joln, 217.

F,'2 L line -1 406

78 - vi, 78

71 407

Rot in, l)ornn, 64.

3,')

40

Sherp, Lawrene Alan, 594.

Sline -7 409

223, 223, 405 (execise 22),

~IL~ 410

Silver, Roland Lazarms, 576.

:?,i""WU Simultaneous comparisons entry 411

add p. 640

7Y "ll L412

StCV4rnn, DaVIdI, 640.

.',WXLnew subentry under Sorting 413

hi:sIori of, 382-3118, 417-418.

414

415

Siugitn, Yncin, 607.

416

Sz.cnerieli, Vndlrc, 528.

417

'Ipe arrhtnp, 41(10-401, 40.S.

~ 'Z.U.~line 2541

~law~iii Av~SItwomir

'C'revc~nrt, tee Trep s'etian ort, IfiajwOt.

'tL 420

STurski entry 421

Wkidyh~w ,.,4 Wh~ldy~Iiw

~ 6i Luirnan ery422

i."Lnew subentry under Thie search 423

424

Wrcqii(r, Rnwil, 614.

117 1:1 " 425

Yan, Fnoij, Fraricr, 232, 422.

' Wrench entry 426

at1, p. (6

, ' . , , Yao, An6rew entry 427

ad lip. 232,422, 479, 549, 678

Z• , - ', L42UO

Y uha, "'ndii1ugiu, 60;.

1a ,429

Zeta f inril ion, 612, 666.

,.'~t just before 2-3 trees entry 430

545 re;,.b, 570.

,"Z' .'i (namely the endpapers of the book) 431

dehte "Table 1"alekn chang, I in italic I in windiemlwr 35

t',L changes to MIX booklet 432

16O, Fit,. 3: Step P3 ,inidlI -aiy "500 fninl?"p34, Fipl. -1: ih.,i raiii ,lnul &jy L EQU 0

143 , hlio 1: 001i/ '\-* 0066743, hli. 2:193,331 ", 133,13t

p04, prbilrin 16, lin' 2: row... haiial I.- row ani l mnunin104, lrololvn 16, line 8: 10 IN.- 9anil hanife "reroril" in "hln,'k" evrrywhere url the ohxriu-ion of MIX 1/0 nprrators.

5-

'LV changes to the book Surreal Numbers 433

p1 ,qIn 2: (4) '\ (3)pill1, lii,; 4 awl 5i, I ittrrilif Inje he nIlle of liclt, 11,14-r:

Q A *-t 3 _ *1 3

r117, pirri I tin 18, hii ma'. 3 awl 4 41ilaId 1w:

571

BIBLIOGRAPHIC DATA 1. Report No. 12. 3. Recipient's Accession No.SHEET STAN-CS-76-551

4. Title and Subtitle S. Report Datejune 1976

THE STATE OF THE ART OF COMFUTER PROGRAIMING 6.J

7. Author(s) 8. Performing Organization Rept.D. E. MUTH No. STAN-CS-76-551

9. Performing Organization Name and Address 10. Project/Task/Work Unit No.STANFORD UNIVERSITYCOMPUTER SCIENCE DEPARTMENT 11. Contract/Grant No.STANFORD, CA. 94305 NOOOIk -76-C-O33O

12. Sponsoring Organization Name and Address 13. Type of Report & PeriodCovered

Office of Naval Research Tena

Department of the Navy Technical

Arlington, Virginia 22217 14

15. Supplementary Notes

16. AbstractsThis report lists all corrections and changes to volumes 1 and 3 of

The Art of Computer Programming, as of May 14, 1976. The changes applyto the most recent printings of both volumes (February and March, 1975);if you have an earlier printing there have been many other changes notindicated here. Volume 2 has been completely rewritten and its secondedition willbe published early in 1977. For a summary of the changesmade to volume 2, see SIGSAM Bulletin 9, 4 (November 1975), P. lOf --

the changes are too numerous to list except in the forthcoming book itself.

17. Key Words and Document Analysis. 17e. Descriptors

17b. Identifiers/Open-Ended Terms

17c. COSATI Field/Group

18. Availability Statement 19. Security Class (This 21. No. of PagesReport) O

Approved for public release;distribution unlimited. 20. Seeuriy Class (This 22. PricePa~UNCLASSIFIED # Y'.0

FOnM NTIS-35 4iO-701 USCOMMk4-OC 40329-P71