Communication Complexity: A New Approach to Circuit Depth

79 Pages·1989·4.312 MB·English

by  Mauricio Karchmer

CommunicatiCoonm plexity
ACM DoctorDails sertaAtwiaornd s
1982
Area-EffiVLcSiIe nCto mputation
CharleEsr icL eiserson
1983
GeneratLianngg uage-BaEsnevdi ronments
ThomasW . Reps
1984
ReduceIdn strucSteitoC no mputeArr chitectfuorrVe LsS I
ManoliGs. HK.at evenis
1985
BulldoAg C:o mpilfeorrVL IW Architectures
JohnR .E llis
1986
ComputatioLniamli tatifoonSrs m alDle pthC ircuits
TorkeHla stad
FulAlb stractainodnS emantEiqcu ivalence
KetanM ulmuley
1987
TheC omplexoiftR yo boMto tioPnl anning
JohnC anny
1988
CommunicatCioomnp lexiAt Nye:w A pproacthoC ircuDietp th
MauricKiaor chmer
CommunicatioCno mplexity:
A New approacth o CircuiDte pth
MauricKiaor chmer
TheM IT Press
CambridgMea,s sachusetts
LondonE,n gland
© 1989 MassachuseItntsst itoufTt eec hnology
Allr ighrtess erveNdo. p arotf t hibso okm ay be reproduciena dn yf ormb y
anye lectroonrim ce chanicmaela ns( includpihnogt ocopyirnegc,o rdinogr,
informatsitoonr aagned r etrievwailt)h oupte rmissiionwn r itifnrgo mt he
publisher.
This bowoaskp rintaendd b oundi n tUhnei teSdta teso fA merica
LibraryC oonfg resCsa taloging-in-PuDbaltiac ation
KarchmerM,a uricio.
Communicatcioomnp lex:i tany ewa pproatcohc ircudietp tIh
MauriciKoa rchmer.
p.c m.- -(ACM doctdoirsasle rtataiwoanr d:s )
Bibliography: p.
Includiensd ex.
ISBN0 -262-11143-8 ISBN 0-262-6118(p8a-p0 erback)
1.B ooleaanl gebr2a..L ogicci rcui3t.sC .o mputational
complexi4t.yA .u tomatic theorem 1p.Tr iotvlieIn.Ig ..S eries:
ACM doctordails sertaatwiaornd .
QAlO.3.K371 989
511.3'24--dc20 89-34499
CIP
To InyP arents
Samuela nd Susana
Contents
SeriFeosr eword
Preface
Acknowledgments
1 Introduction 1
2 DefinituinodnP sre vious War k 5
2.1 CircuCiotm plexity 5
2.1.1D efinitions 5
2.1.2P revi\oVuosr k 8
2.2C ommunicat.Cioomnp lexiotfyR elations 10
2.2.1D efinitions 10
2.2.2P revioWuos rk 16
3 CommunicatCioomnp lexiatnydC ircuDietp th 19
3.1 TheG enerGaalm e 19
3.2 TheM onotonGea me 22
3.3 CommunicatCioomnp lexiatnydF ormula Size 23
3.4S ynchroniPzreodt ocols 25
3,5C onsequences 26
4 MiscellanAepopulsi cations 2!J
4.1O n MonotonCei rcuaintdsS lices 29
4.2 UppeBro unds 31
4.3 UniverPsraolt ocols 33
1.1 Khrapchenbkoou'nsd 37
::; MonotonDee pthL oweBro unds 41
5.1A LoweBro undf ors t-Connectivity 41
5.U Intuition 41
Contents
5.1.2N otatiaonnd D efinitions 43
5.1.3T hep roof 45
5.2L owerIl oundVsi aR eductions 51
5.2.1R eductioannsdl\ lonotoLnoew erD ounds 51
5.2.2R eductioannsdP artiFauln ctions 54
6 Discussuinodn Futnucrsec arch 57
Ilibliography 65
Index 67
SerieFso reword
Thisv olumie st hed octortahle siosfM auriciKoa rchmerw,i nneorf t he1 988
Distinguished DDoicstsoerratla Atwiaornd s,p onsorbeydt heA ssociatoifCo onm ­
putinMga chinearnyd T heM IT Press.
Thea wardf,i rpsrte sentiend1 9 82i,d entiftiheebs e sdto ctotrhaelsi incs o mputer
scienacned e ngineersiunbgm ittiendt hep asty earT.h ea wardh asb ecomve ery
widelkyn owna ndv ercyo mpettiiveT.h isy eatrh ere wmeorree t han n5o0m inees,
9 ofw hom weref roms chooolu tsitdhee U niteSdt atesD.r .K archmert'hse sis,
submittebdy H ebrewU niversiitsty h,efi rsfto reiwginn ner.
\Vritiunngd etrh ed irectoifoP nr ofessAovriW igdersoDnr,.K archmesrt uditehse
circtuc io mplexoift yB oolean fuant cotpiioocfn lso,n g-stainndtienrgei snc to mput er
sciencCel.a udeS hannowna st hef irstot o bservteh ata lthougmho stB oolean
functiomnuss tb ec omplewxe,d on otk nowa nys peciefxiacm ples. pTrhoibsl eomf
findinlgo werb oundos n thec omplexiotfyc ircufioBtros o lefaunn ctihoans s
motivated much reseatrhceph a sot4v 0e yre arsb,u ti th asp rovetdo b ev ery
stubborn.
Int hitsh esiKsa rchmedre veloap nse w stratefgoyrs tudyicnigr cuciotm plexity,
whichh ec aIItsh ec ommunicatcioomnp lexiatpyp roacThh.i sap proaccha sttsh e
computatiodneavli caesa mechanisfmo rs eparatinwgo rdfsr omt hen onw ords of
languagaen,di te xplicciatpltyu rtehsen otioonft hefl owo fi nformaAtpiploony.i ng
thirse formulattoim oonn otoncei rcuihtess h,o wsa tighQtO og2nm)o notondee pth
bounfdo rs t-connecttihvepi rtoyb,l eo_fdl elt ermintihneeg x isteonfca ep ath between
distinguivsehretdi c5e asn dt in aunn directgerda ph.T hiismp liesa super­
polynomi(anlw{ Iolgonw)e)br o undf orm onotoncei rcuciotmsp utisntg- connectivity.
LawrencSen yder
ChairA,C M DistingisDhoecdto raDli ssertation
AwardS electCioomnm ittee
Preface
Int hitsh eswise proposae n ewa pproacthos tudtyh ed eptoh fb ooleacni rcuits:
TheC ommunicatCioomnp lexiatpyp roacThh.ea pproaicshb aseodn a ne quivalence
betweetnh ec ircudietp tohfa givefnu nctiaonnd,t hec ommunicatcioomnp lexity
ofa relatperdo blemT.h e bottom-loifnt eh en ewa pproacihst hati tl ookast
a computatidoenv icaes a separatdienvgi cteh;a ti sa, devicteh ats eparattehse
wordso fa languafgreo mt hen on-wordTsh.i sa llowusst ov iewc omputatiionan
Top-Dfoawshni aonn dm akeesx plitchiietd etah aflt owo fi n/ormiasa t criuocina l
termf or understacnodmipnugt ation.
\Ved emonstrtahtaett h ec ommunicatcioomnp lexaiptpy roaicshb othu sefualn d
intuiti'vVeed. o s ob y
• Givinnge ws implperro oftsoo ldr esulwthsi chhe lups u nderstatnhder esults
int hec orrescett ti.n g
• Provings uap er-logarimtohnmoitco ndee pthl owebro undf ort hef unction
st-connectivity.
\Vep reesnt, ino urn ews ettirensgu,lo tflls e rkowialnzdD unnree latmoinnogt one
andn on-monotone computati\ovnpe.r esAelunsptop ,e bro undtso s omef unctions
byg ivinpgr otocfoolrts h ec ommunication problems waistshto hceimaa,tn edwd e
introdutcheen otioonfu niverrsealla tions iwnha i scehn,s ceo,r resptoonu dn iversal
circuits.
\Vhabte setx empliftiheefs i ristte ma bovies p,e rhapas n,e wp roooff a theorem
ofK hrapchenTkhoe.o riginparlo ooff t het heoregma ven oc luweh atsoevteotr h e
factth ati tst rutsht emfsr oma simplien fmoartiotnh eoreftaicct O:n en eedJso gd
biltso d istinguaimsohn gd possibilities.
n)
\Vep resean tt igh0t( log2 depthl owebro undf or monotcoinrec uciotmsp ut­
ingt hef unctisotn- connecat ifuvnicttyi,wo hni chh asO (n31ong) s izmeo notone
circuiTthsi.si so urm aint echniccoanlt ributAi omno:n otondee pthl ower bound
whichi ss uper-logairnit thhesm iizcoe f t heb estc ircufiotrt he funcctoinosni d­
eredT.h ati so,u rt echniqaupepsl tyo d eptrha thetrh ant os izeT.h uso,u rr esults
complemetnhto sbey A ndreeavn dIl azboro\ ·v..h . oo btaineexdp onentsiiazlle o wer
boundfso mro notone cicorrncpuutinitgss omef unctioinnNs P .A sa consequence,
(nn(logn»)
weg etb oths uper-polynomial sizleo webro undfso rm onotonfeo rmulas
computisnfg- connectainvdia t ys,e paratoifot nh em onotonaen aloguoefsN et
andA Ct.