c&c&ti AlppOpr Cj.$-{or - University of Ioannina
60
Airofte}{vl 0 Ezn A ftaitrli'Are N c&c&ti AlppOpr Cj.$-{or.rz . flo1"/l,lou.^irat=,o- t O, 9 . Ti)e.*- t.-f{** - kleeers - n7-42a$t^ a//o\)e.^f tG,/'1/C - nf " 84"f ^-,"- Ge)za ctz'o lwit"ltz1. , Molalr.F [1lc4te1 , AX?""f ,?'( 6'?s ,.t(rlt+ W t7-l"l,tn . Al)"1+ r iltz{,aau> , ,tlppol. r 6:eiAa4 Drati,*. '%1J , TeIteLa E5e<r-Cer.l
Transcript of c&c&ti AlppOpr Cj.$-{or - University of Ioannina
Airofte}{vl 0 Ezn A ftaitrli'Are N
c&c&ti AlppOpr Cj.$-{or.rz
. flo1"/l,lou.^irat=,o- t O, 9
. Ti)e.*- t.-f{**- kleeers- n7-42a$t^ a//o\)e.^f tG,/'1/C- nf "
84"f ^-,"-
Ge)za ctz'o lwit"ltz1.
, Molalr.F [1lc4te1
, AX?""f ,?'( 6'?s ,.t(rlt+ W t7-l"l,tn
. Al)"1+ r iltz{,aau>, ,tlppol. r 6:eiAa4 Drati,*.
'%1J, TeIteLa E5e<r-Cer.l
o [. rL.or.hc FouqJo.dio,rs
, 6r..pL 6l= (v,e)
Ia3kr
5' =1v,e) anJ 5'=(v', E') a,hc iso,..oryl"ic,
de"olcJ G:G', it 3 -
biie.{ion /: Y +vt.,
C2,,1) ee a G<-.1,{cyt) c.g'{ u1QT.
\Mkr,rcq
)<krrS
A^F&
. No{ e,vcry s^Up.,1L J
Vle d.&.<e ll^t
Gt=(A,Ee),
ut'pt
[clbbA
A cv..io be
r"fnpl i".J,..cJu.rkel-e
Ee = I -2<E lo.eA a^t 7e t l
eG
nI
tfll
lz-T
- t -
o clilre ner..ber wCG)
ttte rru,-.lel .{ verh.cr 't\ . r4.,rlrr..r,.* J,po.{C.
o s{^L;lrl1 qqa.Lct dCG){Lc rrr,,"rloer .{ wHi.es i,. q, :l.blc t€{ "{ u,qx <sd,.r.
a A clilq€. (ovei # .,t. k ll o. f qr+iho1
V= Cr + ca +.---+ Crt'rcL tt o{ C; it rr. Ji1,.^e.
r A prc?cl <- <"lorir1 'r: .r, pn>h ho1
V = Xr.t XL { -- - -tX.s""L Io4 Xi ir o '{"!lc sJ.
j cl irre-covel. Uu*be.h kcC). t - 0 r l | | r r t r .tl!€ tt"€ hf tke s|qq,let+ rosild< ,lirrre <"veh .l-6
r lo ct rou,ealic rrur<^bc; J Cc)
, f r r I A&e. st*lqlfsr{ io>ril(e < (or p,.t.L l,.etr: %s+a- ?to ' -z l ' c- -o lod' i t
" { *- .
-?-
exo-t ?le
o For,'vr7 V^& G , wCG) 3 },G)
/ (c). )ka\; /(c) = o(d) a...J leq..) = /. Ce) .
. L.[ 6]=(v,e) b. o,n- ,"Jir.c{.J V1L,
L(cJ = g jcc) =7
(0 r(G^)=i f (G^)
(?') I (c^) = L (c^) ,r Ae'tf
l"l(G) -- e
t , (e)-- s-|i 1.^a <o.
"tV= [z,sJ r [r, rl + I rl
c-.oloti*trY=lr , l . r l+tz,qt
TAsv
tl
G i' ."lU ferlel.
_z-cr';lqbl*
i l.lers..lio.. Gro?Ls
r Le{ f L. * t^-il1 -1 r.or,rew,ylX :ls.
1l"c lqlcrs.o{r..
hePlcs€hli,^l .nL
/-1 .+
TLe 1..{"}r.ch*,
oh q li.neolly
1i,".) is c..[l.J
AI.FL "{ f is oL{oi<eJ
t"{ ir T \ ^ ver(c"t'
S-f l 3r 4#
nft "l n {*,ly .{ i.[.r^ls.rJor.J sJ Clil- 41" p*l
ba
unrt i\+ehvcl
q',t i..lerval ?t"pL.
rr
r"
o
a fl"F€l i'.LrrJ- f.{t- ho q{eryol. ft?ts"t}
or.r,{ri<S a-rqo {laeh
t lr
-+-
Co..siJer tL. fol\oru.t rclcxqhoq :
i{ uc f.i.1 O" *r" ouJ,
-4 oqr lir.., (l"r-
. lll*,i"1.lo.st
"{
irfepolr urill b..or-c. aFc3 or,r <tla.c. circle.
oL{.',.q
^l l .J
q,|cl {o sli l ovar., b€
i.lerseclio' 1trpLs
(t
ttcirctrlc,F- q,t-c
?ho..?h f .
o €irculaF-o,rc yrfls ?wfarlT .o"k',r fl"<inlervql p*pt"s.
o frcpeF ohcnlcth-q.rt ffht
- f , -
A 1*t^...{"h"ro,n u*'Jn- o{ +ro
lir.r seTucnts
J,"y^"n co'.Lrsls
p"fo,llel li,..r c..J,^^.1.Li"tr(L. ft{r.
o{ n joi.,lr
n s{reiplrl
a
{--ga,t,s.z1
Ct4r7
r t'r{+se..li.t1 .l^o.Jr "'1 a. <ircle-
fro?osilio.r l.l. A{. qJuceJ sultr-.g[ J .^r.rt*vql
f^* ir ar4 i.[,+',.1
T.pL.
P*.1. I{ tlrlr.v is c,. ,.,1.r"*l herrr.lerlelr..r. {
^ pl l c=(v,E). (Lcr,n t l , l , .x i r .r1,tlcrrnl npr.tso.{oh.e I tc^ r.,Jq.eJ s,^LppLGr= (x, E,).
o -Trianl.^lol.J ?*fL
Evc.h3 si.-;le cy.lc
", "L"rJ.
7rrp.vltr"t |o.fl / >: ?o',G'rcr
o Tii+-r7ul*LJ ?*f[! (.r d..*J pftr)
a
-T-
ProfoEil ' ,o* l.g. Ar. ixl+r.l ?^fL
sol,r{r.r
&. hi""1.^l"l"J ?'.qfL fbrr41.
9*"1. Su??ot e G co,a{,r\,"r f, vo, v., -.-, v{-l , yo]I,
l r i t l {>" . L.[ 1r + vr.
Foh i-i.,z,...,1-! , <Loolc * p.',"{ ?r 61,L-Ln fi .
Si,,,.e 1;1 .*J f;11 C. -l
orcrfef , .!4r- pi,"fg
5, <or,*iL[e, a. stri.l'|1 ',1.re'-si"] ot Jccre^si.1
5c1trc'k rc'.
'tLct.lte, il ir rsforible {. fo --J le-r
{ i"[.tt."{, cr,.ly..Ji.hq ([^- orlerin fl.,fV5 V4-1 it o,r,r .).
-L C,
. Tro.hs',live oHe..lohoq frogcr.{1Eo"L e$c c*, Le ..ui1,.eJ r o!rc-r^r.l J,*.ho1
ir rus( q ur.,l tlJ &. n ulli*1 .ricr"{cJ
).r[ cv, r) ,abeF r..J VeF
-) ec€F (ft.e,.etl
-B-
Cr"1tr-s
Pi"p*1.?
orrertJiol
?"f''
,rlt"l- *"ht(6 dlr. +t ..silive
+tc ^llel
.o,^.gonbilit
. froporilio,., 1.3.
4.JD.fir.e
T[e coqlle*o..[ J ... [k*J
tnasill.^e, on,*.{"ho. F?*4.
&. l\!Fv. hepfel. t r G-0,sJ.
F "l E= (v,E) "r fol l"rsrT,^ < Iy (r^X€ F).
plL s.h'fi"r dtt
z'tr < F 44'
hr., l-< T, 'n,ens trwt r7 li.1(* LU -4 15. <lc.+11 lt^+ Le9 i
Sihle f, . T! ( llz. + A.^ 1
4trts. F v a, trrx":ilir^.- oh*.{-ho1
qlt
Ir"],*y L.on er.{ohor
a.{rr..17 i.r :nkr(r .J,
I2. .
.+ a.
-1-
Tl",eorcq J. Q. A'r
i.[,+rJ ?n.?L \11.p.J ilr con..1lc,.-a"{
Er.L J {^ p1t,c.[.r' q^^ { eHA
\,ar",l<,<, 7 = -
,^ l i r-".{"J ar.?[ G'rt q.. ,r{ ,
Gir
Cirhi.^./JeJ T*ll.
a
q. co.-.proLili\ t*?t
.o l. ncJ rrri ur5
e hic.^/e.
e, bc
crc{ +{,i\. I
3
o /-?'e'l.t'l y1*Iy. Ft c-1" iqL.cJ sJ1,. c^'lC
/(c^\= ur( c^)
" ot-?*l"J ?^p*q. Eo c,A uJ.^"aJ "hq+c
/( E^)- L(c,.)
-lo -
g -rle Dcsi "l e4.4,"iqI
c"*.g.thbilil7
c.,.,,?,rl,-bi hl?o( eL,s{s...c , f.s
soLcs jf"fler."
o,JJrcsser
{l.ef e o,r<,
T?
co.. g,^lal, or.f c"n gleri 17
1{531f 4"rl[ {o 1..c{ior5a[*itlu"r t"l.i.L
o A* o/?ohi+l,r.,. lnv 4f rs o" '{c7-bg-rtepfto<eJ;Fc ,rili.L rL"" .ffli"J l. .r1 ivstrrre
"{ 4f 1toJ,^.c.1 e sol, li-r
o Rerrrrile cta ofli...i2.lio" ?roblc..r *, c Jocirr*,fl$|.
Gro-pL CoLri..nrd '
l r , ,r lq,..er Apt"tLG
Quer*.q : wLl il &c suollal
,ru,.!eh J col.rl r..eJ"J {"t., ,e?
4. l}cgel <olovi.r7 wa
G.n1t Gl"ri.^1Iqslqcc: G o*,1
qucrlior '. DoeS
o,i:[ o, fYoFeh
Lcz+lLex-
L?
- t -
.,Ln7 J G
o Delerrair,ri,,rT {lte ce*plexil; "{ otclqiFeS * 4wo - siJeJ c.{lo..l. r
yrcble. 1f
(l) 1L. qTer [o*xJ - tLc rei,rir<urer 6-,glcx\b1.l ott l(r,ourrn- olloni*lrt-s *bi..tr 4l-.
(1) 1t.e lorrer LnqJ {he lcler} f..c.hor/ f* ,h.L il le.rl fccv groveJ (..,oll.t-[,.*lly)
lL^I all |"sciblc. ",Apvithat s"lr;1 zf
4|< reluireJ .b t.,.,ns co*1l.xiJy .rl lort(
"s \lrL *l !.
o Gnp bclrrree,. (rl -(t) + Fcsccrhol
r Ex.*gle t r,t .#tr *..rllit[...1io..,
-g l rasfeq f r rct) o( ,n ' ' t ' )Pa-r t 1r1r13 o(. t r 'ze;
o(rct '6ot '{) k>>
- TX* (..er [o..J h4o{,t {o &le $r lt,lPr"blcq 11
-.$ 0 ( n2 ) t lt ", it.p"Ir, ,ll*r, rr!4,6n!l
-2-
o Tbe bl'22"s1 oper laeshon irvol,,v1 \+. ?oFbc{,"ec^r' ulpef a.J lor,.et oo,^.jleirsl b"u*ll
\q,ofuer ttae so c.ll.J h/P- crrr.Tlcl Twblaurs.
o 1fe al?-<on1Ac '+ o,l? €xgotreu{rc4-Ii,.a-
"ly"r.. tlu 4t< tr,.or,a,r,
y"l +l.c bal h.,er b*"Js Fet 5. 4,n ar-'.
golyrot^,,ry' f...<ho*s.
. 11- e P i( qt*n err:$r o, t"kn^,i,"irhc"
p.lyr^or..'r l-li..a ^hu f* ,L;rLsolrq 4f .
. A v*"J"lerr"i.rishc a{or-i+hwr il o.<e $.r,Lr.L o. :{q{e r."y J.loh.,ike h,c,.? wtl
o."{ *[n',-( {lL*t'ut o" .cA J tL ,^4t
s[o, {er sih.rtel{rhg.r..l l3 .
[ol;r.to.^i nl - {i.-r-
s.b4 ,If .
s{"leS
. 1fe NP il nde erir4r o, Lqler."i.ri r*iC'
-3-
"'l1ti.tl^ ,A*L
. clco,.ly ., P g tl P .
- ] .Y1-
- isTI
I
o Ageu 1,^e:1.o1 ir
d Pir NPirtI(it
{
\
, 1" t{^et^
fo?er
co.r{ cih r..€.. {
?+N? ?
. 4l e uP- c",^n7Ltc r( dcp? V q, p?-L,qv! .
. PcpeJ dr^r- f"ll.-,.^1 rrrlxrdiou!:0l FrhJ ". cor.J;J.4e nf ,'L.L,*,JL+ be n?-<.'&1.(&l Sel.J 1[' fi.*. tr*- b: J [P-c.qLle &r({ sfot^' (tJ lte NP q^.J .ff'<'tl"
Ql ltt <f +. &tA ,.A .
Ctq.ichf,.|
^4'
Tt^coror.r ( ?.11.1. Crr+q):sTAEtE stT ( srrrnre ser ol
TFIAdGEE.FPEE G?A?H'
, ' - . - | - .
LeI o be o Xr.gL n,,cwe cok:bq.( 4ro-
^'tria..lle-{,*. *r^tt6 n.tl. pprlT {^,.{:Kuowir^1 X(H) t",ll
-,fl.
vL
6H
vevhtcz) urJ 'n .*t.
i*rrn Jtrly'2 XG).c-?
uSl*"s*LJiriJe e.A 4u J G ',qlo n ?o+|."llnrtz
H ir h,orl"'1r..0
112v vehhcel
3 rra el1ts.a
Aln, H @.^ k ctqilrr^(teJ 4*," C ,q Okfui.
Fi,".f f), Sir-cs /( l{) = A(6) + vi , n Jrkre,iuirl;cpolyro*.iJ hi,e J2.rtlL u.k;d^ ylret l"rt(fuil yi"lt; n rl,llr.t t" KG)
4-t
ortL, nu" J
I t
- Si\ce il is o,cll b.*,, fl",l s7ABIE sET
is il?- a*plele, q)(e obhiq (Le
lq e* lr*, >to ll.
".*lf.l? r s?ARLE tET oN tHAr'lE -FPEE
G&APH5 is tl?' aa1lelk.
. 4*r.* tzul;"t E tr+cr?') :STABLE SET < 68A?II COLORING.
l"lL"'^1
'l'Z
. S.{.4 N?- coqgle{c- ?r.ablc*s
- 6*1L colori*;
!r:{r,urcc : G .
Qteerhc.l , ul.{ r. /rc11
- Clr lue
hs{ter<e. r G.
S{o,ble selt ! r r+q,r( g : G.
e.^erho.r, ,rla..{ i(
- <(i1*e Gvci
t"rt{r<aqe : C.
t*echo'r : J^4 it kcC ) ?
1.^,ertto., : ul^.o4 ir oCGJ ?
dG)?
o ?er(J Z^ht"s 7 o;htnu.A.,, ?*llo?
'5-
x <- choice(S) creates I S I copies ofthe algorithm, and assigns every memberof the set S to the variable x in one of the copies.
failure causes that copy of the algorithm to stop execution.
success causes all copies of the algorithm to stop execution and indicatesa "yes" answer to that instance of the problem.
A no_ndrterministic polynomial-time algorithm for the decision version ofthe CLIQUE problem is the following: Let G = (V, E) be an undirectedgraph and let k > 0.
procedure CLIQUE(C. k)rbegin
l . A*@:2. for all r e ,/do.,t € cholcr(i.{ + 1r.},,{}):i3. lf l,{ | < A' then frilurci4. foral l r , veA,t + t | do5. lf rx' + E thcn fsilure::6, success:
erd
The loop in line 2 nondeterministically selects a subset of vertices ?4 s y;lines 4-6 decide if I is a complete set. If success is reached in one of thecopies, then the final value ofl in that copy is a clique ofsize at least k. Usingthe above procedure we obtain a nondeterministic polynomial+ime algolrithm for th_e optimization version of rhe CLIeUE pioUiem as follows: LetG be an undirected graph with n vertices.procedure MAXCLIQUE(C):be8ln
fork-21elsaep- ldolf CLIQUE(G, ,t) ther rerurn k:
end
r Arno,lsis o{ ?omllel ,{l ritlur J
PFo(estcl-s , ly jr <"113
r"{erco,."e.{el i1 o,
*Le <oorJl..qfi..
J {t.. ,o*. Iy?.,ccr{oi,< f,rsl^i.., .{. allo,,
J fl+ir cchviher o^,1 ,fu-
{. lrcro.t o,lln*tlwis e''li,.ll.*e-{o.Lor oq T"t..llel
ttr"1 ot T(..) "{ ^ trnllelq! tlec }r...c |e1uircl
rlvc a co...rgulalo,"l
o. p.r.llel .(,'titt *rJar.lct ba
. A B^nllel c"*j..,ler is s,"flg o coll."ko,,r u{
eroLo,^;e J &1".
04r xrr,i.. 1oo,( ir
qlt s,,iLLle. f,tcrr.,rfula{.
fl.e rrerrriqi li.<re.0
^lpnittrr is Je{,reJ
bq lte trlttoriftr,. 1o, l
1wl lerr .
o Fot o. proble,^a -t ,,a. w, \l *Le hqr,.rbeF.
u{ f...s:sF! . relqirc{
\ir .'' {,...1io.r .{ r+, .{Lor ,1 il
P(?') oF PCr,.1.
o llea-sreti,^ .il"e ge>Potu,r..,.n.e J -
( t cos[ : c(q) = 1tq0. ?cn)
Ar:*wrc tt r ^ loroe.r bo.rd ol 9 (/c"rJ i,r 1.,o",,
o,r .f(c ,{a..bg} J "+.?l
}cluireJ i,. ,l!.e,
worr{-cqiq {o solrrc o.re fFLb^ -1, , rr . w.
- If +L. oJl "f c Terclfel"1,Tarit:],,'. is 0(tc"tl,
1l^A't lLe c{tr.ri,t{-< il tciJ {" be
asy..f"*<dl; coS{ og{'rrrol.
o Spce"Ju; : sCt p) = {t(r.)+t ('u)
- A 7."J p.rnlfci ol1orill,.". is ohc f"r- ,l*A
'tlir rrho ir hle.
- l -
6 Effrcie'.tcy : ECr,?) = S
*Gql tl^e r.lua.dirl
tr li,.*. .jli,.-,{.
(il rf E(r,p) 1L , l\etr&c lrntlel "(y.v,tL"ir r..ol <"r{ o7lir.^..1.
(l) r( E(r,r) = L, {Leq &c Tonlfel cl;orilf,-.is coS{
"pli**f proriJed
Jp*tu
A) l ( E(tp)>1, tLer o, (o i ler re luo.. l tJ
Jy*b <cq bc oLlqi,.eJ
b? sir.trlcli,el f,l^e prrlle(
Otr€.
6 WorE , W (rt) n4c4sqtc> {r. {"{.1 rrqrsba}
.-{ "getrlior5 ,.rcJ b1 e. J1"ritt.
- t -
r 'TLe uollr w(r) h.rt r^otl,i*1 .to Jo u,,&
'lle hu-bcb of flo(€rt€l, o,v.ib.Lle.
. 1iLe corl c(r,{.) t",recsqFe, fLte cFr{ ,.1.l[c
"lpriil,- tclq,li'e 1o .tt. vruzrbeF pG,)
"f 16.!51orl avoiLbfe.
S+cps
a Wotk y?Frr t Gs[.
h q. h rr{ ?r?c
Pro<elfo tS
+
(
r
aa
't
t
I
/ , ,
/, /,
/ / t/ r /
,,
//
'/t,,
/,
il,//
//l hry;
-1-
MoJels o4 G*.fl^lo.lio..,
Ro.r.Jo.- Accest Ma.Li,.e Ctru)
E.cL slcl J +t^.(fl Re^o jLcre
Jyill* .rrirl, "4 u? +.) r fla:
(z) corrnrre il^rle
e) WglTg ft"'.<
f{Er\rctY A c.it, uNFr
frrtcsrcts 3
- lo -
. ?o!"llcl P*.lo,.r A<esf Ua4i-e ( PPArr)
MEf,{oEyEtrt trr
liErrogy Accc'l ut,|1
P.(
. (eAD pl'.t:
E
pro(cti.rr (,,ry {o
4r.,." (u1 +. "c)
,c) rc^l
u*nS,..,f"U.,rn-{t
lrtrho,+).
. (oH?oaE l*ra, ?ltcasr.}t ( r'rp lo rt) gct{o;r.r
bctr'< a,li&,-cli. or lol,cJ
otclolic.r, .1 llrit L.J L,{r.
lLn.: ?ln<<r€l, (,^lk,r.) wri{c li,*^h.,r{o (ql {.tc) ,.4u-}a In-,{i..lt.
- l t -
. UUBITE
. Uew',vl Acccrr
o EBEW
r cE EhJ
. E:c(|.,
. c&ct!
r Tlte, C.u, \qtlvuch"'r
0l ?gtogtry cw
(t) Cotrroil Cw
(rt AIBITFTEy crr,
. Bosic Tecltril.^el
- ?tcfix 3qu{s
- fqro,llel ?rc*x- u€Fli|.lt- Cor^r1uti*t F^< fra,Xrrrrr.lu.r
- fqrcrlior ir.fro lr7 lvaeE
- G"r-t-r, Lrll
- <o[ovi[ t{,G r,G].tiea t{
^ fn
- r t -
I
I Bcrsic 'TCcl , r .n i j r reS
r t/e irlto&(€ so.,4- U.ri. lc.l.hilrrol *i "121y
Itd.. 1. o. ,J..{.J sg{ J qo,.1Li..lort*( proLlcr.s,
orLi.L are irtel-*? * ltcin oo.1 -.J
.1U,,
clpccl. 4) sob.prcbler.l it rrtrrac,>oss co"rfq[r[io.r.
(r) ALs Pqrli lio",
Cirq, . -r.rc.lcJ
y€Y we J.f i ,".
t{"c se+ V (rlc
l.eL 6=tv,el *.J o. rcrke
A grr^li ho.1 l(e ,") oltL.ll r^rc tte [cr.. pcr^|.[i-.r
,^,i& rcr;.<{ + ta wtkr
= [ /tco
-Lare
"4 W /'.rt^ c),v o's 6ll.-l t
tcn,q I " .V, osleLr, ls L, <l . . l l
lV;C"),
ir lle.
ale fl.c o.Ji"..er..5-L*l
leuTtL .i tl'* fqFhA"-'
s.JS,*t L,
-t -
t "G.vl .
o T[e .Ji..e,,rg-level :cb of n, p*kh*, tc6,,O
1 t,g1l "=cv,e), €u3 {.r-.1t1 J.li,".J
. l{ G ir o, Jir-.....kf ?.+ +
il"k Svl : fv(v) = o{3 cv'1rv[ ' ] l = l" lu!( ' ) .
/6,t,1, I te'V I dc",a) = lJ
,J"oe Jcvla1 Jd..lc, lta Jislq,.".e y-r.r iq G.
o J(vr,r) >,o , o^.J JcguJeo .1,.r"< vEq.
JCqq) = o4 v1,4'r? ze C<i o..l ? * <j , Lfi .
e Obr1ourll t Lv = r.,^r [J<.r.,a) lqeTJ
"^.Jtr"c") = 1r1 cr.J Ur (rJ =
^/(vl
-2-
. L"{ 6 bc h^. t"l|q ,f""(.r
. TL'c ALr o{ 6 4.c,
$6t &Ql u"bl ItQl t4bl ,{ttl ,tr(otl
3
t
-3-
. lU€ ltS hrate ,ttc 4ff.,,-6 7rc7crfic.r:
Ntj ln 4t")=1 +Lf it@) n ilt-'(il if tf *e/iCv\M(41 n il;-7cvl =f $,^e lre C")
*-tY= llelvJ + ,vr(| {. -- + ltlr,C'l
r TLo Att rl {Cqr), ccr ba cn-plcJ rct.rrrinb4, {.1[."a,
l / " (v)=lv l , veT
l4@lz oJ;c")
"^.J/V6 Cv) = I * I o,..t. E, *e (tcqt -X
af -- tlurcl u fi.-r(cv), 2sL € Lv1v.
" ALs <-,. oh. bc <o*7ckJ \ {,^ Ji*.rceu^a+}ix J tL Tna- G.
- lt-
(Il) FSA g"co,-t"ri h"r
c Gver ^ l*tL G, an, e,t7e. c2.,,6) = (2.2i
J e it .fcrrifi eJ q dll"-r .e.vJio/,)4e rol.h*.Lrp
'4 .lor.J hcilhlacurL.Jr,
cx,N) \s {rn- ',{ {Ea.j= ttEaTC?,f ) ll g,.i-{,.^ it ,Vta] < NCIT(rVI it ccLJ ',{ /trl\ UEyp t' (
o.,rJ XCtr)\vcal{{.
r Wt d"a'L , Fg {L., lcl .{ ln<-r. elytSE ., So..i-f,c,q
A E r, ."c(r.t "r
. 4Lr4r,E(cJ= FE +SE+AE
a
. A dr.F.J;e .4
6 ir e.,llcJ o* A -{r.. i{ crc.r.1
6 ir eZ tl,cl 9:n< or- se,*t- (r+c .
o fu. J/;"e tk ,.torl "( " ,ra.f t e "4 t ll..nr,
o4Lsru^ !. Lst G fs o ri.rf le y.1L. '[^c {^f.l(.-,.,11 st le,".c..6 q^^L epir^le.{.
CI G ir ,. l-k^ r,-fuiGil G l^r, r,.. tn&.eJ suLyqts rror.,.Fpltc
{o ?1 o| Lq)(iil) E *l cotr{cc6eJ r.rJq.eJ s-Ly-ft^
GEsl , S svC6), snkrfia" <o,t(Gc$)lf.
c*,lc c) -f a-el"ltel I nr"l {Cc)l
4,-
(E) G- Occo...l"rilio.r
. lit. J.{iqe
ct7or .1 "+
il{n,rr:
ab y'h.
lLe '!,llcfi"e
"lor. r. ;-*
rrc c.ll [12
( 7 e aCt) , *e2() , l tvqsihvZ
"4 F jar^t i*o,er Et..{o
inplic-(;orn olo.lte.l
l- .r dtc
C;=(w,E) - l
ei l l€r 4=o/ aJ t t ' f e
ot lo=b' aq! xt'(e.
d^c bit,rfT
.qJtrc, c&.J
*(ohor
f'4'L
trc s.tr tlerl o&xb I at,l'.
<bf lbt ,UI
bbl.eis inelbri4 r
Jr r."{11(oa.es a'bt qLeqll,,ci
a,=a "4 bb{€
(',rnlla.,. *( aa)Sircs E
loclAtrf-
',
Abfatb;
,'h]Je.
-1-
o 1L"r ut*ot ab utJ .A qFG ir fire rqr.,<6i-1{ic.lior .frrJ \U tk .. e*it+,
, , I l|.|f .vanz c'l | 5t, 4 Scrqg^. et,
/+ orrt
ab= 4bo f d1f, r . .. r6b; ; wtlL l->o
o gv,eL e lalua.(G l, c,"lleJ q FCLqi.r 1..-
ab Jo cJ, nJ wG En"l tlt( xb {.vre.5ct vlcw+v<Y 4b ft.J.
" lnl 7Cc) Je.otc ftla. ctllechot J
i..^plie.|tc1 cle$et "1 G, W. J.A-+
,1t"*<
f iq=Li l t r lcc>yA= AuA'L
. iiLe u^o-bqt '-4 iC"l ahe a,tlJ ltAcofof clarseS J G.
-a '
r Exo-r^.ples: L"[ G n,! (l le tt^ l"fnaryL4'
.'lL. 7-at^ G hq etglt i,-pfrerltor Jsa:
l-.Z"Jl , /s,lrr,, at,aeJ , tt=ltr.,Uq
AiL--lra,J"t,.
={A, i , , i r ,
AJL.I*1 tl A;4.L.tdi
o o, tb o&"r lr.-rt, lt lnt orh nr,
i.,;fi.'1r.1 ,luSt
l1=f tt,"L,c!, "1.uL, Ll, bt, f,c.Jc,L,t",hy
fu.laLl
At .lt l
A=t '
So e,e L-,c lCqi
Aq I
Or,"t I
-1 -
o Lel G--(v,el b. *r .ar4rec+.t V-7C.A g.*i ho1 J tl,< ely *I
i l*-- ' l 0,€G) = tr+
6rr b4r+ -- . + 6r.
ir <^fl.J a. G-J.o,^,,',ri{ior "( e1e) i4
81 rt qn i...,,plic.,fto., .lt) "+
{"n "tl i--L,4.-., L.
o A rc1uo-ct J "!ft
Etrqr,4?.,-..,drtr1il c*lleJ a Jccor,.76j',4o.t F.lrc..r"e {.r. G
il tl,er air{5 (b G'Jecnn'?.,ri fto1
E(c)= ir+ t.*--+ Gr s.hrfyrT r;)x e Bt{* , l t iv! , t r1-.-rL.
elO-
' AfytirL 6 -Jaq..rfosi dtol
l - .p<{ : G. (vF)
l r i f tof f ; i - - t , €t .8.
J. lrbi*r"trll fi.6 nt eJ1rr. eL,4;?e e €;-;
1. €us^cro,k dtc i.yl <1.s5 Bi -{
4 6r{rihiyer2g
?. O.Ar.<. Ei+t -- E; - ttt(. I E;+t = y' ttc.t lel L=i a,! S4"f ;
o(4|u\se, i= i t t etJ yk fk? t .
--+-a
o<C(<b
I Ire \l-l
Je aa
a
a
+-.a\L
8t
+<e
.t/\a
81
-tl -
B,
(!Vf E*r 0.co,..?ori ttor __
o BF: a^al OF, er.c. trro V*f\-&nverlr,(
r^,.lr.Js
d^.4 l.-<- Lo--r {-."J {o bc. e(leehrt
ir L.*Jli*1 t,.'.rl nt-:lq..yc{L T*,Ll*r.
o lLv**er, ln ,]A.rnI p*-llJ i-ple,^c..{rrlic.13
-1 t|lc,r +,- \' ell-h l,-o-r a{ &:1 p":..{.
. W. i.,&- J".,c fQ. k.L.i1"e -1 "-n Jeo-1.s',lto.,
41"[ 4.r h* o" e((..*{ g,tlrtLl n/r.r.h.rnr.
. 4". ear J.-,.np.rilio1 ir elrc..hcff; o.1 orJercJ
prrlihon -1 n w( ELe) irfo 9i,.?f< ?q&s(.rL,J* ,...1.J. si...ple <7.ler ) -
. Lc.l 6.-(v,E) be a,. ,r*.Jirc.feJ tt.7Q,
or fL lf l -- ,r- -rf lEl = v-r .
-l1L e
A1 ea,F Jecohposilio.i "{ G shrti.; .ritt Po
i5 a.r "rJ*e.l prFlthr.r -1 ttn tn+ ECG)
g=?ou7Lu.- .o7*
t"o( 14, {o, a&, r s! e h ,
Uef"r; {o ?o V ?t U - -. U
-1 qLrc irlc.t r,.,r|' rnarti cer
?i, it 4 li.- pte. gof{^ trL.r. et Jrir{r
. €.4 si..7fc. ?.fL ft it c*tlJ ecF.
ll , 1., enA Llo, ?; it n l a,bo-1otilic.1 tt c*lleJ e1 olht
?i-1, Lu.* h.€{o
&er.
cyele, ltt-
* Lrr,q-tih"t-
a
-11 -
o TltepvttA t A't ,r.Jrc"lcl tr?|.
G=Cv,E) l.*)
dr! eo:? Jec,o*Tosihoq tl{ tl it bri{eles!.
tLe fr*?\
a b,ag o1 o?det Co.r Je.tay,9.filior
( ii is b,rn.+,e.1+J. "4. -ll"vr(
, Lel G L,.tE a... o?c.tr *r Jc.cot"yosilio.l
' E-?o U Pt u. , .u fe
esl co.l + Sjalrri,,rT **e I G.
o 4het<(orc, tle nu*bet ol 6nrt it ?"'{4o ehc, hren beF (wr-rt) +t ol vror.ln-eJXes.
. A.1'*,1$, lg,r.rcvir+? o, orbil^r7 eQo to-^
o llare, Po it a. si*y'c Trlcft tt a si,-ple. f A; cal7oi1fu s fo?a l! o r inpl. f .A; a.JVzals €fcU?t
- ,q-
o 0r^ ftLe ",tl.eF L...J , tee f.",r, {t r{ a, c,)r.le
bc3i, cc'\ bc tqcqlot Qto^ c.rt a.rli{to;r1
sfqhhiql lrcc. '4 G b2 ,|1.e ao,., 4vez elycs.
. llir obrernl.o., ruTc,clr nr tlf.*y *e[.1{o oLlri,a a,.. ecl. Jecoarfcri li o\.
- Wc InLcf edlL '^vttt ee- etye e= ( tr,v)
4r f.l[rs ,
. f erel Cc) = leuol "( +Le lo,^,erl coh,ru.orr
qcr<cf+oF " f
l t a\ ! v.
,^tl.nl< 9(e) it tl.<- g"n o,f nuL\J*.4 e. *tr t t 1 s(e) 3 rr4.
. Inbel (e) = ( l"".f ce) , s(c))
" 6r otL h* * t, 14 lob"(c,1). Le aln N4_t*lael. J *? u^r,,,"tr^+e 4- t'4r.'re .y4e-a"-{'i-J
A"' _lr_
LrLal(o, tt)
(o, to)
'Ct't)( l r t )
(o, tr)
( t , ?)( orto)
(o.ttl{ f , t )
{o'e)'(', f r)( , r7|
1ori l)
ete3
%?1
@
€1
€2
oC11
J*"r"rk t}€/q4t.
u I peT t l"t.( c7) = l ' l , '(
- l(-
o €xa,taplg
(d l ,r P"=l<]
tv) fulcr - Toqr TAcLnt
o L.{ T--Cv.E) be, q 7,iYc^.,{"cc' n l lrl
Tt (v,e') bc. ,t{e Jirc.{-cJ /^?L qL{.',.rcJ
lro,*t T &o. a.O^ (a,v) eE is r^.pl,.ceJ
btr {u'o 4r<, 1u,v) a.xJ (v,u) .
o Si,a<€ hJe6t<-c- (u) = ou(h.y.-.
#t^e T' + 'r' lr o,,1 Ller"ia.r,r
F. lcr" cir<uil ol -l | cn
"y'i*.f 7"roilel co*7uhlic,r
lq,..A.,rt o 4 -I'.
be
a
(,..)
t*tt'
urcJ tor 4e
u'nil{J
r Le{ oJ;Cv) = 1uo,ttt,..., ult T
'\tVc JJv" 4Le sacccssor '4 oz €(Fc
? =4(t i ,v) o;5, {o l foos:
E(1ur,v)) = (,v, t{i+r) r,^-r J )
tn, o: i7J- | .
-t? -
o Er.r .p le
4'?,r) .+ < l.t> a 11rr7''s 1c'77
23f? + ZL. | ) +(t r7 lq17,9z-. t
14,11 + 14,1>+a1,3> +2qt>+
leg> *1s,1) -> 2l,t) -t 2a,l
<t.J>
t
.t
Icltl
t,rr,6j ,q, l
Itl
,
t rzt I
I23tlrc+t
/1,1>/1,t)
ltl,7>ItrrT
/?,.1>z7'11z\\ )at,r.,zq,t)zrl,r>lq,tl(qpl
2rrtl2t?>
24r>1l,ll
4.1,"t>
zql>
17,ttl
zt1rz7
1qt,t7
Ll t '7
lal,t l
z4ul.4t>17t>/c,l>c?,tt7
ta,s>
l ' l , t )
/ l . t)
.f,,
- t t -
O . ilthttaut*t b"yG-7t-aar(1.
v
C;,wre.l< I Ca*t 1ou.au*l
. A ps",^J.for..{ ir o J,r-c$cJ 7'pL i., 'Lt.Le.d vqkr l,a *, *r,lJty-cz l"rr tL-., t
"yJ t" t.
1i--- \ ' - ' - ' - ---- '
A. o..Lihorl (,*rh..., U:1r---r1r J.{i."1 q
pr...Jofo.r,i (\r,F) , 'Lare F'[<v,D(v)) l"ev\,t l We cr,^vqt? \s r^-t r^ccettcrri ly *r..c.
ll/c : Gu,ae.le,t 6,Vnath
ltrx,) , Bicor.e.kl
t .
.r Le{ f Le ahe J; r.ahr J G-- Cv,e)
; c:1r-21f : C(v)=l t r i r . r I r r , le(u,v)=t ] ,*J
'rl v il crr E la{J Farlsx .tla*l
c Cv) --V.
r Le*,,^u.: Lel 6=(qE) b" ^ y^yL *A c Lc&'+ ('.nohoz J.f,qeJ p)^.v,o,*rl-J.fu*r, c J"];r.+tn pr"nJ-(llr.* I tt"-a p*thhol5 at--Tr+1fil--+n;
'Ai+a. V; is tltc t4 J \q\ce, \", Ti "1 9.
a" 1.1[.-2 .l.lq l.lJ,L. All ve*a."t J V; il^) 1o tk lo.,,-c ce.i. G,qf.g. €^L "y.b in F e;ltJ is n r"l.l-lo.p
ot a-lteg .rc.lp &a"-o arcs.
".414 .7rL
-I *L &tc- T iz # anfc,i--1[Le sr*.lles( vqkx i{ Ts.
a.
. e--c% E) ;
oc{"711oc/o*)
o--- - ---.-oLZ
nf 'J2
I 'tg
. 4'114. Tl.alotor^cl f :
2
. fl< ,lrri,.h*,t J fL J,r"c.FeJ keett
(
f
t
!
a o6
Q"
( ;
")
7.
Mrr.rwt -tt
Spenni*| -Trce5
c Lewrws Le,I 6=.(v,€) Ve cr *ar6h+<t ,y.1L.
Rr eool tLeT, lel Cctr) €T le s*4 tlol
(q, (,tu)) is tte va\urwtut<tt-t'catL+ "*o'rurci Jc,n I o'-:. t+, -(Le,n ,
1. All tt- 4q (u,1cv)) d-t 4. tk lt4sT,
Z. IL Nn Ch". 4 JJ,,*t a ptutJ-o /''<t+ fsuoL ttJ .-rL J;rckJ tTnR- L4 a-
oyJ. a'.k -7t .c^(9 1"- a'tc1.
3. L-l 1fr,Vr,--tY+ be tL+ rcrL<t s€'/ t '{
dl*- twel - lr,Tu ,--rT+ e €.
For eJ i, lel ec b. dV- t*tit+ttttttr-,*i/tt
eJTe (.q+eeh+ & vcrki i,, V; +o a.vt*I<p i r T-Tt, l - .1- . t .
@, il/ rt- .4.t €; L'/"tt lo rn tttr
-+ W b,,(L G., ( .
. 411 \tyu/. yr"(L G=(v,E); a,.J ?
n fu *qrJ
c,
.L@
Cl)
q
@(rJ
=tl.. 4l*" lrarJofl.ros| (r'J i4cr"hor)
C}
( ' )
. O(bt'^)oLi'/ti")
.d-
+
- Bicptr.<e.te,J Go,-po,e.,. {f
. c,+{pir,rtJ
. 'TL.{orurl
,,/.eq
a,lle"Ii.,q6 cycle
QFs hee -{
G
a
a irlerr.l uelc u.
loJ F
. b) T= ({, Er) b"' o s6,.. tt e "/ C.
E".L.y.(e
et6e-Ca
e€ E- Et
"Jle.J
lc . Ila'rtl
cfc.r,fe.l
-lo T.
eeE-Er1{". G.
bn.sis cy.le,.
a. trniltee
E-L ce it a*lleJ L
-(!Ch-q-= ewe'cx*t7n l l ' \ ,4{h.f(h-7/ '
covty.ec.let cs\.nh
' o(hq ,(fi) cPEr^r
, 0LL-trn) o (S) EE€wt0" '
. D(ln\ o (4++a) c}1<W
, frtuve rfl.lorttr* .. G ) d = cl
conncc-le,! etpy.t*l> iY G'
' (oh q1rou$ru!q
I
Co- C.s'at1 ecieJ
. o.rdr"iqtl - bl..lr
G:A'-)
D.o tV"a- o
n.q
4 yl.&t + G
r Ce,3)b.l.*1 {o c. c€t^a,a..o.l
+l; J.{crr-l.ecJ Va, rr.".r{r.+<- 4y. :
\-'<.-:
G
a
. 9gl"ho* ?-. ", E
"Jtr, ,7
sgtq. i
1L. rc{L',i,e. *v.nsol 4,. c..rsir{J Jlot
0)Gi/ tkrc >roQ
..,trot,ettrer
t ' ,t
1,, t,,.-,1, e. l
-
o Ler,r t "^c: L l T L. ^
sg. trc<. G,o"^I b{ 4. L. ,h. r . lohol J**L p' .v4.
4L*, nb -- Q: , theve. 2b rr ,t'|..e
e1,^',v^f err.a14 G.
ralc4iol J-4,"1.1 die, tf-.[1
r='l(cXlve, Ay,<ote(ttt a,'.ttr.+rci live.
-Lic,l^ eit{€}
"=3rL.4c..l
, o?
elill
sQ"
\Q"
i l i , . .Lr ," ,cc'tl f.rr'5 .'{
-A:eJtr.t (e,y)
E.t*
\r
r epivJcrl.e Ea,
. TLe J G c-o,r be. J.{.rr-tq"Jbl..Lrd({oo r :a,t
L'l G' = (v', Et )
Vr = .tLe r.{ .{
wLev<,
E -{ elVct " t G
(e,7) e Et a+ -8.7.
acr
. 'fLe corauec{cJ cou.,psna*lz ol GlC-eb"et?.4J -{o &l^e eluivrlcrace. Jqi,i,el
.l q2 , o,-! L..lt*.e u.ni1ue,l.1i l.nt {y tLe bl" .Er I G .
dcr- €?o
4
1.
o fu O?h,^t(.9"wld Co-vvuaAv'\ &lyr'&
. Le**o- l: L"l G aet u-^,u) tr 1h.il v 'fs tL.e vu^k^ -'l ntn J1fee. n*7GE Ntvtl hns t<lru (1,-^ {zrt" velkcsl.
GuqeClFJ coqr"u".f
St,..<
0/*Ln-)
iv t G;
iiJr'<a*r<ckt ronf,
6f k-ao*p-@
N t"l/N
@si"r.e drcr) r-.i.r +
T +.dt ?z A.Je{r) o
T4J"(rJ < ZAI(.:.) Lrur 0t XNc"l la'{E"
v+ @.*2 -:-
I
l r .
. l-gva*,*z: LJ A =vCc') a*'! B evG)
^t lnB =f (Jrr ;" i ' t )
a*1tv\<- \er*q
-t A b"f.g {o
dlc S..,.e' CO-Co'tph
!o L lb .e)aoa .-+ E.
Tl M har-rb:l J $ts -{ G ''i lt. o..r'
a..lp.i..{ 1.,r A n'..J lt^ 'tter i'r B "t
lcst +l /a*, l * l . lg l
rt,.+iu, 4 *ua ..4 H.2 $
A - a.'r*nos4 'd^c'r,fudk n-"
ohAJte-
t ( .
o &lVri (W ?nr-Co-o"4","<h
skpl , a*f.+ v w',tt, pnik Jq* i1
tnsky z : t( rnr < r.r -l ol J6 6"1 =
aLaa4"r e*L ue v CGl,
@-cbu?Er.J +sl.?.
, Is-t€f 3: C-owgu*e
- ,(-!krr^K,t-
G *rp+{
G,
'q pJlel
GE uc">)d arron C.) , **e?frc,D
corue.!+,! aor-7'-.h ''l Zt*{)
o
&v)
U(w)
|-g -tg'12M01
dz.
5q q ' For e^cl ,* e NC") ,,\ G, b Iq f*lblco - co'npluf o r tt_g_s u' khyS
IJ
I ote ee-co"q'u4(
'{ cep.u>l
e4 u-e G , Jo 11 g-nlU
T4 wv f, ecc\ , t,e. ue U0)+1e*,
c.o-cn,1f.ul -
v )
elrc ! ,.e ttL'll
t€ ,L;J* & U*fial Jotu)*Jer*nCr) < h-L{1,'g"'-- r^"hk +t+- iz4rglhhvg 4 u;
SLg r: For
l r /
) a.L
:@6
1=r-:]\,r,..
,l tv* r^apa,lcarto H.,c
iJ be.*e,J
<w
*1 n
/
/, 't@@ -@4C*)+-v;
{
\ W s^'<- <i-(r.f..#
t>.
Le,va s'. A vqk.'.1 w€ [t/Cv) is urol -nJlncotl{
b J las( oue vqk)( lq- v16l - ME')--HQI' \1{ dntn) *dGcyon(u) < k-l .
pJ' Lei #- xurrbcl tI u.itLb*le{ rr ,t;cl UlUgf,o il(y'l a,-l t{(v)luO
l. =,
e:
tbq, . le*12, ATcpc,i l(ul = lrcr) l- lc - I , : t t
dea) -- E+ L +t Cl.r\
{-$r M(')\ J\ \,-/
/ r \
(lucol-r) - r
4ta,, , It<, cn.Ji h ot,
J6 (.") + J 69r",1: (" ) <7 L
i t oyti .J*[ 1P
lNt t l4, .e. < u-L 4
11u-r l n/ot
4. l^* ("1/"n1, livtt : h-!-luc't1= | Ucol
