Algorithms From THE BOOK by Kenneth Lange
July 6, 2022 | Author: Anonymous | Category: N/A
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p h p . d s c i / s n d a r u i c / r g i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` a s ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . 8 4 ? . 4 4 . 5 ; 4 . 5 2 4 i t ; ? / 8 4 / 0 ; e ` e d i n a w i E
ngirbthlsfril Dngirbthls fril
p h p . d s c i / s n d a r u i c / r g i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` a s ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . 8 4 ? . 4 4 . 5 ; 4 . 5 2 4 i t ; ? / 8 4 / 0 ; e ` e d i n a w i E
p h p . d s c i / s n d a r u i c / r g i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` a s ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . 8 4 ? . 4 4 . 5 ; 4 . 5 2 4 i t ; ? / 8 4 / 0 ; e ` e d i n a w i
ngirbthlsfril Dngirbthls fril
J`aa`th Ndag` Pabv`rsbty if Mdnbfirabd Nis Dag`n`s, MD
E ]imb`ty fir Baeustrbdn dae Dppnb`e Ldth`ldtbms [hbnde`nphbd
p h p . d s c i / s n d a r u i c / r g i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` a s ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . 8 4 ? . 4 4 . 5 ; 4 . 5 2 4 i t ; ? / 8 4 / 0 ; e ` e d i n a w i
Mipyrbght © ?;?; oy th` ]imb`ty fir Baeustrbdn dae Dppnb`e Ldth`ldtbms 4; 2 8 7 6 0 5 = ? 4 Dnn rbghts r`s`rv`e. [rbat`e ba th` Pabt`e ]tdt`s if Dl`rbmd. Ai pdrt if thbs oiij ldy o` r`prieum`e, stir`e, ir trdaslbtt`e ba day ldaa`r wbthiut th` wrbtt`a p`rlbssbia if th` puonbsh`r. Fir bafirldtbia, wrbt` ti th` ]imb`ty fir Baeustrbdn Baeustrb dn dae Dppnb`e Ldth`ldtbms, Ldt h`ldtbms, =6;; Ldrj`t ]tr``t, 6th Fniir, [hbnde`nphbd, [D 424;5-?688 P]D. Ai wdrrdatb`s, `xpr`ss ir blpnb`e, dr` lde` oy th` puonbsh`r, duthirs, dae th`br `lpniy`rs thdt th` prigrdls miatdba`e ba thbs t hbs vinul` dr` fr`` if `rrir. U`y shiune ait o` r`nb`e ia ds th` sin` s in` odsbs ti sinv` d prion`l whis` bamirr`mt sinutbia miune r`sunt ba bacury bacur y ti p`rsia ir prip`rty prip`rt y. Bf th` prigrdls dr` `lpniy`e ba sumh d ldaa`r, bt bs dt th` us`r‒s iwa rbsj rbs j dae th` t h` puonbsh`r, duthirs, dae th`br `lpniy`rs ebsmndbl dnn nbdobnbty fir sumh lbsus`. rde`ldrj`e adl`s ldy o` us`e ba thbs oiij wbthiut th` bamnusbia if d trde`ldrj syloin. U`s` rde`ldrj`e adl`s dr` us`e ba da `ebtirbdn miat`xt iany: ai bafrbag`l`at if trde`ldrj bs bat`ae`e. Cunbd bs d trde`ldrj if CunbdNdag. Ubs oiij bs typ`s`t usbag th` DL]-e`v`nip`e DL ]-e`v`nip`e Nd`X Nd`X styn` n`s.
_iydntb`s fril th` sdn` if thbs oiij dr` pndm`e ba d fuae ti h`np stue`ats dtt`ae ]BDL
l``tbags ]BDL-r`ndt`e fuaefir bs delbabst`r`e de lbabst`r`e oy ]BDL, dae qudnb`e baebvbeudnsdae dr`ith`r `amiurdg`e ti wrbt` dmtbvbtb`s. ebr`mtny tiUbs ]BDL gube`nba`s. [uonbmdtbias Ebr`mtir @x`mutbv` @ebtir E`v`nipl`atdn @ebtir Ldadgbag @ebtir [rieumtbia @ebtir Mipy @ebtir [rieumtbia Ldadg`r [rieumtbia Miirebadtir Milpisbtir Grdphbm E`sbga`r
Jbvldrs H. Oiwnbag @nbzdo`th Gr``aspda L`nnbsd [dsmdn` J`nny Uilds Edvbe _b`g`nhdupt Mndueba` Eugda Eiaad Wbtzn`o`a Mdnny D. ]hrde`r Mh`ryn Hufadgn` Eiug ]limj
Nbordry if Miagr`ss Mdtdnigbag-ba-[uonbmdtbia Edtd Adl`s3 Ndag`, J`aa`th, duthir. btn`3 Dngirbthls fril H@ OIIJ / J`aa`th Ndag`. E`smrbptbia3 [hbnde`nphbd 3 ]imb`ty fir Baeustrbdn dae Dppnb`e Ldth`ldtbms, Z?;?;\ | Bamnue`s obonbigrdphbmdn r`f`r`am`s dae bae`x. Be`atb`rs3 NMMA ?;42;02;5; (prbat) | NMMA ?;42;02;54 (`oiij) | B]OA 2784644276468 (pdp`rodmj) | B]OA 2784644276470 (`oiij) ]uoc`mts3 NM]H3 Dngirbthls. | Milput`r dngirbthls. | Milput`r smb`am`--Ldth`ldtbms. Mndssbmdtbia3 NMM TD2.08 .N=6 ?;?; (prbat) | NMM TD2.08 (`oiij) | EEM 048.4--em?= NM r`mire dvdbndon` dt https3//nmma.nim.giv/?;42;02;5; NM `oiij r`mire dvdbndon` dt https3//nmma.nim.giv/?;42;02;54
E
bs d r`gbst`r`e trde`ldrj.
p h p . d s c i / s n d a r u i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . 8 4 ? . 4 4 . 5 ; 4 . 5 2 4 t i ; ? / 8 4 / 0 ; e ` e d i n a w i E
Miat`ats
[r`fdm`
Mhdpt`r 4. Damb`at Dngirbthls 4.4. Batrieumtbia 4.?. [`dsdat Luntbpnbmdtbia 4.=. Odoyniabda L`thie 4.5. Tuderdtbm @qudtbias 4.0. @umnbe‒s Dngirbthl 4.6. ]b`v` if @rdtisth`a`s 4.7. Drmhbl`e Drmhbl`e`s‒ `s‒ Dpprixbld Dpprixbldtbia tbia if ϋ ϋ 4.8. [rion`ls
4 4 4 ? = 5 0 0 8
Mhdpt`r ?. ]irtbag ?.4. Batrieumtbia ?.?. Tubmjsirt ?.=. Tubmjs`n`mt ?.5. H`dpsirt ?.0. Obs`mtbia ?.6. [rbirbty Tu`u`s ?.7. [rion`ls
44 44 44 4= 45 46 48 48
Mhdpt`r =. Grdph Dngirbthls =.4. Batrieumtbia =.?. Fril Decdm`amy ti A`bghoirhiies =.=. Miaa`mt`e Milpia`ats =.5. Ebcjstrd‒s Dngirbthl =.0. [rbl‒s Dngirbthl =.6. [rion`ls
?4 ?4 ?4 ?= ?5 ?6 ?8
Mhd5p.4t`. r 5B.atri [erbul mtdbnibtay U`stbag
? ?2 2 ?2 =; =4 =4 =? =5
5.?. 5.=. 5.5. 5.0. 5.6. 5.7.
[`rf`mt [iw`rs Lieundr Drbthl`tbm dae Griup Uh`iry @xpia`atbdtbia ba Lieundr Drbthl`tbm F`rldt‒s Nbttn` Uh`ir`l Lbnn`r”_doba U`st [rion`ls
Mhdpt`r 0. ]inutbia if Nba`dr @qudtbias 0.4. Batrieumtbia 0.?. NP E`milpisbtbia dae Gdussbda @nblbadtbia v
bx
=7 =7 =7
vb
p h p . d s c i / s n d a r u i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L D ] B i t t m ` c o u s a i b t u o b r t s b e ` _ . 8 4 ? . 4 4 . 5 ; 4 . 5 2 4 i t / ? ; 8 4 / 0 ; e ` e d i n a w i E
MIAU@AU]
0.=. 0.5. 0.0. 0.6.
Mhin`sjy E`milpisbtbia T_ E`milpisbtbia dae Grdl”]mhlbet Irthigiadnbzdtbia Miacugdt` Grdeb`at L`thie [rion`ls
Mhdpt`r 6. A`wtia‒s L`thie 6.4. Batrieumtbia 6.?. 6.=. 6.5. 6.0.
54 55 57 0; 0= 0=
_iit Fbaebag oy A`wtia‒s L`thie A`wtia‒s L`thie dae Iptblbzdtbia Vdrbdtbias ia A`wtia‒s L`thie [rion`ls
Mhdpt`r 7. Nba`dr [rigrdllbag 7.4. Batrieumtbia 7.?. Dppnbmdtbias if Nba`dr [rigrdllbag 7.=. _`vbs`e ]blpn`x L`thie 7.5. _`vbs`e ]blpn`x Mie` 7.0. Jdldrjdr‒s Dngirbthl 7.6. [rion`ls
0= 6; 6= 67 74 74 74 75 76 78 84
Mhdpt`r 8. @bg`avdnu`s dae @bg`av`mtirs 8.4. Batrieumtbia 8.?. Dppnbmdtbias if @bg`a-e`milpisbtbias 8.=. Uh` [iw`r L`thie dae Ldrjiv Mhdbas 8.5. _dyn`bgh Tuitb`at L`thie 8.0. Hius`hine`r Urdasfirldtbias 8.6. Ebvbe` dae Miaqu`r ]p`mtrdn E`milpisbtbia 8.7. Cdmiob‒s L`thie 8.8. @xtrdmtbia if th` ]VE 8.2. [rion`ls
80 80 80 86 82 2; 2= 26 4;; 4;?
Mhdpt`r 2. LL Dngirbthls 2.4. Batrieumtbia 2.?. Ldcirbzdtbia dae Miav`xbty 2.=. ]dlpn` LL Dngirbthls 2.5. [rion`ls
4;0 4;0 4;6 4;8 446
Mhdpt`r 4;. Edtd Lbabag 4;.4. Batrieumtbia 4;.?. j -L`das Mnust`rbag 4;.=. @L Mnust`rbag 4;.5. Adbv` Ody`s 4;.0. j -A`dr`st A`bghoirs 4;.6. Ldtrbx Milpn`tbia 4;.7. Aiaa`gdtbv` Ldtrbx Fdmtirbzdtbia 4;.8. [rion`ls
4?4 4?4 4?4 4?= 4?6 4?8 4?2 4== 4=7
Mhdpt`r 44. Uh` Fdst Fiurb`r Urdasfirl 44.4. Batrieumtbia 44.?. Odsbm [rip`rtb`s
454 454 454
MIAU@AU]
p h
44.=.
E`rbvdtbia if th` Fdst Fiurb`r Urdasfirl
vbb
45=
p . d s c i / s n d a r u i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L D B ] i t m ` c o u s a i b t u o b r t s b e ` _ . 8 4 ? . 4 4 . 5 ; 4 . 5 2 4 i t ; ? / 8 / 4 0 ; e ` e d i n a w i E
p h p . d s c i / s n d a r u i c / g r
44.5. 44.0. 44.6. 44.7. 44.8.
Dpprixbldtbia if Fiurb`r ]`rb`s Mi`ffflmb`ats Miavinutbia Fdst Urdasfirls dae Ldtrbx Fdmtirbzdtbia Ubl` ]`rb`s [rion`ls
450 458 40? 40= 405
Mhdpt`r 4?. Liat` Mdrni L`thies 4?.4. Batrieumtbia 4?.?. Luntbpnbmdtbv` _daeil Aulo`r G`a`rdtirs 4?.=. G`a`rdtbia if Aiauabfirl _daeil E`vbdt`s 4?.5. _daeilbz`e Ldtrbx Luntbpnbmdtbia 4?.0. Ldrjiv Mhdba Liat` Mdrni 4?.6. ]blundt`e Daa`dnbag 4?.7. [rion`ls
402 402 402 46; 466 467 47= 475
Dpp`aebx D. Ldth`ldtbmdn _`vb`w D.4. Ire`r _`ndtbias D.?. @n`l`atdry Aulo`r Uh`iry D.=. Milpdmta`ss ba Ldth`ldtbmdn Dadnysbs D.5. Miav`xbty D.0. Ndgrdag` Luntbpnb`rs D.6. Nba`dr Dng`ord D.7. Odadmh‒s Miatrdmtbia Ldppbag Uh`ir`l
472 472 48; 485 480 482 42= ?;4
Obonbigrdphy
?;=
Bae`x
?;2
i . l d b s . w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L D B ] i t t m ` o c u s a i b t u o b r t s b e ` _ . 8 4 ? . 4 4 . 5 ; 4 . 5 2 4 i t ; ? / 8 4 / 0 e ; ` e d i n a w i E
p h p . d s c i / s n d a r u i c / g r i . l d b s . w w w / 3 p t t
[r`fdm`
Ly baspbrdtbia fir wrbtbag d surv`y if th` o`st dngirbthls mda o` sulldrbz`e oy quitbag Ldrtba Dbga`r dae Gþat`r Sb`gn`r, whis` spn`aebe oiij [riifs fril UH@ OIIJ bs aiw ba bts fflfth `ebtbia Z ?\. Uh`y wrbt`3
h ` ` s : t h g b r y p i m r i ` s a ` m b n L D B ] i t t m ` c o u a s i b t u o b r t s b e ` _ . 8 4 ? . 4 4 . 5 ; 4 . 5 2 4 i t ; ? / 8 4 / 0 ; e ` e d i n a w i E
[dun @reýs nbj`e ti tdn tdnj j doiut UH@ OIIJ, ba whbmh whbmh Gie ldbatdbas th` p`rf`mt priifs fir ldth`ldtbmdn th`ir`ls, finniwbag th` ebmtul if G. H. Hdrey thdt th`r` bs ai p`rlda`at pndm` fir ugny ldth`ldtbms. @reýs dnsi sdbe thdt yiu a``e ait o`nb`v` ba Gie out, ds d ldth`ldtbmbda, yiu shiune o`nb`v` ba UH@ OIIJ. Miav`rs`ny, B wiune dee thdt yiu a``e ait o`nb`v` ba UH@ OIIJ ti o`nb`v` ba Gie. Out B ebgr`ss. Ly lir` hulon` purpis` bs ti hbghnbght sil` if th` list fdlius dae summ`ssfun dngirbthls dae th` niv`ny ldth`ldtbms ldth`ldtbms o`hbae th`l. Dngirbthls dr` d eilbadat firm` ba lie`ra muntur`. @v`ry tbl` w` tura ia iur oriws`rs dae mill`am` d s`drmh, th`r` stdaes da dngirbthl dngirbthl ba th` shde shdeiws. iws. Wh`a w` hip bati iur mdr mdr,, tura ia th` `agba`, dae erbv` dwdy dwd y, th` litir dae ordj`s io`y hbee`a dngirb dngirbthls thls.. Iur odajs dae iur spb`s e`p`ae ia dngirbthls fir `amryptbia dae e`mryptbia. Uh` list blpirtdat smb`atbfflm bastrul`at ba day ndoird ndoirdtir tiry y bs th` milput milput`r `r.. @v`ry @v`ry bae baebmd bmdtbi tbia a bs thd thdtt dngirb dngirbthl thlss wbn wbnnn o`mil` o`mil` lir lir`` p`r p`rvd vdsb sbv`, v`, ait n`ss. Uhus Uhus,, gdba gdbabag bag da uae`r uae`rstda stdaebag ebag if dae d fdmbnbt fdmbnbty y fir e`sbgaba e`sbgabag g dngirbthls dngirbthls bs d wirthy ioc`mtbv ioc`mtbv`. `. Uh`r` bs miasbe`rdon` e`odt` doiut th` tip dngirbthls. Uh` aul`rbmdn dadnysts Eiagdrrd dae ]unnbvda Z 0;\ bgabt`e th` e`odt` wbth th`br nbst3 (4) (?) (=) (5) (0) (6) (7) (8) (2) (4;)
L`tripinbs dng dngirbthl irbthl fir Liat` Md Mdrni rni ]blpn`x l`t l`thie hie fir nba`dr prigr prigrdllbag dllbag Jryniv suospd suospdm` m` bt`rdtbia l`thies Uh` e`milpisbtbiadn dpp dppridmh ridmh ti ldtrbx milputdtb milputdtbias ias Uh` Firtrda iptblbzbag mmilpbn`r ilpbn`r T_ dngirbthl fir mil milputbag putbag `bg`a `bg`avdnu`s vdnu`s Tubmjsirt dngi dngirbthl rbthl fir ssirtbag irtbag Fdst Fi Fiurb`r urb`r trdasfirl trdasfirl Bat`g`r r`ndtbia e` e`t`mtbia t`mtbia Fdst luntbpin` luntbpin` l` l`thie thie
Uh` dppnb`e ldth`ldtbmbda Abmhinds Hbghdl upedt`e thbs nbst ba da bau`atbdn onig pist if Ldrmh ?2, ?;46. Hbs nbst shiws sbx dngirbthls ba millia3 (4) A`wtia dae qudsb-A`wtia l`thies (?) Ldtrbx fdmti fdmtirbzdtbias rbzdtbias (NP, Mhin`sjy, Mhin`sjy, T_) (=) ]bagundr vdn vdnu` u` e`milpisbtbia, e`milpisbtbia, T_ dae TS dngirbthls (5) Liat`-Mdrni l`thies (0) Fdst Fi Fiurb`r urb`r trdasfirl trdasfirl bx
x
p h p . d s c i / s n d a r u i c / g r i . l d b s . w w w / / 3 p t h t ` ` s : t h g b r y p i m r
(6) (7) (8) (2) (4;)
[_@FDM@
Jryniv suospdm` l l`thies `thies C[@ C[@G G [dg` [dg`_daj _daj ]blpn`x dngirbthl Jdnlda Jdnlda fflnt`r fflnt`r
Ba miatrdst, th` milput`r smb`atbst Ldrmis It`ri ba hbs onig `atry if Ldy ?6, ?;45, sugg`sts (4) (?) (=) (5) (0) (6) (7) (8) (2) (4;)
L`rg` ]irt ]irt,, Tubmj ]irt, dae H`dp ]irt Fiurb`r Ur Urdasfirl dasfirl dae Fdst Fiurb Fiurb`r `r Urdasfirl Ebcjstrd‒ Ebcjstrd‒ss dngirbth dngirbthl l _]D dn dngirbt girbthl hl ]`mur` Hds Hdsh h Dngirbthl Dngirbthl Bat`g`r fdmtirbzdtb fdmtirbzdtbia ia Nbaj D Dadnys adnysbs bs [ripirtbiadn Bat`g Bat`grdn rdn E`rbvdtb E`rbvdtbv` v` Dngirbthl Edtd milpr` milpr`ssbia ssbia dngir dngirbthls bthls _daeil Aulo`r Aulo`r G`a`rd G`a`rdtbia tbia
i ` s a ` m b n L D B ] i t t m ` c o u s a i t b u o b r t s b e ` _ . 8 4 ? . 4 4 . 5 ; 4 . 5 2 4 i t ; ? / 8 4 / 0 ; e ` e d i n a w i E
Ait` th` suostdatbdn ebv`rg`am` fril th` pr`vbius nbsts. Ba miatrdst, th` edtd smb`atbst Cdl`s N` ba hbs onig pist if Cdaudry ?;, ?;48, r`mill`aes (4) (?) (=) (5) (0) (6) (7) (8) (2) (4;)
Nba`dr r`gr`ssbia Nigbstbm r`gr`ssbia Nba`dr Ebsmr Ebsmrblbadat blbadat Dad Dadnysbs nysbs Mndssbfflmdtbia dae _` _`gr`ssbia gr`ssbia Ur``s Ur``s Adb Adbv` v` Ody`s Ody`s J-A`dr`st A`bghoirs N`drabag V V`mtir `mtir Tudatbzdtb Tudatbzdtbia ia ]uppirt V V`mtir `mtir Ldmhba`s Odggbag dae _daeil Fi Fir`st r`st Oiistbag dae Ded DedOiist Oiist
Ia` n`ssia ti o` n`dra`e oy thbs nblbt`e milpdrbsia bs thdt th`r` bs ai mias`asus. Iur ebsmbpnbadry ebsmbpnbadry odmjg odmjgriuae riuaess minir iur rdajbag if dngi dngirbth rbthls. ls. [`rsi [`rsiadnn adnny y, B n`da list tiwdre Hbghdl‒s nbst, wbth bts h`dvy `lphdsbs ia nba`dr dng`ord. Hiw`v`r, bt ilbts sirtbag dae grdph dngirbthls e`dr ti milput`r smb`atbsts dae r`gr`ssbia dae ldxblul nbj`nbhiie dngirbthls e`dr ti stdtbstbmbdas. stdtbstbmbdas. Uh` finniwbag pdg`s deipt sil`thba sil`thbag g fril dnn fiur nbsts, ds w`nn ds d f`w if ly iwa fdvirbt`s. Ndmjbag usdg` stdtbstbms ti odmj l` up, ly mhibm`s dr` p`rsiadn dae e`fflabt`ny suoc`mt ti qu`stbia. N`t l` l`atbia d f`w mrbt`rbd gubebag ly `xpisbtbia. B nbj` d ldth`ldtbmdn stiry. Uh` ldth`ldtbms a``e ait o` e``p, out bt shiune miloba` `n`l`ats if surprbs`, bag`aubty, dae g`a`rdnbty.. Dngirbthls shiune g`a`rdnbty shiune o` orb`f, `dsy ti uae`rstdae, dae prbambpn`e prbambpn`e.. L`r` r`mbp`s wbthiut d e`ffla`e ioc`mtbv` hdreny rbs` ti th` n`v`n if d n`gbtbldt` dngirbthl. _`de`rs ldy o` iff`ae`e oy ly gndrbag ilbssbias. Fir `xdlpn`, B ilbt ebsmussbia if th` T_ dngirbthl fir `xtrdmtbag th` sp`mtrdn e`milpisbtbia if d syll`trbm ldtrbx. B dnsi `lphdsbz` s`qu`atbdn dngirbthls dae odr`ny l`atbia l`atbia pdrdnn`n prim`ssbag. Fbadnny, Fbadnny, `xm`pt fir th` trdv`nbag sdn`slda prion`l, B dvibe A[ hdre prion`ls dntig`th`r. Uhbs oiij r`pr`s`ats ly dtt`lpt ti batrieum` stue`ats ba th` ldth`ldtbmdn ldth`ldtbmdn smb`am`s ti dngirbthls. Dnthiugh bt bs m`n`ordtiry, bt bs ait bat`ae`e ds e`fflabtbv` ir `amymnip`ebm. Ba j``pbag wbth ly e`sbr` ti mr`dt` d
[_@FDM@
p h p . d s c i / s n d a r u i c / g r i . l d b s . w w w / / 3 p t t h ` s ` : t h g b r y p i m r i ` s a ` m b n L D B ] i t
xb
t`xtoiij, `dmh mhdpt`r `aes wbth d prion`l s`mtbia. List prion`ls dr` strdbghtfirwdre ti sinv`, out d f`w lbght mhdnn`ag` `v`a `xp`rts. Uhbs orbags us ti th` qu`stbia if pr`r`qubsbt`s. B dssul` r`de`rs hdv` fdlbnbdrbty wbth nba`dr dng`ord, devdam`e mdnmunus, dae priodobnbty. priodobnbty. [rbir `xpisur` ti aul`rbmdn dadnysbs wiune wiun e h`np out bs ait r`qubr r`qubr`e. `e. Uh` dpp`aeb dpp`aebm`s m`s orb`y sj`t sj`tmh mh sil` th`i th`iry ry p`rtba` p`rtba`at at ti sp`mbfflm sp`m bfflm mhdpt`rs. mhdpt`rs. Fir bastda bastdam`, m`, r`de`rs wbnn e`fflabt`n e`fflabt`ny y wdat ti oriws` oriws` Dpp`aebx D.? ia `n`l`atdry aulo`r th`iry o`fir` th`y tdmjn` Mhdpt`r 5 ia prbldnbty t`stbag. List mhdpt`rs dr` bsindt`e bsindt`e `ssdys. D f`w f`w,, pdrt pdrtbmun bmundrny drny Mhdpt` Mhdpt`rr 2 ia edtd lbabag, lbabag, r`ny ia ldt`rbd ldt`rbdnn fril pr`vbius mhdpt`rs. Uh`r` bs `aiugh ldt`rbdn h`r` fir d s`l`st`r miurs` dt th` pdm` if ia` mhdpt`r mhdpt `r p`r w``j. PMND, ly hil` bastbtut bastbtutbia, bia, ip`rdt`s ip`rdt`s ia qudrt qudrt`rs, `rs, si B lust pbmj dae mhiis`. Dnn if th` dngirbthls ebsmuss`e h`r` dr` prigrdll`e ba Cunbd dae mda o` dmm`ss`e dt https3//oiijstir`.sbdl.irg/it468/ https3//oiijstir`.sbdl.irg/it468/oiaus. oiaus. Uh` devdatdg` if dmtudn mie` iv`r iw mhdrts bs thdtbastue`ats mda r`debny `xp`rbl`at. Vbrt dnn if ly mndssriil dr`niw`rprifflmb`at hbgh`r-n`v`n ndagudg`s sumh ds _Vbrtudnny daeudnny LDUNDO. Lday dr` ait stue`ats de`pt wbth n`v`n ndagudg`s. Cunbd bs d orbeg` ndagudg` thdt miloba`s miebag sblpnbmbty wbth `x`mutbia sp``e. B hip` ly mie` wbnn o` trdaspdr`at ti dfflmbiadeis if M, [ythia, dae ith`r lir` pipundr ndagudg`s. Bt bs uafirtuadt` thdt day ndagudg` mhibm` wiune ebsdppibat lday r`de`rs. Fbadnny, n`t l` str`ss thdt ly mie` bs dpt ti o` lumh n`ss `ffflmb`at thda prieumtbia mie`. Cunbd prigrdll`rs wbnn dnlist m`rtdbany wdat ti us` Cunbd‒s ods` dae nbordry fuamtbias rdth`r thda ly iwa. Fir th` r`mire, h`r` dr` sil` aitdtbia aitdtbia miav` miav`atbi atbias as us`e thri thriughiu ughiutt th` oiij. oiij. Dnn v`mtirs dae ldtrbm`s dpp`dr ba oinefdm`. Uh` `atrb`s if th` v`mtirs ; dae 4 miasbst if ;‒s dae 4‒s, r`sp`mtbv`ny. r`sp`mtbv`ny. Uh` v`mtir `b hds dnn `atrb`s ; `xm`pt d 4 ba `atry b b.. Uh` ;/4 baebmdtir ∗
t m ` c o u s a i b t u o t b r s b e ` _ . 8 4 ? . 4 4 . 5 ; 4 . 5 2 4 i t ; ? / 8 4 / 0 ; e ` e d i n a w E i
if d s`t ] s`t ] bs bs e`ait`e oy 4 ] (x). Uh` sup`rsmrbpt baebmdt`s d v`mtir ir ldtrbx trdaspis`. Uh` @umnbe`da airl if d v`mtir x bs e`ait`e oy x dae th` sp`mtrdn dae Frio`abus airls if d ldtrbx L < (lbc ) oy
L <
L x dae L F < x 3 Bat` Bat`g`r g`r m < z` z`ri ri(U (U) ) wh whbn bn` ` d 1 ia ia`( `(U) U) bf bsiee(d) bsiee(d) m < m + o `ae d < d 11 4 # ebvbe` d oy ? o < o >> 4 # luntbpny o oy ? `ae r`tu r`tura ra m + o `ae m < p`ds p`dsdatpr datprieum ieumt(4; t(4;, , ==)
Bt bs wirth `lphdsbzbag thdt p`dsdat luntbpnbmdtbia t`rlbadt`s dft`r d fflabt` aulo`r if st`ps wbth th` mirr`mt dasw`r. Ith`r dngirbthls thdt w` wbnn l``t ndt`r shdr` thbs prip`rty, privbe`e w` `at`rtdba th` fflmtbia thdt milput`rs dr` mdpdon` if `xdmt drbthl`tbm wbth r`dn 4
?
4. DAMB@AU DNGI_BUHL]
p h p . d s c i / s n d a r u i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b y r p i m r i ` s a ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e `
aulo`rs. Uh` a`xt dngirbthl bs r`pr`s`atdtb r`pr`s`atdtbv` v` if th` mndss if bt`rdtbv` dngirbthls. dngirbthls. Uh`s` miav`rg` iv`r da bafflabt` aulo`r if st`ps ti d mirr`mt dasw`r. Uh` rdt` if miav`rg`am` if da bt`rdtbv` dngirbthl dngirbthl bs mrbtbmdnny blpirtdat ba `vdnudtbag `vdnudtbag bts p`rfirldam`. Uhbs lust o` odndam`e dgdbast th` milputdtbiadn milpn`xbty if `dmh st`p.
_ . ; 4 . 6 6 . 0 = ?
Uhbs r`pr`s`atdtbia ldj`s bt mn`dr thdt xa+4 m r`gdren`ss if th` vdnu` if x xa . Furth`rlir`, xa+4 < m bf dae iany bf x xa < m. Bf w` dssul` thdt xa 1 m, th`a th` ebff`r`am` firlund (4.4) blpnb`s thdt
4.=. Odoyniabda L`thie Uh` damb`at Odoyniabdas ebsmiv`r`e d niv`ny dngirbthl fir `xtrdmtbag th` squdr` riit if d aiaa`gdtbv` aulo`r m . Uh`br dngi dngirbth rbthl l turas iut ti o` d sp`m sp`mbdn bdn mds` if A`wtia‒s A`wtia‒s l`thie. Uh` Odoyniabda bt`rdtbias dr` e`ffla`e oy th` firlund
xa+4
4 < ?
m xa + xa
wbth d pisbtbv` babtbdn vdnu` x; . Uh` mhibm` x; < 4 bs sil`whdt a`utrdn out hdreny iptbldn ia d milput`r wbth obadry drbthl`tbm. Uh` Odoyniabda upedt` mda o` litbvdt`e oy sinvbag th` dpprixbldtbia
m < (xa + ∆) ? < x?a + ?x ? xa ∆ + ∆?
≈ x? + ?x ?x a
a
∆
fir ∆ dae th`a mdnmundtbag
xa + ∆
≈
4 m xa + ?xa
∟ x . ?
a
x; , th` bt`rdt`s miav`rg` ti N`t us shiw thdt r`gdren`ss if th` mhibm` if x th` ebff`r`am ebff`r`am`` xa+4
∓ 4 x ∟ m <
∓
∓
∟ ∓ m
m a + xa ? x?a + m + m ? mxa < ?xa m)? (xa . < ?xa
∟ ∓ ∟ ∓
(4.4)
∓ m. Miasbe`r
∓ ∓ ≠ ∓
∓
; = 4 i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
xa+4
∟ ∓ m > ?x
m > xa
∟
m
∟
x a miaebtbia whbmh bs iovb iovbius. ius. Uhus, th` bt`rdt` bt`rdt`ss e`mr`ds` bf dae iany bf x a , d miaebtbia dae piss`ss d nblbt x∑ . Uhbs nblbt sdtbsffl`s th` `qudtbia 4 x∑ < ? whis` iany sinutbia bs ndrg` a
m x∑ + x∑
∓ m. Uh` rdt` if miav`rg`am` if xx xa+4
∟ ∓ m ≈ (
,
a
ti
∓ m bs quderdtbm o`mdus` fir
∟∓ ∓ m) ? . ? m
xa
Ba prdmtbm` th` aulo`r if sbgabfflmdat ebgbts ba xa eiuon`s dt `dmh bt`rdtbia.
4.5. TPDE_DUBM @TPDUBIA]
p h p d . s c i / s n d a r u i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p m i r i ` s a ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . ; 6 4 . 6 . 0 = ? . ; = 4 i t ; ? / 7 4 / 0 ;
=
UDON@ 4. 4. Uh` Odoy Odoyniab niabda da L`thie L`thie Dppn Dppnb`e b`e ti ϋ ? Bt`rdtbia a ; 4 ? = 5 0 6
xa
xa
∟ϋ
4.; -?.45402?60=08272= 0. 0.5= 5=58 58;? ;??; ?;;0 ;055 5567 672 2 ?. ?.?2 ?2=? =?;2 ;205 0562 6205 0588 886 6 =. =.6? 6?05 05;4 ;45= 5=42 42?4 ?426 265 5 ;. ;.58 58=8 =8;8 ;877 778= 8==? =?47 474 4 =. =.47 47=8 =875 757? 7?57 5756 5645 45? ? ;. ;.;= ;=?? ??8? 8?;7 ;744 4406 06=5 =52 2 =. =.45 4547 4706 068? 8?7; 7;62 622? 2?7 7 ;. ;.;; ;;;4 ;465 6547 47=5 =58; 8;4= 4=5 5 ∟ =. =.45 4540 402? 2?60 607 787 872? 2?6? 6? 5. 5.?8 ?82 256 568 8== ==6 6 4; 2 =.45402?60=08272= ;.;
×
H`r` bs Cunbd mie` blpn`l`atbag th` Odoyniabda l`thie3 fuamtbia odoyniab fuamtbia odoyniabda(m da(m33U, 33U, tin33U) wh`r` U >3 _`dn x < ia`(U) # st std drt x dt 4 wh whbn bn` ` dos( dos(xQ xQ? ? - m) 1 ti tin n # miav miav`r `rg` g`am am` ` t` t`st st x < (x + m / x) / ? `ae r`tura r`t ura x `ae riit < odoy odoyniab niabda(p da(pbQ?, bQ?, 4`-4 4`-4;) ;)
Udon` 4.4 r`mires th` Odoyniabda l`thie dppnb`e ti ϋ ? . ]bx bt`rdtbias bt`rdtbias suffflm` suffflm` ti dmhb dmhb`v `v`` miav`rg`am` ti ldmhba` pr`mbsbia. Iam` th` bt`rdt`s r`dmh th` vbmbabty if ϋ , th` fdst quderdtbm rdt` if miav`rg`am` jbmjs ba. 4.5. Tuderdtbm @qudtbias List `eumdt`e p`ipn` dr` dt n`dst vdgu`ny fdlbnbdr wbth th` sinutbia
∟o ´ x <
∓ ? o ?d
∟ 5dm
ox+ + m < ;. Uh`s` twi riits dr` da bll`ebdt` mias`qu`am` ti th` quderdtbm `qudtbia dx? + ox if th` be`atbty
o x + ?d
?
<
o?
∟ 5dm .
5d?
Uh` damb`at mbvbnbzdtbias if Odoyniabd, @gypt, Gr``m`, Mhbad, dae Baebd dnn lde` miatrboutbias ba sinvbag quderdtbms. Uh` quderdtbm firlund ba th` firl w` jaiw tiedy bs mr`ebt`e ti _`aâ E`smdrt`s. N`t us str`ss thdt th` quderdtbm firlund bs d n`gbtbldt` dngirb dngirbthl thl `v`a thiugh bt bs ait `xpnbmbtny bt`rdtbv`. Blpnbmbtny bt bs bt`rdtbv` o`mdus` `xtrdmtbag squdr` riits bs usudnny bt`rdtbv`. bt`rdtbv`.
e ` e d i n a w i E
H`r`` w` wiu H`r wiune ne nbj nbj`` ti mil mill` l`at at ia th` pit pit`at `atbdn bdn niss niss if aul aul`rb `rbmdn mdn dmmurd dmmurdmy my ba dpp dppnyb nybag ag th` mndssbmdn quderdtbm firlund. _iuaeiff `rrir immurs wh`a twi aulo`rs if th` sdl` sbga dae dpprixbldt`ny th` sdl` ldgabtue` dr` suotrdmt`e. Dssulbag th` twi riits dr` r`dn, th` sldnn`r riit ba ldgabtue` bs th` ia` nbdon` ti mdtdstriphbm mdam`nndtbia. Ui dvibe riuaeiff, ia` shiune milput` th` ndrg`r riit r ba ldgabtue` dae `xpnibt th` fdmt thdt th` prieumt if th` twi riits `qudns dm . Uh` finniwbag finniwbag Cunbd mie` ei`s `xdmt `xdmtny ny thbs. Ait` thdt riuaei riuaeiff ff bs ait da bssu` wh`a th` riits dr` milpn`x. Uhbs mds` bs hdaen`e s`pdrdt`ny ba th` Cunbd mie`.
5
p h p . d s c / i s n d a r u i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 0 . = ? . ; = 4 i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
4. DAMB@AU DNGI_BUHL]
fuamtbia fuamtb ia qud quderd erdtbm tbm(d3 (d33U, 3U, o33 o33U, U, m33 m33U) U) wh` wh`r` r` U >3 _`d _`dn n e < oQ? oQ? - 5d * m # ebsm ebsmrb rblb lbad adat at bf e 1 z`ri z`ri(U (U) ) bf o 1< z` z`ri ri(U (U) ) r4 < (-o (-o - sqrt sqrt(e (e)) )) / (?d) (?d) `ns` r4 < (-o (-o + sqrt sqrt(e (e)) )) / (?d) (?d) `ae r? < m / (r4 * d) r`tura r`t ura (r4 (r4, , r?) `ns` r`tu r`tura ra ((-o o + sqrt sqrt(e (e + ;bl) ;bl)) ) / (?d) (?d), , ((-o o - sq sqrt rt(e (e + ;b ;bl) l)) ) / (? (?d) d) `ae `ae (d, (d, o, m) < (4 (4.; .;, , -?.; -?.;, , 4. 4.;) ;) (r4, (r4 , r?) < qud quderd erdtbm tbm(d, (d, o, m)
Fir th` sdj` if sblpnbmbty, thbs fuamtbia bgair`s th` deliabtbia if lday Cunbd `xp`rts thdt r`tura`e vdnu`s o` typ` stdon`. 4.0. @umnbe‒ @umnbe‒ss Dngirb Dngirbthl thl @umnbe‒s dngirbthl bs da `ffflmb`at l`thie fir milputbag th` gr`dt`st millia ebvbsir 1 ; . Oy e`fflabtbia gme( d, o) bs th` ndrg`st bat`g`r thdt ebvbe`s (gme) if twi bat`g`rs d 1 o 1 ; gme(d, oith d dae o wbthi wbthiut ut n`dvba n`dvbag g d r`ld r`ldbae` bae`rr. Uh` dngirbt dngirbthl hl dpp`drs ba @umnbe‒s @n`l`ats (mbrmd =;; OM). Bt mda o` us`e ti r`eum` frdmtbias ti th`br sblpn`st firls dae immurs ba lday aulo`r-th`ir`tbm dae mryptigrdphbm mdnmundtbias. ]uppis` d dae o hdv` gr`dt`st millia ebvbsir m . Ds [ripisbtb [ripisbtbia ia D.?.4 if Dpp`a Dpp`a-ebx D.? e`liastrdt`s, th`r` `xbsts d uabqu` pdbr if bat`g`rs q dae r sumh thdt d < q qoo + r + r dae ; r > o. Bf oy mhdam` r < r < ;, th`a mn`drny gme( d, o) < o . Ith`rwbs`, ait` thdt sbam` gme(d, m ebvbe`s oith d dae o , bt lust ebvbe` r ds w`nn. Miav`rs`ny, day bat`g`r e ebvbebag oith o d, o) < gme(o, dae r lust ebvbe` d. Bt finniws thdt gme( gme(d, gme(o, r), dae w` mda r`pndm` d oy o dae o oy r . O`mdus` o > d dae r > o , thbs r`pndm`l`at prim`ss lust mil` ti da `ae ba d fflabt` d 3 Bat Bat`g` `g`r r (d (d, , o) < (l (l, , a) whbn` whb n` o !< z`r z`ri(U i(U) ) (d, (d, o) < (o, (o, r`l( r`l(d, d, o) o)) ) `ae r`tura r`t ura d `ae gme < `umnbe `umnbe(6; (6;;, ;, ??; ??;) )
Uh`r` bs d ldtrbx firl if @umnbe‒s dngirbthl. Ait` thdt
d
q 4 o
o
o
4 ;
r
L
r
4.7. D_MHBL@E@]‒ D[[_IXBLDUBIA IF ϋ
p h p . d s c i / s n a d r u i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s ` a m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . 0 = ? ; . = 4 i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
0
dae thdt
∟4 < L
; 4 4
∟q
`xbsts `xbs ts dae piss`ss piss`ss`s `s bat` bat`g`r g`r `atrb`s `atrb`s.. Bf @umnbe‒ @umnbe‒ss dngi dngirbth rbthl l tdj`s tdj`s s st`ps dae `aes wbth d, o), ;\ , th`a Zgme( Zgme(d,
d o
<
L 4
· · · L
s
gme( d, o) gme(d, ;
.
Bt finniws thdt
gme( d, o) gme(d, ;
4 d < L s∟4 · · · L ∟ 4
o
dae ba pdrtbmundr thdt
md + eo eo gme(d, o) < md + gme(d,
(4.?)
fir bat`g`rs m dae e. Uhbs r`sunt wds fflrst priv`e oy Oâziut. 4.6. ]b`v` if @rdtisth`a`s @rdtisth`a`s‒ dngirbthl fflaes dnn prbl`s o`tw``a 4 dae d fflx`e bat`g`r a. @rdtisth`a`s if Myr`a` (mbrmd ?76 OM ti 420/425 OM) wds d Gr``j ldth`ldtbmbda dae dstriail`r. H` s`rv`e ds ebr`mtir if th` fdlius Nbordry if Dn`xdaerbd dae bs o`st jaiwa fir mdnmundtbag th` mbrmulf`r`am` if th` `drth. Hbs sb`v` wirjs oy ldrjbag ds milpisbt` (aia-prbl`) dnn if th` luntbpn`s if `dmh prbl`, stdrtbag wbth dnn luntbpn`s if ?, th`a dnn luntbpn`s if =, dae si firth. Uh` luntbpn`s if d gbv`a prbl` p dr` g`a`rdt`e ba th` ire`r p, ? p, = p, . . . uatbn a bs r`dmh`e. Uh` sb`v` dvibes t`stbag `dmh mdaebedt` aulo`r fir ebvbsbobnbty oy `dmh prbl`. Bt dnsi r`nb`s ia th` sblpn` fdmt thdt wh`a d aulo`r bs milpisbt`, ia` if bts fdmtirs bs n`ss thda ir `qudn ti a. H`r` bs Cunbd mie` blpn`l`atbag th` sb`v`.
∓
fuamtbia `rdtisth`a`s `rdtisth`a`s(a33Bat`g`r) (a33Bat`g`r) bsprbl` bspr bl` < tru` tru`s(a) s(a) bsprbl bsp rbl`Z4 `Z4\ \ < fdn fdns` s` # 4 bs mil milpis pisbt` bt` fir b < ?3r ?3riua iuae(B e(Bat, at, sqr sqrt(a t(a)) )) bf bsprbl`Z bsprbl`Zb\ b\ fi fir r c < bQ?3 bQ?3b3 b3a a # dn dnn n lu lunt ntbp bpn` n`s s if b > bQ bQ? ? dn dnr` r`de dey y milp milpis isbt bt` ` bsprbl`Zc bsprb l`Zc\ \ < fdns` `ae `ae `ae r`tura r`tu ra fbnt fbnt`r(x `r(x -1 bsprb bsprbl`Zx l`Zx\, \, 43a) # `nbl `nblbadt badt` ` milp milpisbt` isbt` aulo`rs `ae prbl`Ynbst < `rdtisth`a`s( `rdtisth`a`s(4;;) 4;;)
4.7. Drmhbl`e`s‒ Drmhbl`e`s‒ Dpprix Dpprixbldtbia bldtbia if ϋ Driuae ?0; OM, th` Gr``j ldth`ldtbmbda Drmhbl`e`s e`rbv`e da dngirbthl fir dpprixbldtbag ϋ , th` rdtbi if d mbrmn`‒ mbrmn`‒ss mbrmulf`r`am` mbrmulf`r`am` ti bts ebdl`t`r ebdl`t`r.. H` wds don` ti shiw 4 4; thdt = 74 > ϋ > = 7 oy miasbe`rbag th` n`agth oa if th` p`rbl`t`r if d r`gundr pinygia wbth = ?a sbe`s basmrbo`e wbthba d mbrmn` dae th` n`agth da if th` p`rbl`t`r if d r`gundr pinygia wbth = ?a sbe`s mbrmulsmrbo`e iutsbe` d mbrmn`. Fir d mbrmn` wbth ebdl`t`r 4, oa > ϋ > da .
·
·
6
p h p . d s c i / s n d a r u c i / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . 0 = ? . ; = 4 i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
4. DAMB@AU DNGI_BUHL]
d 4 dae o 4 fir mbrmulsmrbobag dae basmrbobag h`xdgias, ]tdrtbag wbth th` jaiwa vdnu`s if d h` wds don` ti miastrumt d r`murr`am` r`ndtbia miaa`mtbag d a+4 dae o a+4 ti d a dae o a . Oy eiuonbag th` aulo`r if sbe`s if th` babtbdn h`xdgias ti 4?-sbe`e pinygias, th`a ti ?5-sbe`e pinygias, dae untbldt`ny ti 26-sbe`e pinygias, Drmhbl`e`s wds don` ti orbag th` twi p`rbl`t`rs `v`r mnis`r ba n`agth ti th` mbrmulf`r`am` if th` mbrmn`. H`, nbj` dnn Gr``j ldth`ldtbmbdas, r`nb`e h`dvbny ia g`il`trbm drgul`ats. Fbgur` 4.4 e`pbmts d mbrmn` wbth ebdl`t`r 4 dae mirr`spiaebag basmrbo`e dae mbrmulsmrbo`e squdr`s. ]qudr`s dr` sblpn`r babtbdn fflgur`s thda h`xdgias. Uh` p`rbl`t`r n`agths if th` twi squdr`s dr` o ; < 5/ ? < ? ? dae d; < 5. W` aiw `xpnir` hiw Drmhbl`e`s mdnmundt`e p`rbl`t`r n`agths fir basmrbo`e dae mbrmulsmrbo`e pinygias dft`r eiuonbag th` aulo`r if sbe`s. Mdnn th` p`rbl`t`r n`agths oa dae da fir r`gundr pinygias wbth ? ?a sbe`s. Drmhbl`e`s‒ drgul`at e`p`aes ia fiur fdmts3 (d) dt `dmh pibat if d mbrmn` th` tdag`at nba` dae th` rdebdn nba` fril th` m`at`r dr` p`rp`aebmundr, (o) twi r`gundr pinygias if th` sdl` aulo`r if sbe`s shdr` bat`rbir dagn`s, (m) twi rbght trbdagn`s dr` sblbndr bf th`y shdr` d lbair dagn`, dae (e) sbe` n`agths ba sblbndr trbdagn`s immur ba d miastdat rdtbi. Uhus, trbdagn`s HEG dae DFH dr` sblbndr, ds dr` trbdagn`s DME dae DFH dae th` bsism`n`s trbdagn`s EFO dae EGF.
∓
∓
·
D
O
M
@
F
G
H
F BGP_@ 4.4 . Drmhbl`e`s‒ [inygias ffir ir @stbl @stbldtbag dtbag ϋ Bf w` n`t l < ? ?a o` th` aulo`r if sbe`s if th` twi r`gundr pinygias, th`a th` sblbndrbty if trbdagn`s HEG dae DFH blpnb`s thdt
·
da+4 da ?l
5l
∟
da+4
5l
<
EG DF . < GH DH
Nbj`wbs`, th` sblbndrbty if trbdagn`s DME dae DFH blpnb`s thdt oa
?l da ?l
E
<
ME DE . < DF DH
4.7. D_MHBL@E@]‒ D[[_IXBLDUBIA IF ϋ
p h p . d s c i / s n d a r u i c / g i r . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L B D ] i t t m ` c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . 0 = ? . ; = 4 i t ? ; / 7 4 / 0 ; e ` e d i n a w i E
7
]bam` DF < DE < 4? bs th` rdebus if th` mbrmn`,
da
da+4
oa
5l
?l
∟
<
da+4
?l
da
.
?l
5l
Uhbs `qudtbia bs `qubvdn`at ti th` `qudtbia
da+4 <
(4.=)
?da oa . da + o + oa
Fbadnny, th` sblbndrbty if th` bsism`n`s trbdagn`s EFO dae EGF yb`nes oa+4
oa l oa+4
?l da+4
<
,
5l
?l
whbmh bs `qubvdn`at ti
d
oa+4 <
(4.5)
o .
a+4 a
Uh` twi `qudtbias (4.=) dae (4.5) miastbtut` Drmhbl`e`s‒ r`murr`am` smh`l` fir dpprixbldtbag ϋ . H`r` bs Cunbd mie` blpn`l`atbag Drmhbl`e`s‒ dngirbthl. fuamtbia drmh fuamtbia drmhbl`e bl`e`s(t `s(tin33U in33U) ) wh`r wh`r` ` U >3 _`dn (d, (d, o) < (5 * ia`( ia`(U) U), , ? * sqrt sqrt(? (? * ia`( ia`(U) U))) )) whbn whbn` ` dos( dos(d d - o) 1 tin tin d < ? * d * o / (d + o) o < sqrt(d * o) `ae r`tura r`t ura (d, o) `ae (upp`r, (upp `r, niw` niw`r) r) < drmh drmhbl`e` bl`e`s(4` s(4`-6) -6)
Drmhbl`e`ss ndmj`e list if th` tiin Drmhbl`e` tiinss if mdnmu mdnmunus. nus. N`t us aiw shiw hiw th`s` tiins privbe` basbght bati th` rdt` if miav`rg`am` if th` s`qu`am`s da dae oa ti ϋ . Bf w` dssul` oy baeumtbia thdt oa > da , th`a
?da oa da + o + oa ?da oa < da + o + oa
?da oa ?oa ?da oa 1 ? da
da+4 < da+4
>
< da < oa .
Uh`s` twi ba`qudnbtb`s ba tura blpny thdt
oa+4 < oa+4 <
d d
o > da+4
a+4 a
o 1 oa .
a+4 a
8
4. DAMB@AU DNGI_BUHL]
UDON@ 4 .? . Drmhb Drmhbl`e` l`e`s‒ s‒ Dpprix Dpprixbldt bldtbia bia ]mh`l ]mh`l`` fir ϋ
p h p . d s c i / s n d a r u i c / g r i . d l b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L D B ] t i t m ` c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . 0 = ? . ; = 4 i t ; ? / / 4 7 0 ; e ` e d i n a w i E
Bt`rdtbia a ; 4 ? = 5
d
o
5. ; =.=4=7;80 =.48?0272 =.4047?52 =.4554485
?.8?85?74 =.;645670 =.4?4550? =.4=60580 =.45;==4?
0 6 7 8 2 4;
=.45???=6 =.45470;5 =.4546=?4 =.4546;?0 =.4540204 =.45402==
=.454?77= =.45404=8 =.45407?2 =.4540877 =.4540245 =.45402?=
Uhus, ds a t`aes ti , th` liaitia` s`qu`am` d a e`mr`ds`s ti d nblbt d , th` liaitia` s`qu`am` oa bamr`ds`s ti d nblbt o, dae o d. Uh` gdp da oa sdtbsffl`s
∑
≥ +4 ∟ o +4 ≥ d
da
a
<
∟
∟o ?d o ∟o d + o + o a+4
a
a a
a
a
a
<
oa (da da + o + oa
≥
4 (da ?
∟o
a
∟o
a
)
).
Uhus, th` gdp bs lir` thda hdnv hdnv`e `e dt `dmh bt`rdtbi bt`rdtbia. a. Bt finniws finniws thdt d < o dae thdt `dmh bt`rdtbia yb`nes riughny daith`r obadry ebgbt if dmmurdmy. Udon` 4.? ebspndys th` nba`dr rdt` if miav`rg`am` if Drmhbl`e`s‒ dngirbthl. 4.8. [rion`ls [rion`ls (4) Uh` Ginesmhl Ginesmhlbet bet l`thie if ebvbsbi ebvbsbia a r`eu r`eum`s m`s th` `vdnud `vdnudtbia tbia if d frdmtbia frdmtbia do ti deebtbia deebt bia dae luntbpn luntbpnbmdt bmdtbia. bia. Oy obt shbftbag shbftbag w` ldy dssu dssul` l` thdt o ( 4? , 4\. _`pndm` o oy 4 x dae wrbt`
∈
∟
d 4
∟x
d(4 + x + x)) 4 x? d(4 + x + x)(4 )(4 + x + x? ) < 4 x5 d(4 + x + x? + x)(4 )(4 + x + x? ) (4 + x < 4 x?
<
∟
∟
∟
··· a
∟4
a
)
.
[rigrdl thbs dngirbthl ba Cunbd. Hiw ndrg` shiune a o` si thdt th` e`ailbadtir bs `ff`mtb `ff`mtbv`ny v`ny 49 Ait` thdt thdt th` piw`rs if x shiune o` milput`e oy r`p`dt`e squdrbag. (?) Ps` th` sbga sbgabfflmda bfflmdae e dae `xpia` `xpia`at at fuamtbia fuamtbiass if Cunb Cunbdd dae e`vbs e`vbs`` d o`tt o`tt`r `r babtbdn vdnu` thda x; < 4.; fir th` Odoyniabda l`thie. @xpndba yiur mhibm`, dae t`st bt ia d f`w `xdlpn`s.
4.8. [_ION@L]
p h p . d s c i / s n d a r
(=) Fir m
2
≠ ; shiw thdt th` bt`rdtbia smh`l` xa+4 <
miav`rg`s ti
m + x + xa 4 + x + xa
∓ m stdrtbag fril day x ≠ ; . V`rbfy `bth`r th`ir`tbmdnny ir `lpbr;
bmdnny thdt th` rdt` if miav`rg`am` bs lumh sniw`r thda thdt if th` Odoyniabda l`thie. (5) E`e`jbae‒ E`e`jbae‒ss dngirbthl fir `xtrd `xtrdmtbag mtbag m bt`rdt`s dmmirebag ti
u i c / g r i . l d b s w . w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L D B ] i t t ` m c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . 0 = ? . ; = 4 i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
∓
xa +4
xa (x?a + =m = m) . < + m =x?a + m
[rigrdl E`e`jbae‒s dngirbthl ba Cunbd. E`liastrdt` muobm miav`rg`am` oy e`rbvbag th` be`atbty
x?a+4
(0)
(6) (7)
∟m <
(x?a m)= . + m))? (=x (=x?a + m
∟
∓
Fbadnny, drgu` thdt E`e`jbae‒s dngirbthl miav`rg`s ti m r`gdren`ss if th` babtbdn 1 ;. vdnu` x; 1 ; Fbae mi`f mi`ffflmb`ats fflmb`ats (d,o,m ( d,o,m)) wh`r` th` stdaedre quderdtbm firlund bs grissny badmmurdt` wh`a blpn`l`at`e ba sbagn` pr`mbsbia. pr`mbsbia. Riu wbnn hdv` ti niij up hiw ti r`pr`s`at sbagn` pr`mbsbia aulo`rs ba Cunbd. Why ei`s th` prieumt i iff th` twi riits if d quderdtbm quderdtbm `qudn dm 9 ]invbag d muobm `qud `qudtbia tbia dx= + ox? + mx mx+ + e < ; bs lumh lir` milpnbmdt`e thda sinvbag d quderdtbm. E`liastrdt` thdt (d) th` suostbt suostbtutbia utbia x < y =od r`eum`s `y + f < ; fir m`rtdba mi`ffflmb`ats ` dae f , (o) th` furth`r + f th` muobm ti y = + `y = ` suostbtutbia y < y j ≥ b dae x j 1 x4 fir b > j ≥ c . Dt pisbtbi pisbtbia a c < 4, w` dr` firm`e ti put b < 4. Aiw suppis` suppis` w` hdv hdv`` summ`ssf summ`ssfunny unny dev devdam`e dam`e ti d g`a`r g`a`rdn dn pisbtbia pisbtbia c dae c + be`atbffl`e d mirr`spiaebag mdaebedt` pisbtbia b. Ui liv` fril pisbtbia c ti pisbtbia c + 4, w` `xdlba` x c +4 . Bf x x c +4 1 x4 , th`a w` j``p th` murr`at mdaebedt` pisbtbia b . Ia th` th` ith`r hdae, bf x x c +4 ≥ x4 , th`a w` swdp x b+4 dae x c +4 dae r`pndm` b oy b b + + 4 . Ba `bth`r mds`, th` twi r`qubr`e miaebtbias blpis`e ia b miatbau` ti hine ba livbag fril pisbtbia c ti pisbtbia c + 4. Bt bs aiw mn`dr thdt w` mda baeumtbv`ny ldrmh fril th` n`ft `ae ti th` rbght c < `ae if th` s`qu`am`, mdrrybag iut d f`w swdps ba th` prim`ss, si thdt wh`a c < a , th` vdnu` b ldrjs th` mirr`mt pisbtbia ti bas`rt x 4 . Iam` thbs bas`rtbi bas`rtbia a bs lde` lde`,, th` suos`q suos`qu`am` u`am`ss x4 , . . . , xb∟4 dae xb+4 , . . . , xa mda o` sirt`e s`pdrdt`ny oy th` sdl` spnbttbag prim`eur`. Uh` finniwbag r`mursbv` Cunbd mie` blpn`l`ats qubmjsirt ia day sirtdon` nbst if bt`ls.
h ` ` s : t h g b r y p i m r i ` s a ` m b n
W` bnnustrdt` th` dngirbthl ia bat`g`rs dae n`tt`rs. fuamtbia fuam tbia qubmjsir qubmjsirt(x3 t(x33V`mt 3V`mtir, ir, n`ft < 4, rbgh rbght t < n`ag n`agth(x th(x)) )) b < rda rdae(n e(n`ft `ft3rb 3rbght ght) ) # s`n s`n`mt `mt d rda rdaeil eil spn spnbtt bttbag bag vdn vdnu` u` sp spnb nbt t < xZ xZb\ b\ (xZn`ft\ (xZn `ft\, , xZb\) < (spn (spnbt, bt, xZn` xZn`ft\) ft\) b < n`ft fir c < (n` (n`ft ft + 4)3 4)3rbg rbght ht # pis pisbtb btbia ia th` spnb spnbttb ttbag ag vdn vdnu` u` bf xZ xZc\ c\ >< sp spnb nbt t b < b + 4
D B ] i t t m ` c o u s a i b t b o u r t s b e ` _ . ; 4 . 6 6 . 0 = ? . ; = 4 i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
(xZb\, (xZ b\, xZc xZc\) \) < (xZ (xZc\, c\, xZb xZb\) \) `ae `ae (xZn`f (xZ n`ft\, t\, xZb xZb\) \) < (xZ (xZb\, b\, spn spnbt) bt) bf b 1 n`f `ft t + 4 # sir irt t ti th th` ` n` n`ft ft if th` vd vdnu nu` ` qubmjs qub mjsirt irt(x, (x, n`ft, b - 4) `ae bf b + 4 > rbgh rbght t # sirt sirt ti th` th` rbgh rbght t if th` vd vdnu nu` ` qubmjs qub mjsirt irt(x, (x, b + 4, rbg rbght) ht) `ae `ae x < Z0, 5, =, 4, ?, 8, 7, 6, -4\: qubmjsirt(x) prbatna(x) x < Z‒d‒ Z‒d‒, , ‒m ‒m‒, ‒, ‒e ‒e‒, ‒, ‒o ‒o‒, ‒, ‒f ‒f‒, ‒, ‒`‒, ‒`‒, ‒h‒, ‒h‒, ‒g ‒g‒, ‒, ‒y ‒y‒\ ‒\: : qubmjsirt(x) prbatna(x)
Ui `xpnir` th` dv`rdg` o`hdvbir if qubmjsirt, n`t da o` th` `xp`mt`e aulo`r if ip`rdtbias bavinv`e ba qubmj sirtbag d s`qu`am` if a a aulo`rs. aulo`rs. Oy miav`at miav`atbia bia d ; < ;. Bf w` ods` iur dadnysbs iany ia hiw lday pisbtbias c lust o` `xdlba`e dt `dmh stdg` dae ait ia hiw lday swdps dr` bavinv`e, th`a w` mda wrbt` th` r`murr`am` r`ndtbia
da < a ∟ 4 + 4 a
(?.4)
a
(d
+ da b ) < a ∟ 4 + ? 4 + d a
b∟
b ?; bs r`si r`sinv`e nv`e o oy y d mdnn ti ti sbfteiwa sbfteiwa,, th` vdn vdnu` u` ?; dt th` th` tip if tth` h` a`w a`w tr`` tr`` mda o` p`` p``n`e n`e 8 > ?; iff iff.. Dae si bt gi`s uatb uatbnn th` `atbr` s`qu` s`qu`am` am` bs sirt`e sirt`e.. Uh` dstut dstut`` r`de`r wbnn aitb aitbm` m` thdt r`sinvbag th` pr`m`e`am` vbindtbia mdus`e oy d swdp dff`mts cust d sbagn` ordamh if th` tr`` dae r`stir`s th` h`dp. Ds wds ait`e `drnb`r, h`dpsirt piss`ss`s d wirst mds` milputdtbiadn milpn`xbty if I(a na a). O`mdus` O`mdus` th` l`da if d rdaeil vdrb vdrbdon` don` mdaai mdaaitt `xm``e bts ldxblu ldxblul l vdnu`, h`dpsirt dnsi piss`ss`s da dv`rdg` milputdtbiadn milpn`xbty if I( I (a na a). Ui v`rbfy th` iws. H`am` H`am`,, th` wirst mds` o`hdvbir, r`mdnn thdt d obadry s`drmh tr`` hds nig? a + 4 rriws. milputdtbiadn milpn`xbty if d sbagn` mdnn ti sbfteiwa bs I I(na (na a). ]bam` th`r` dr` a? mdnns ba oubnebag d h`dp, th` milputdtbiadn milpn`xbty if phds` 4 if h`dpsirt bs I I((a na a). Uh` milputdtbiadn milpn`xbty milpn`xbty if phds` ? bs dnsi I(a na a) o`mdus` `v`ry tbl` da bt`l bs p``n`e iff, sbfteiwa lust o` mdnn`e iam` ti r`stir` th` h`dp.
?.0. Obs`mtbi Obs`mtbia a Iam` da drrdy bs ire`r`e, bt bs `dsy ti s`drmh fir sp`mbfflm sp`mbfflm `atry vdnu`s. vdnu`s. Fds Fdstt s`drmh e`p`ae e`p `aess ia th` prb prbamb ambpn` pn` if obs`mt obs`mtbia bia,, ia` if th` `dr `drnb` nb`st st `xdlp `xdlpn`s n`s if th` eb ebvbe vbe`` dae mia miaqu` qu`rr strdt`gy ba dppnb`e ldth`ldtbms dae d eilbadat th`l` ba s`drmh dngirbthls. Ba aul`rbmdn dadnysbs, obs`mtbia bs us`e ti fflae th` riit if d smdndr `qudtbia f f ((x) < ;. Mia Miasbe sbe`r `r da da bat`rvdn Zd, o\ wh`r` f (d) > ; > f ( f (o) ir f ( f (d) 1 ; 1 f f ((o). Bf f f ((x) bs miatbauius, th`a th` bat`rl`ebdt` vdnu` th`ir`l gudrdat``s th` `xbst`am` if d riit ia Z d, o\. Aiw n`t Zd, l < (d + o)/? o` th` lbepibat if Zd, f (l) < ;, th`a w` dr` eia`. Ith`rwbs`, `bth`r Zd, o\. Bf f (
E
f f ((d) dae f ( f (l) dr` if ippisbt` sbga, ir f ( f (o) dae f ( f (l) dr` if ippisbt` sbga. Ba th` firl`r l, o\ ei`s. Ba `bth`r mds`, th` bat`rvdn Zd, Zd, l\ ordmj`ts d riit: ba th` ndtt`r mds`, th` bat`rvdn Z Zl, mds`, w` r`pndm` Zd, Zd, o\ oy th` mirr`spiaebag ordmj`tbag bat`rvdn dae miatbau`. Bf w` obs`mt
?.0. OB]@MUBIA
47 a
p h p . d s c i /
a tbl`s, th`a th` ffladn ordmj`tbag bat`rvdn hds n`agth ? Zd, o\ d titdn if a
n d s a r u i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` a s ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . 0
fuamtbia obs`mt(f fuamtbia obs`mt(f33Fu 33Fuamtbi amtbia, a, d33U, d33U, o33U o33U, , tin3 tin33U) 3U) wh`r wh`r` ` U >3 _`dn (fd, (fd , fo) < (f( (f(d), d), f(o)) f(o)) Kd Kdss ss`r `rt( t(d d > o && fd * fo >< z`ri z`ri(U (U)) )) # mh mh`m `mj j fi fir r ba bapu put t `rri `rrir r fir bt` bt`rdt rdtbia bia < 434 434;; ;; l < (d + o) / ? fl < f(l) f(l) bf do dos( s(fl fl) ) > tin tin r`tura r`tu ra (l, bt`r bt`rdtbia dtbia) ) `ae bf fd * fl > z`ri z`ri(U (U) ) (o, (o, fo) fo) < (l, (l, fl) fl) `ns` (d, (d, fd) fd) < (l, (l, fl) fl) `ae `ae r`tu r`tura ra ((d + o) / ?, 4;; 4;;) )
? = . ; = 4 i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
`ns`bf xZl\ > vdn `ns`bf vdnu` u` d < l + 4 `ns` r`tura r`t ura l `ae `ae r`tura r`t ura ; `ae
∟
d)). Fi (o ∟ d Firr a
ndrg` `aiugh, w` mda stip dae dpprixbldt` th` ordmj`t`e riit oy th` lbepibat if th` ffladn ordmj`tbag bat`rvdn. Uh` finniwbag Cunbd fuamtbia blpn`l`ats thbs strdt`gy.
`ae f(x) < xQ= - 0x + 4.; (x, bt` bt`rdt rdtbia bia) ) < obs obs`mt `mt(f, (f, ;.; ;.;, , ?.; ?.;, , 4`4`-45) 45)
Uh` prim`ss if obs`mtbia `xt`aes bll`ebdt`ny ti s`drmhbag d ebsmr`t` ire`r`e nbst. Uh` iany ebff`r`am` bs thdt w` hdv` ti dnniw fir th` pissbobnbty thdt th` mhis`a vdnu` ei`s ait dpp`dr ba th` nbst. Uh` finniwbag Cunbd mie` r`turas th` pisbtbia if th` vdnu` ba th` nbst. D r`tura`e ; baebmdt`s thdt th` vdnu` bs lbssbag fril th` nbst. fuamtbia obadryYs`drmh(x33V`mtir, obadryYs`drmh(x33V`mtir, vdnu`) d < 4 o < n`a n`agth gth(x) (x) wh whbn bn` ` d >< o l < ebv(d + o, ?) bf xZ xZl\ l\ 1 vdnu vdnu` ` o < l - 4
x < Z‒d‒ Z‒d‒, , ‒o ‒o‒, ‒, ‒e ‒e‒, ‒, ‒f ‒f‒, ‒, ‒g ‒g‒\ ‒\: : prbatna(obadryYs`drmh(x, prbatna(obadr yYs`drmh(x, ‒f‒)) x < Z4, ?, 5, 7, 2\: prbatna(obadryYs`drmh(x, prbatna(obadr yYs`drmh(x, =))
48
p h p . d s c i / s n d a u r i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` n b m L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . 0 = ? . ; 4 = i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
?. ]I_UBAG
?.6. [rbirbty Tu`u`s Ba milput`r smb`am`, d prbirbty qu`u` bs da dostrdmt edtd typ` miasbstbag if j`ys dae prbirbtb prbi rbtb`s. `s. [rbi [rbirbty rbty qu`u` qu`u`ss dr` typbmdnn typbmdnny y ods`e ia h`dps dae obadr obadry y s`drmh`s s`drmh`s.. @dmh j`y hds da dssimbdt`e prbirbty thdt nbv`s nbv`s ia th` h`dp. Bt`ls mda o` `ffflmb`atny `at`r `at`r`e `e bati th` qu`u` dae `xtrdmt`e dmmirebag ti th`br prbirbtb`s. Ba thbs s`mtbia w` orb`y e`smrbo` Cunbd‒s blpn`l`atdtbia blpn`l`atdt bia if prbirbty qu`u`s wbthiut iff`rba iff`rbag g mie`. Ndt`r mhdpt`rs ia grdph th`iry (Ebcjstrd‒ss dngirbthl dae [rbl‒s dngirbthl) dae nba`dr dng`ord (Cdmiob‒s dngirbthl) f`dtur` (Ebcjstrd‒ sil` bat`r`stbag bat`r`stbag dppnbmdtbias. Cunbd‒ Cunbd‒ss prbirbty qu`u`s iff iff`r `r thr`` fuamtbias3 `aqu`u` fir bas`rtbag d a`w bt`l, e`qu`u` fir e`n`tbag th` niw`st prbirbty bt`l, dae p``j fir `xpisbag th` niw`st prbirbty bt`l. Fir `xdlpn`, th` milldaes usbag Edtd]trumtur`s Edtd]trumtur`s pq < [rb [rbirb irbtyT tyTu`u u`u`() `() # `lp `lpty ty qu` qu`u` u` pqZ‒ pqZ‒d‒ d‒\ \ < 4; # `a `aqu qu`u `u` ` ir push push pqZ‒ pqZ‒o‒ o‒\ \ < 0 pqZ‒ pqZ‒m‒ m‒\ \ < 40 p``j(pq) e`qu`u e`q u`u`!( `!(pq) pq) # e`q e`qu`u u`u` ` ir pip
s`t up d prbirbty qu`u` wbth th` j`ys ‒d‒, ‒o‒, dae ‒m‒ dae th` prbirbtb`s 4;, 0, dae 40, r`sp`mtbv`ny. Uh` p``j milldae r`turas (‒o‒, 0). Uh` e`qu`u`! milldae e`n`t`s th` pdbr (‒o‒, 0) fril th` qu`u`. ?.7. [rion`ls [rion`ls (4) Whdt bs th` priodobnbt priodobnbty y thdt d rdaeil p`rlutdtb p`rlutdtbia ia if a a ebstbamt aulo`rs miatdbas dt n`dst ia` pr``xbstbag spnbtt`r9 Whdt dr` th` l`da dae vdrbdam` if th` aulo`r if pr``xbstbag spnbtt`rs9 a? ip`rdtbias. (?) ]hiw thdt th` wirst mds` if qubmjsir qubmjsirtt tdj`s ia th` ire`r if a (=) Wh`a da bt`l bs p``n`e iff th th`` tip if th` h`dp ba h`dpsirt, priv` firldnny firldnny thdt dt list d sbagn` ordamh if th` a`w tr`` lust o` suoc`mt`e ti sbfteiwa ti r`stir` th` tr`` ti d h`dp. (5 (5)) Gbv` Gbv`a a d sirt sirt`e `e drrd drrdy y if aulo aulo`r `rss if n` n`ag agth th a dae dae d au aulo lo`r `r m, wrbt` wrbt` d Cunbd Cunbd prigrd prigrdl l ti fflae th` pdbr if aulo`rs ba th` drrdy whis` sul bs mnis`st ti m . Da `ffflmb`at `ffflmb`at sinutbia mda fflae th` pdbr ba I( I (a) tbl`. (0) ]uppi ]uppis` s` x dae y dr` dr` tw twi i sirt sirt`e `e drrd drrdy y if au aulo lo`r `rss if n` n`ag agth th l dae a, r`sp`mtbv`ny. E`sbga dae blpn`l`at d Cunbd prigrdl ti l`rg` x dae y bati d sbagn` sirt`e drrdy if n`agth l + milpn`xbty l + a a. Riur dngirbthl shiune hdv` milputdtbiadn milpn`xbty I (l + a + a)). (6) ]uppi ]uppis` s` yiu dr` gbv` gbv`a a twi ire`r ire`r`e `e drrdys x dae y if bat`g`rs. bat`g`rs. Bf th`s` r`pr r`pr``s`at bat`g`r s`ts, wrbt` Cunbd fuamtbias ti fflae th`br uabia, bat`rs`mtbia, dae s`t ebff`r`am`. Ei ait us` `xbstbag Cunbd fuamtbias fir s`t ip`rdtbias. (7) Uwi `atrb`s x b dae x c if d aul`rbmdn s`qu`am` x < (x4 , . . . , xa ) r`pr`s`at da bav`rsbia bf x x b 1 xc dae b > c . Wrbt` Wrbt` da `ffflmb`a `ffflmb`att Cunbd fuamtb fuamtbia ia ti miuat th` aulo`r if bav`rsbias ba x . (8) Wrbt` d obs`mtbia dngirbthl ti fflae th` qudatbn`s if d gdlld ebstrbout ebstrbout`e `e rdaeil vdrbdon`. (2) Wrb Wrbt` t` da `ffflmb `ffflmb`at `at Cun Cunbd bd fuamtb fuamtbia ia ti `xt `xtrdm rdmtt th` mnis`s mnis`stt `atry `atry if d ire`r` ire`r`e e aul`rb aul`rbmdn mdn s`qu`am` x ti d gbv`a aulo`r m . (Hbat3 Ps` obs`mtbia.)
?.7. [_ION@L]
p h p . d s c i / s n d a r u i c / r g i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . 0 = ? . ; = 4 i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
42
j (4;) Miuatbag Miuatbag sirt dssul dssul`s `s thdt dnn bt`ls ba d s`qu` s`qu`am` am` o`niag o`niagss ti d sldnn nbst if j `xbstbag `xbs tbag bt`ls. bt`ls. E`sb E`sbga ga dae blpn blpn`l`a `l`att d Cunb Cunbdd fuamtbia fir miuatbag miuatbag sirt wbth I (a). milputdtbiadn milpn`xbt milpn`xbty y I( (44) Oumj`t Oumj`t sirt dssul`s thdt th` `atrb`s if d s`qu` s`qu`am` am` x dr` erdwa bae`p`ae`atny dae uabfirlny fril th` bat`rvdn Z;, Z;, 4\. E`sbga dae blpn`l`at d Cunbd fuamtbia fir oumj`t sirt wbth milputdtbiadn milpn`xbty I I((a) ia dv`rdg`.
p h p . d s c i / s n d a r u i c / g r i l . d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . 0 = ? . ; = 4 i t ; ? 7 / 4 / 0 ; e ` e d i n a w i E
MHD[U@ MHD [U@_ _ =
Grdph Dngirbthls
=.4. Batr Batrieumt ieumtbia bia
Grdph th`iry wds babtbdt`e oy N`iahdre @un`r ba hbs 47=6 stuey if th` ]`v`a Orbeg`s if Jöabgso`rg prion`l Z 4?\. Ba th` bat`rv`abag m`aturb`s, grdph th`iry hds onissil`e bati ia` if th` list f`rtbn` ordamh`s if ebsmr`t` ldth`ldtbms. Uh` dppnbmdtbias ba milput`r smb`am`, nbagubstbms, physbms, mh`lbstry, th` simbdn smb`am`s, obinigy, dae vdrbius ordamh`s if dppnb`e ldth`ldtbms dr` sblpny tii aul`rius ti de`qudt`ny sulldrbz` h`r`. Uh` lie`ra syaiayl a`twirj sugg`sts sil` if th` pit`atbdn if grdph th`iry. Grdphs dr` ldth`ldtbmdn strumtur`s miasbstbag if aie`s dae `eg`s. Da `eg` miaa`mts twi aie`s. ]il`tbl`s aie`s dr` r`f`rr`e ti ds v`rtbm`s ir pibats dae `eg`s ds drms ir nba`s. D grdph ldy o` uaebr`mt`e ir ebr`mt`e: ba th` ndtt`r mds` `dmh `eg` hds da irb`atdtbia dae pibats piba ts fril bts tdbn ti bts h`de. W` wbnn r`s`rv` r`s`rv` th` t`rl grdph fir uaebr`m uaebr`mt`e t`e grdphs dae mdnn ebr`mt`e ebr`mt`e grdp grdphs hs ebgrdp ebgrdphs. hs. Ba d w`bgh w`bght`e t`e grdph ir ebgrdp ebgrdph, h, `dmh `eg` bs dssbga dssbga`e `e d aiaa`gdtbv` w`bght. Uh` decdm`amy ldtrbx if th` grdph ir ebgrdph `amie`s th`s` w`bghts. Uh` dos`am` if da `eg` bs baebmdt`e oy d z`ri `atry if th` ldtrbx. Uhbs ldtrbx bs syll`trbm fir d grdph dae miasbsts `atbr`ny if ;/4 `atrb`s ba th` dos`am` if w`bghts. Uh` a`bghoirs if d aie` dr` this` bll`ebdt`ny decdm`at ti bt. Uh` aulo`r if a`bghoirs if aie` b b ba ba d grdph ir ebgrdph bs mdnn`e th` e`gr`` if b. b. Uh` ebgrdph if d ebsmr`t`-tbl` Ldrjiv mhdba hds `eg` w`bghts gbvbag th` priodobnbtb`s if livbag fril ia` stdt` ti th` a`xt Z 06\. Uh` ebgrdph if d mhdba bs uausudn ba th` s`as` thdt th` tdbn dae h`de h`de if da `eg` mda mibamb mibambe`. e`. Uhbs immu immurs rs wh`a th` mhdba mda r`ld r`ldba ba ba pndm` fir d rdaeil rdaeil aulo`r if `pimh `pimhss (g`a`rdt (g`a`rdtbias bias). ). Uh` w`bght` w`bght`e e decdm decdm`amy `amy ldtrb ldtrbx x [ < ( pbc ) if d Ldrj Ldrjiv iv mhdba bs r`f` r`f`rr`e rr`e ti ds d priodobn priodobnbty bty trdasb trdasbtbia tbia ldtrb ldtrbx. x. Bts riw suls c p bc `qudn 4. ]`mtbia 8.= `xpnir`s th` piw` piw`rr l`thie fir fflaebag th` `qubnborbul ebstrboutbia if d Ldrjiv mhdba. Uhbs mhdpt`r fimus`s ia fiur fuaedl`atdn dngirbthls if grdph th`iry, riughny ba ire`r if suotn`ty Z=0\. Uh` fflrst shiws hi hiw w ti pdss fril th` decdm decdm`amy `amy ldt ldtrbx rbx e`smrb e`smrbptbi ptbia a if d grdph ir ebgrdph ti bts a`bghoirhiie e`smrbptbia. Uh` s`miae minn`mts th` aie`s if d grdph
bati miaa`mt`e milpia`ats. milpia`ats. Uh` thbre, Ebcjstrd‒ Ebcjstrd‒ss dngirbthl, fflaes th` shirt`st pdth fril d siurm` aie` ti `v`ry `v`ry ith`r aie` if d w`bgh w`bght`e t`e grdph ir ebgrdph. Uh` fiurth fiurth,, [rbl [rbl‒s ‒s dngirbthl, fflaes d lbablul spdaabag tr`` if d w`bght`e grdph. Uh` ndtt`r twi dngirbthls ip`rdt` ba d gr``ey fdshbia. Uh`s` summ`ss`s dr` `xm`ptbias ti th` run` if thulo thdt list gr``ey dngirbthls dr` uaeia` oy th`br shirtsbght`ea`ss.
=.?. Fril Decdm`a Decdm`amy my ti A` A`bghoirhiies bghoirhiies
Uh` grdph e`pbmt`e ba Fbgur` =.4 ?4
??
p h p . d s c
=. G_D[H DNGI_BUHL]
6 7
?
5
i / s n d a r u i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . 0 = ? . ; = 4 i t ; ? / 7 4 / ; 0 e ` e d i n a w i E
4
0 =
F BGP_@ =.4. =. 4. D Grdph w wbth bth 7 Ai Aie`s e`s dae = M Milpi ilpia`ats a`ats
hds th` syll`trbm decdm`amy ldtrbx
D
<
; 4 4 ; ; ; ; 4 ; 4 ; ; ; ; 4 4 ; ; ; ; ; ; ; ; ; 4 4 ; . ; ; ; 4 ; 4 ; ; ; ; 4 4 ; ; ; ; ; ; ; ; ;
H`r` dae ba th` s`qu`n, w` deipt th` miav`atbia thdt th` aie`s if d grdph ir ebgrdph wbth a aie`s dr` aulo`r`e 4 thriugh thriugh a a.. Uh` finniwbag Cunbd mie` miav`rts D bati d syst`l if a`bghoirhiies dae w`bghts stir`e ds twi v`mtirs if v`mtirs. fuamtbia decdm`amyYti decdm`amyYtiYa`bghoirhiie( Ya`bghoirhiie(D33DostrdmtLd D33DostrdmtLdtrbx) trbx) (aie`s (ai e`s, , U) < (sb (sbz`( z`(D, D, 4), `nt `ntyp` yp`(D) (D)) ) a`bgho a`b ghoir ir < ZV` ZV`mti mtir{B r{Bat} at}() () fir b < 43a 43aie` ie`s\ s\ w`bght w`b ght < ZV` ZV`mti mtir{U r{U}() }() fir b < 43a 43aie` ie`s\ s\ fi fir r b < 43 43ai aie` e`s s fir fir c < 43 43ai aie` e`s s bf DZb, DZb, c\ !< z` z`ri ri(U (U) ) push!(a`bghoirZb\, push!(a`bghoi rZb\, c) push!(w`b push! (w`bghtZ ghtZb\, b\, DZb, c\) `ae `ae `ae r`tura r`tu ra (a`b (a`bghoi ghoir, r, w`bg w`bght) ht) `ae D < ZZ ; 4 4 ; ; ; ; ;\ \: Z 4 ; 4 ; ; ; ;\: Z 4 4 ; ; ; ; ;\: Z ; ; ; ; 4 4 ; ;\ \: Z ; ; ; 4 ; 4 ;\: Z ; ; ; 4 4 ; ;\: Z ; ; ; ; ; ; ;\\: (a`bghoir, w`bght) < decdm`amyYtiYa`bghoirhiie(D decdm`amyYtiYa`bghoirhiie(D): ):
Uh` mie` bs lir` ir n`ss s`nf-`xpndadtiry. Ia iutput th` c th a`bghoir if aie` b aie` b bs bs dmm`ss`e ds a`bghoirZb\Zc\. Bf j j bs thbs a`bghoir, th`a th` mirr`spiaebag w`bght if th` `eg` ( b, j) bs b.. Bf stir`e ba w`bghtZb\Zc\. Uh` milldae n`agth(a`bghoirZb\) e`nbv`rs th` e`gr`` if aie` b th` ldtrbx D `at`rs th` fuamtbia wbth r`dn `atrb`s, th`a th` `atrb`s if w`bght dr` r`dn-vdnu`e rdth`r thda bat`g`rbat`g`r-vdnu`e. vdnu`e.
=.=. MIAA@MU@E MIL[IA@AU]
p h p . d s c i / s n d a r u i c / g r i . l d
?=
@xdmtny @xdmtn y th` sdl` mie` bs dpp dppnbm nbmdon don`` ti ebgrdphs ebgrdphs.. Ba thbs s`ttb s`ttbag ag th` mill milldae dae n`agth(a`bghoirZb\) n`agth(a`bghoir Zb\) e`nbv`rs th` aulo`r if aie`s r`dmhdon` fril b ba ia` st`p. Uhbs aulo`r bs mdnn`e th` iut-e`gr`` if b. b . Ba-e`gr``s if `dmh aie` mda o` milput`e fril th` trdaspis` if th` decdm`amy ldtrbx. =.=. Miaa` Miaa`mt`e mt`e Milpia`ats Milpia`ats
Ba grdph th`iry d pdth bs d fflabt` s`qu`am` if aie`s d ∟ o ∟ m ∟ · · · miaa`mt`e oy mias`mutbv` mias`mutb v` `eg`s. ]il` duthi duthirs rs furth`r stbp stbpundt undt`` thdt ai aie` shiune o` r`p` r`p`dt`e dt`e.. W`
b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L D B ] i t t m ` o c u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . 0 = ? . ; = 4 i t ; ? / 7 4 / 0 ; e ` d i n a w i E
ilbt thbs `xtrd r`qubr`l`at. Ba d grdph twi aie`s dr` miaa`mt`e bf th`r` bs d pdth o`tw``a th`l. Uh` r`ndtbia if pdthwbs` miaa`mt`ea`ss spnbts th` aie`s if d grdph bati `qubvdn`am` mndss`s. _`mdnn thdt da `qubvdn`am` r`ndtbia ≤ bs mhdrdmt`rbz`e oy th` prip`rtb`s (d) d (d) d ≤ d o ≤ m ea∟4,c . Ui milpn`t` th` baeumtbv` drgul`at, w` shiw thdt th` ebstdam` e ebstdam` e ab < ea∟4,b bs lbabldn. lbabldn. Iur drgul drgul`at `at milpdr milpdr`s `s d shirt`st shirt`st pdth ti ti b wbth th` shirt`st pdth thriugh Ga∟4 ti ti b. Uh` s`miae s`miae pdth hds n` n`agth agth ea∟4,b . Uh` fflrst fflrst p pdth dth G a∟4 . Bt lust th`a babtbdnny pdss`s thriugh G thriugh G a∟4 dae `xbts bt ti sil` sil` aie` c aie` c iutsbe` iutsbe` G trdv`rs` da deebtbiadn suopdth fril c fril c ti b ti b.. Uh` titdn ebstdam` ebstdam` if th` twi pdths dr` dr` e e a∟4,b
?6
p h p . d s c i / s n d a r u i c / g r i . l d b s . w w w / / 3 p t t h ` s ` : t h g b r y p i m r i ` s a ` m b n L D B ] i t t m ` c
=. G_D[H DNGI_BUHL]
=
=
7
?
0
0
?
8
2
4
6
5
F BGP_@ =. ?. Lbab Lbablul lul ]pd ]pdaabag aabag U Ur`` r`` if d Miaa`m Miaa`mt`e t`e Grdp Grdph h b.. Uh` ba`qudnbtb`s m dae ea∟4,c + mcb , wh`r` m dae e wh`r` mcb bs th` n`agth if th` suopdth fril c fril c ti ti b ba`qudnbtb`s mcb ≠ ; dae e dae e a∟4,b ≥ ea∟4,c aiw priv` iur dss`rtbia dae milpn`t` th` baeumtbia. =.0. [rbl‒ [rbl‒ss Dngir Dngirbthl bthl
[rbl Z455\, Cdraìj Z82\, dae Ebcjstrd Z58\ shdr` th` mr`ebt fir thbs odsbm grdph dngirbthl. rbth l. Bt bs ba bavij vij`e `e oy d a`twirj a`twirj riut`r ti lbablbz lbablbz`` th` riut`s ti ith`r milpia`a milpia`ats ts ba d a`twirj. a`tw irj. D tr`` bs d miaa`mt`e miaa`mt`e grdph wbth ai sup`r sup`ruius uius `eg`s `eg`s.. Ba ith`r wire wires, s, e`n`tbag day `eg` ebsm ebsmiaa`m iaa`mts ts th` tr``. tr``. O`mdus` O`mdus` if th` dssbga dssbga`e `e `eg` w`bghts, tr``s ldy vdry wbe`ny ba th`br `eg` w`bght suls. Firtuadt`ny, d gr``ey dngirbthl ebsmiv`rs th` lbablul spdaabag tr`` (L]U). Fbgur` =.? e`pbmts th` L]U if bts w`bght`e grdph oy thbmj `eg`s. Uh` iv`rdnn dngirbthl o`drs d striag r`s`londam` ti Ebcjstrd‒s shirt`st pdth dngirbthl.
o u s a i b t u o b t s r b e ` _ . ; 4 . 6 6 . 0 = ? . ; = 4 i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
@dmh aie` bs trdmj`e ds vbsbt`e ir ait y`t vbsbt`e. Dt stdg` a if th` dngirbthl, da `eg` if lbablul w`bght bs dee`e ti th` murr`at suotr`` U a∟4 ti prieum` da `andrg`e suotr`` U a wbth ia` lir` aie`. Ia` aie` if th` `eg` bs dttdmh`e ti U a∟4 , dae ia` aie` fdnns iutsbe` ebsmiaa`mts U a . U a∟4 . E`n`tbag thbs `eg` ir day pr`vbiusny dee`e `eg` ebsmiaa`mts U D prbirbty qu`u` dgdba pndys d mrbtbmdn rin`. Hiw`v`r, Hiw`v`r, th` j`ys if th` prbirbty qu`u` dr` aiw `eg`s rdth rdth`r `r thda aie`s, aie`s, dae prbirb prbirbtb`s tb`s dr` w`bghts rdth`r thda ebstda ebstdam`s. m`s. @eg`s @eg`s dr` pipp`e iff th` qu`u` ds th`y dr` dee`e ti th` lbablul spdaabag tr`` ir miaa`mt twi dnr`dey dnr` dey vbsbt` vbsbt`e e aie`s. Uh` tr`` griws fril bts tbp, whbmh bs upedt`e upedt`e ds th` dngir dngirbthl bthl prigr`ss` prigr `ss`s. s. Uh` `eg`s a`bghoirba a`bghoirbag g da uavbsb uavbsbt`e t`e tbp dr` push`e push`e iati th` qu`u`. Uh` tr`` bts`nf bs r`mire`e ds d s`qu`am` if `eg`s. Uh` mie` fir [rbl‒s dngirbthl finniws. usbag Edtd]trumtur`s Edtd]trumtur`s fuamtbia prbl(a`bghoir33Drrdy{Drrdy{Bat, prbl(a`bghoir33Drrdy{Drrdy{Bat, 4}, 4}, w`bght33 w`bg ht33Drrdy Drrdy{Drr {Drrdy{U dy{U, , 4}, 4}) wh`r wh`r` ` U >3 Aulo Aulo`r `r # aie`s aie` s < n`agt n`agth(a` h(a`bgho bghoir) ir) vbsbt`e vbsb t`e < fdns fdns`s(a `s(aie`s ie`s) ) (l (lst stYa Yaie ie`s `s, , b) < (4 (4, , 4) # ba babt btbd bdnb nbz` z` L]U L]U wbth wbth aie` aie` 4 ds tb tbp p j`y < Drr Drrdy{ dy{Uup Uupn`{ n`{Bat Bat, , Bat Bat}, }, 4}( 4}(uae uae`f, `f, ;) # e`f e`fba` ba` j`y j`ys s prbirbty < Drrd prbirbty Drrdy{Fn y{Fnidt65 idt65, , 4}(u 4}(uae`f, ae`f, ;) # e`fba e`fba` ` prbi prbirbtb` rbtb`s s pq < [rbi [rbirbty rbtyTu`u Tu`u`(zb `(zbp(j`y p(j`y, , prbi prbirbty) rbty)) ) # babtb babtbdnbz dnbz` ` qu`u` lst < Drrdy{Uupn`{B Drrdy{Uupn`{Bat, at, Bat}, 4}(uae`f, ;) # lbablul spdaabag tr``
=.0. [_BL‒] DNGI_BUHL
p h p . d s c i / s n d a r u i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e `
?7
whbn` lst whbn` lstYai Yaie`s e`s > aie aie`s `s bf !vb !vbsbt sbt`eZ `eZb\ b\ # mh` mh`mj` mj`e e bf th` aie aie` ` hds o``a vbs vbsbt` bt`e e vb vbsb sbt` t`eZ eZb\ b\ < tr tru` u` # ldrj ldrj th th` ` aie` aie` ds vbsb vbsbt` t`e e fir j < 43n 43n`ag `agth( th(a`b a`bgho ghoirZ irZb\) b\) # dee a`w `eg `eg`s `s ti th` qu`u qu`u` ` c < a`bg a`bghoir hoirZb\Z Zb\Zj\ j\ pqZ(b, pqZ(b , c)\ < w`bgh w`bghtZb\ tZb\Zj\ Zj\ `ae `ae ((j, ((j , n), vdn) < p`` p``j(p j(pq) q) # mhi mhiis` is` nbg nbght` ht`st st `eg `eg` ` e`qu`u e`q u`u`!( `!(pq) pq) # pip `eg `eg` ` iff qu` qu`u` u` bf !v !vbs bsbt bt`e `eZj Zj\ \ # bf ai ait t pd pdrt rt if tr`` tr``, , pu push sh th th` ` `e `eg` g` iati iati tr`` tr`` push!(lst push !(lst, , (j, n)) lstYaie`s < lstYaie`s + 4 b < j # j bs th` a`w tbp `ns`bf !vbsbt`eZn\ push!(lst push !(lst, , (j, n)) lstYaie`s < lstYaie`s + 4 b < n # n bs th` a`w tbp `ae `ae r`tura r`tu ra lst `ae D < Z; ? ; 6 ;: ? ; = 8 0: ; = ; ; 7: 6 8 ; ; 2: ; 0 7 2 ;\: (a`bghoir, w`bght) < decdm`amyYtiYa`bghoirhiie(D decdm`amyYtiYa`bghoirhiie(D): ): lst < prbl(a`bghoir prbl(a`bghoir, , w`bght)
Uh` mirr`mta`ss if [rbl‒s dngirbthl st`ls fril d sblpn` prip`rty if spdaabag tr``s dae mymn`s. _`mdnn thdt d mymn` bs d pdth thdt stdrts dae `aes wbth th` sdl` aie`. [ _I[I]BUBIA =.0.4. ]uppis` w` tdj` d spdaabag tr`` U dae dee day `eg` ait ba U ti bt. Uhbs dmtbia dmtbia mr`dt` mr`dt`ss d mymn mymn`. `. Bf w` r`liv` day `eg` fr fril il th` mymn`, th`a w` dr` n`ft wbth d pissbony a`w spdaabag tr``.
_ . ; 4 . 6 6 . 0 = ? . ; = 4 i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
[riif3 Uhbs priif bs r`n`gdt`e ti prion`l (0).
[ _I[I]BUBIA =.0.?. Fir d w`bght`e miaa`mt`e grdph, [rbl‒s dngirbthl t`rlbadt`s dft`r d fflabt` aulo`r if st`ps wbth d lbabldn spdaabag tr``. [riif3 W` fflrst priv` thdt [rbl‒s dngirbthl miastrumts d spdaabag tr``. Bt stdrts wbth d tr`` U 4 wbt wbth h d sbag sbagn` n` aie`. ]up ]uppis` pis` dt stdg stdg`` a ∟ 4 bt prieum prieum`s `s d tr`` tr`` U a∟4 . Dt stdg` a bt dees H`am`, U a < U a∟4 ∣ {`} iutsbe` U a∟4 . H`am`, U da `eg` ` `eg` ` thdt thdt hds ia` aie` ba U ba U a∟4 dae ia` aie` iutsbe` U dnsi bs d tr``. Wh`a bt `xhdusts dnn aie`s, [rbl‒s dngirbthl th`r`fir` yb`nes d spdaabag tr`` U U .. Bf th` tr`` U tr`` U bs bs ait lbabldn, n`t a n`t a o` th` fflrst stdg` dt whbmh U a∟4 mda o` `lo`ee`e ba d lbabldn spdaabag tr`` ] tr`` ] ,, out U out U a mdaait mdaait.. H`am` H`am`,, th` `eg` `eg` ` ` dee`e ba gibag fril U fril U a∟4 ti ti U a fdnns iutsbe` iutsbe` ] . [ripisbtb [ripisbtbia ia =.0.4 blpnb blpnb`s `s thdt thdt ] ∣ ∣ ` miat miatdbas dbas d mymn`. O`mdu O`mdus` s` ` hds ia` `aepibat ba U ba U a∟4 dae ia` `aepibat iutsbe` U iutsbe` U a∟4 , th`r` lust o` daith`r `eg` f `eg` f dniag th` mymn` wbth `xdmtny ia` `aepibat ba U a∟4 . Uh` dssulptbia w dssulptbia w ` ≥ wf bs ebmtdt`e f fril ] dae dee ` dae stbnn wbae up wbth oy [rbl‒s [rbl‒s dngirbthl. dngirbthl. Uhus, w` mda e`n`t` e`n`t` f fril ]
d lbabldn lbabldn spdaab spdaabag ag tr``. Uhbs miatrd miatrdebmt ebmtss iur dssulptbia dssulptbia thdt thdt U U a bs bamiasbst`at wbth d lbabldn spdaabag tr``.
?8
h p . d s c i / s n d a r u i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r p y i m r i ` s a ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ ; . 4 . 6 6 . 0 = ? . ; = 4
=. G_D[H DNGI_BUHL]
=.6. [rio [rion`ls n`ls (4) [riv [riv`` thdt bf x x bs d aie` if d grdph hdvbag iee e`gr``, th`a th`r` `xbsts d pdth fril x fril x ti ti daith`r aie` y aie` y if if th` grdph hdvbag iee e`gr``. (?) [riv` [ripisbtbia =.5.4. (=) Miasb Miasbe`r e`r d miaa` miaa`mt`e mt`e grdph wbth wbth a aie` aie`s. s. E`lia E`liastrd strdt` t` thdt bt bs d tr`` bf dae iany bf bt hds a hds a ∟ 4 `eg`s. (5) [riv` thdt o`tw` o`tw``a `a day twi aie`s if d tr``, th`r` bs ia` dae iany ia` miaa` miaa`mtbag mtbag pdth. (0) [riv` [ripisbtbia =.0.4. (6) Bf dnn if th` `eg` w`bght w`bghtss if d miaa`mt` miaa`mt`e e grdph dr` uabqu`, th`a priv` thd thdtt th` grdph hds `xdmtny ia` lbablul spdaabag tr``. (7) N`t D o` th` decdm`amy ldtrbx if d grdph. N`t th` ath a th piw`r hdv` `atrb`s d `atrb`s d abc . ]hiw ]hi w thdt thdt dabc `qudns `qudns th` aulo`r aulo`r if pdths pdths fri fril l b ti c ti c wbth wbth `xdmtny `xdmtny a st`ps. V`rbfy yiur r`sunt `lpbrbmdnny fir d sblpn` grdph. (8) [riv [riv`` thdt Ebcjstrd‒ Ebcjstrd‒ss dngir dngirbthl bthl hds ruaab ruaabag ag tbl` tbl` I(@ nig nig A ), wh`r` wh`r` @ @ bs th` aulo`r if `eg`s dae A dae A bs bs th` aulo`r if aie`s.
(2) [riv` thdt [rbl [rbl‒s ‒s dngir dngirbthl bthl hds ruaabag tbl` tbl` I wh`r` @ bs bs th` aul I (@ nig nig A ), wh`r` @ o`r if `eg`s dae A dae A bs bs th` aulo`r if aie`s. (4;) [rigrdl [rigrdl `bth `bth`r `r Jis Jisdrdc drdcu‒ u‒ss dngi dngirbth rbthl l Z400\ ir Udrc drcda‒ da‒ss dngirb dngirbthl thl Z46=\ fi firr fflaeb fflaebag ag th` striagny miaa`mt`e milpia`ats if d ebgrdph. (44) [rigrdl Jrusjdn‒s Jrusjdn‒s dngirbthl Z4;5\ fir fflaebag th` lbablul spdaabag tr`` if d w`bght`e grdph.
i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
p h p . c s d i / s n d a r u i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i r m i ` s a ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 . 0 = ? . ; = 4 i t ; ? / 7 4 / 0 ; e ` e d i
MHD[U@ MHD [U@_ _ 5
[rbldnbty U`stbag 5.4. Batr Batrieumt ieumtbia bia
Uwi g`a`rdtbias dgi aulo`r th`iry wds miasbe`r`e ia` if th` pur`st ordamh`s if pur` ldth`ldtbms. Bt hds aiw o``a sunnb`e oy dppnbmdtbias ti hbgh-sp``e drbthl`tbm ia milput`rs, aul`rbmdn dadnysbs ba g`a`rdn, mryptigrdphy, dae `sp`mbdnny s`mur` milluabmdtbia. [rbl` aulo`rs hdv` o``a th` ios`ssbia if aulo`r th`irbsts sbam` @umnbe. Uh`s` lyst`rbius `atbtb`s dr` s``lbagny smdtt`r`e dt rdaeil thriughiut th` adturdn aulo`rs. R`t prbl`s shiw sil` surprbsbag r`gundrbtb`s. Fir bastdam`, th` prbl` aulo`r th`ir`l sdys th` aulo`r if prbl`s n`ss thda ir `qudn ti a bs dsylptitbm ti a . Uh`r Uh`r`` dr` lday uar`sin uar`sinv`e v`e a na qu`stbias doiut prbl`s. Uh` twba prbl` miac`mtur` e`dns wbth prbl`s a dae a + ?. Bs th`r` da bafflabt` s`qu`am` if sumh prbl`s9 Uh` `lpbrbmdn `vbe`am` iv`rwh`nlbagny suppirts th` miac`mtur`, out ai ia` hds priv`e bt. Ba thbs mhdpt`r w` tdj` up th` sblpn`r prion`l if e`t`rlbabag wh`th`r d aulo`r a bs prbl`. Uh` ]b`v` if @rdtisth`a`s sinv`s th` prion`l wh`a a bs sldnn. Hiw`v`r, th` sb`v` bs fdr tii mulo`rsil` fir th` ndrg` prbl`s if murr`at bat`r`st. Uh` Lbnn`r”_doba dngirbthl Z4=;, 450\ stueb`e h`r` bs fdst, strdbghtfirwdre ti blpn`l`at, dae bah`r`atny priodobnbstbm. Uh`r` dr` e`t`rlbabstbm dngirbthls fir th` sdl` purpis`, out th`y dr` lir` milpnbmdt`e ti uae`rstdae dae lumh sniw`r ba prdmtbm` Z 4, 452\. Pafirtuadt`ny, th` Lbnn`r”_doba ei`s ait e`nbv`r th` prbl` fdmtirbzdtbia if d milpisbt` aulo`r. Uhbs bs d lumh hdre`r prion`l. _`de`rs whi ldst`r th` ldt`rbdn ba thbs mhdpt`r wbnn o` ba d giie pisbtbia ti tdmjn` th` _]D (_bv`st”]hdlbr”Den`lda) dngirbthl Z404\ fir edtd `amryptbia. Uh` _]D dngirbthl wirjs o`mdus` fdmtirbag milpisbt` aulo`rs bs si mhdnn`agbag. Dpp`aebx D.? bs bat`ae`e ti orbag r`de`rs up ti sp``e ia `n`l`atdry aulo`r th`iry. _`de`rs wbnn dnsi o`a`fflt fril th` `airlius drrdy if oiijs ia aulo`r th`iry. W` `sp`mbdnny r`mill`ae th` mndssbms Z 0, 7?, 4=0\. Uh`s` dr` oiuae ti stblundt` yiur dpp`tbt` fir thbs o`dutbfun dae drmda` ordamh if ldth`ldtbms. 5.?. [`rf [`rf`mt `mt [i [iw`rs w`rs
W` o`gba iur `xpisbtbia if prbldnbty t`stbag oy tdmjnbag d lumh sblpn`r prion`l. D pisbtbv` bat`g`r a bs d p`rf`mt piw`r bf bt mda o` `xpr`ss`e ds j c fir bat`g`rs c 1 4 dae j 1 4 . Ia` mda e`sbga d fdst dngirbthl fir t`stbag thbs prip`rty oy r`nybag ia twi mrumbdn basbghts. Fbrst, c nig? a iwbag ti th` ba`qudnbty ? c jc a fir dnn r`n`vdat c dae j . ]`miae, th` `xpia`at c mda o` furth`r furth`r r`strbmt` r`strbmt`e e ti th` s`t if prbl prbl`` aulo`rs. aulo`rs. Bae` Bae``e, `e, bf a < j c dae c < pq wbth p prbl`, th`a (j q ) p < a . Ui bsindt` th` p`rtba`at prbl`s, w` mda
≥
≥ ≥
n a w i E
mdnn ia th` ]b`v` if @rdtisth`a`s ebsmuss`e `drnb`r. Fir d gbv`a prbl` c , th` iany pissbon` j bs c a. H`r` bs Cunbd mie` blpn`l`atbag th` p`rf`mt piw`r dngirbthl.
∓
fuamtbia p`rf`mtpiw`r p`rf`mtpiw`r(a33Bat`g`r) (a33Bat`g`r) l < Bat(fniir(nig Bat(fniir(nig(?, (?, a))) # bat`g`r pdrt if nig ods` ? if a prbl`Ynb prbl `Ynbst st < `rdt `rdtisth` isth`a`s( a`s(l) l) ?2
=;
p h p . d s c i n / s d a r u i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i s ` a ` m b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . = 0 ? . ; = 4 i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
5. [_BLDNBUR U@]UBAG
fir c ba prb prbl`Y l`Ynbs nbst t j < Bat Bat(ri (riuae uae(aQ (aQ(4 (4 / c)) c))) ) bf bs`qudn( bs`qudn(jQc, jQc, a) r`tura r`tu ra tru` `ae `ae r`tura r`tu ra fdns fdns` ` `ae p`rf`mtpiw`r(4;;;)
∓
4
@xdlpn` 6.? sj`tmh sj`tmh`s `s d o`tt`r o`tt`r dngirb dngirbthl thl fir `xBa th` mie` c a bs mdnmundt`e ds ` c na a . @xdlpn` trdmtbag riits ods`e ia A`wtia‒s A`wtia‒s l`thie. Uh` Odoyniabda l`thie fir `xtrdmtbag squdr` riits bs d sp`mbdn mds`. 5.=. Lieund Lieundrr Drbthl`tbm dae Gri Griup up Uh`iry
_`de`rs uafdlbnbdr wbth `n`l`atdry aulo`r th`iry shiune r`vb`w Dpp`aebx D.? dt thbs cuamtur`. cuamtur`. H`r` ia` fflaes th` e`fflabtb e`fflabtbia ia if th` dng`ord dng`ordbm bm strumtur` strumtur` Sa . Uhbs Uhbs ioc` ioc`mt mt bs cust th` s`t if bat`g`rs ;, 4, . . . , a 4 `qubpp`e wbth lieundr deebtbia dae lieundr
{
∟ }
luntbpnbmdtbia. Bf d d dae o o`niag ti Sa , th`a d + o lie a dae do lie a dr` e`ffla`e ds th` d + o dae do ia ebvbsbia oy a. Ait` thdt Sa bah`rbts th` usudn millutdtbv` r`ldbae`rs if d dae dssimbdtbv` ndws if drbthl`tbm. Ds `xp`mt`e, th` deebtbv` dae luntbpnbmdtbv` luntbpnbmdtbv` be`atbtb`s r`sp`mtbv`ny v`ny.. Ebv Ebvbsbi bsbia a bs ait dnwdy dnwdyss pissbon` pissbon`.. Bf gme( gme(d, a) < 4, th`a if Sa dr` ; dae 4 , r`sp`mtb ∟4 ∟4 th`r` `xbsts d r`mbprimdn d sumh thdt d d < 4 lie a . Ba th` sp`mbdn sp`mbdn mds mds`` wh`r` a bs ∟4 prbl`, d `xbsts fir dnn d < ; lie a , dae Sa bs dng`ordbmdnny d ffl`ne. Uh` s`t if bat`g`rs u Sa wbth gme(u, a) < 4 bs e`ait`e Pa . Uh` bat`g`r bat`g`rss ba Pa dr` sil`tbl`s r`f`rr`e ti ds uabts: Pa bs mnis`e uae`r luntbpnbmdtbia dae miatdbas 4 dae Bae``e, bf u u 4 dae u ? shdr` ai aiatrbvbdn ebvbsirs wbth a , th`a th`br 4 < a 4 lie a . Bae``e, prieumt u 4 u? dnsi shdr`s ai aiatrbvbdn ebvbsirs wbth a . Uh` r`mbprimd r`mbprimdnn if `v`ry uabt u bs dnsi d uabt. Uhbs fdmt bs priv` priv`e e ba [ripi [ripisbtb sbtbia ia D.?.7. Ia` mda sulldrb sulldrbz` z` iur fflaebags fflaebags oy aitbag thdt Pa miastbtut`s d fflabt` millutdtbv` griup. W` wbnn wbnn a``e a``e ti ebp orb orb`y `y bati th` th`iry th`iry if fflabt` fflabt` griu griups. ps. D griup griup G bs d s`t sumh ds P a `qubpp`e wbth luntbpnbmdtbia dae piss`ssbag da be`atbty `n`l`at 4 sdtbsfybag d4 < 4d < d fir dnn d G. Furth`rlir`, `v`ry `v`ry griup `n`l`at d G hds d n`ft bav`rs` o sdtbsfybag od < 4. _`gdren` _`gdren`ss ss if wh`th`r G bs millutdtbv`, ia` mda shiw thdt d n`ft
∈
∟
∟
∈
∈
bav`rs`` bs dnsi d rbght bav`rs`. bav`rs bav`rs`. Uh` ire`r if d fflabt` griu griup p G bs cust bts mdrebadnbty G . D suogriup H if d griup G bs d aia`lpty suos`t if G G mnis`e uae`r luntbpnbmdtbia dae th` firldtbia if r`mbprimdns (bav`rs`s). (bav`rs`s). D suogriup dutildtbmdnny miatdbas th` be`atbty `n`l`at 4 if th` griup. W` wbnn a``e th` finniwbag twi prip`rtb`s if d suogriup, th` s`miae if whbmh bs jaiwa ds Ndgrdag`‒s th`ir`l.
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[ _I[I]BUBIA 5.=.4. Bf d aia`lpty suos`t H if d fflabt` griup G miatdbas 4 dae bs mnis`e uae`r luntbpnbmdtbia, th`a H bs bs d suogriup. Uh` ire`r H if d suogriup H ebvbe`s ebvbe`s th` ire`r G if th` griup.
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[riif3 Ui priv` th` fflrst dss`rtbia, w` lust shiw thdt th` bav`rs` d∟4 if d H o`niags ti H . Miasbe`r Miasbe`r th th`` ldp H H e`ffla`e oy o do. Uhbs ldp ldp bs ia`-t ia`-ti-ia i-ia`` do < dm, th`a o < m. ]b o`mdus` bf do ]bam am`` H bs fflabt`, th` ldp bs dnsi iati. iati. H`am` H`am`,, th`r` `xbsts o H wbth wbth do < 4. Fir th` s`miae dss`rtbia, w` batrieum` th` aitbia if d n`ft mis`t. Miasbe`r th` ldp o do fir d ait a`m`ssdrbny ba H . Bt ldps H bati d s`t dH . _`gdren`ss
↟
∈
↟
↟
∈
5.0. F@_LDU‒] NBUUN@ UH@I_@L
p h p . d s c i / s n d r a u i c / g r i . l d b s . w w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a m ` b n L D B ] i t t m ` c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . 0 = ? . = ; 4 i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
=4
d , th` s`ts dH dae H hdv` th` sdl` mdrebadnbty if th` mhibm` if d mdrebadnbty.. Uwi mis`ts dH dae mH `bth`r `bth `r mibambe mibambe`` ir dr` titd titdnny nny ebscib ebscibat. at. Bf th`y iv`rnd iv`rndp, p, th`a dh 4 < mh? fir sil` h 4 dae ∟4 ∟4 h? ba H . H`a H`am`, m`, m d H dae d m H , blpnybag thdt dH mH dae mH dH . Furth`rlir`, `v`ry d H bs miv`r`e oy th` mis`t dH . Uhus, Uhus, th` mis mis`ts `ts if H firl d
∈ ∈
∈
⊏
⊏
pdrtbtbia if G G wbth `qudn sbz` onimjs. Uhbs mda iany o` tru` bf H ebvbe`s G .
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5.5. @xpia @xpia`atbdtb `atbdtbia ia ba Lieundr Drbthl` Drbthl`tbm tbm
Uh` dngirbthls fir prbldnbty t`stbag e`p`ae ia milputbag ndrg` piw`rs ol lieuni a `ffflmb`atny dae mirr`mtny. Uh` adbv` dppridmh if fflrst milputbag o l ba irebadry drbthl`tbm dae th`a ebvbebag oy a ruas th` rbsj if aul`rbmdn aul`rbmdn iv` iv`riw riw.. Bf o bs miav`rt`e ti d r`dn aulo`r aulo `r dae th`a `xpia`at `xpia`atbdt` bdt`e, e, niss if dmmur dmmurdmy dmy bs da bssu` bssu`.. D o`tt`r be`d bs ti mdrry mdrry iut dnn milputdtbias ba S a wbthba th` miat`xt if d ebvbe` dae miaqu`r strdt`gy strdt`gy.. Bf l l < ?j bs l < ?j + 4 bs iee, th`a o l < (oj oj ) o lie a . `v`a, th`a ol < o j oj lie a. Bf l Uh` finniwbag r`mursbv` Cunbd fuamtbia blpn`l`ats thbs strdt`gy. strdt`gy. Wbth lbair liebfflmdtbias, th` sdl` nigbm dppnb`s ba firlbag piw`rs if ldtrbm`s.
×
×
×
fuamtbia piw`rli fuamtbia piw`rlie(o3 e(o33U, 3U, l33U, a33U a33U) ) wh`r wh`r` ` U >3 Bat` Bat`g`r g`r bf l 3 Bat` Bat`g`r g`r
D ] B i t t m ` c o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . 0 = ? . ; = 4 i t ; / ? 7 4 / 0 ; e ` e d i n a w i E
bf a ?) || bs` bs`v`a v`a(a) (a) || p`r p`rf`m f`mtpi tpiw`r w`r(a) (a) r`tura r`t ura prb prbl` l` < fdn fdns` s` `ae s < trd trdbnb bnbagY agYz`r z`ris( is(a a - 4) e < eb ebv( v(a a - 4, ?Q ?Qs) s) fi fir r tr trbd bdn n < 43j 43j d < rdae rdae(? (?3( 3(a a - ?) ?)) ) x < pi piw` w`r rli lie( e(d, d, e, a) # x < dQ dQe e lie a bf x < ; r`tura r`t ura prb prbl` l` < fdns` fdns` `ae x < r` r`l( l(xQ xQ?, ?, a) bf x Pa . Bt finniws thdt
∟
∈ ∟
∟
∈
| |
´
| | |G|| | ≥ | |P| | ≥ a ∟ 4 . ? ? a
d wbth priodobnbty oiuae`e doiv` oy th` Uhus, a bs pdss`e fir `dmh bae`p`ae`at mhibm` if d 4 qudatbty ? . 5.7. [rio [rion`ls n`ls
(4) „Mds „Mdstba tbag g iut aba` aba`s" s" bs da `n`l`a `n`l`atdr tdry y l`thi l`thie e if mh` mh`mjb mjbag ag drb drbthl thl`tb `tbm. m. Fir da bat`g`r a n`t e(a) `qudn th` sul if th` e`mbldn ebgbts if a. ]hi ]hiw tthd hdtt
5.7. [_ION@L]
p h p . d s c i / s n d a r u i c / g
=0
e(l + a) < e (l) + e(a) lie 2 dae thdt e(la) < e (l)e(a) lie 2. Bnnustrdt` hiw th`s` be`atbtb`s mda s`rv` ds sblpn` mh`mjs ia deebtbia dae luntbpnbmdtbia dae oy `xt`asbia if suotrdmtbia dae ebvbsbia. (?) N`t a hdv` e`mbldn `xpdasbia e l el∟4 e; . [riv` thdt a bs ebvbsbon` oy 44 bf dae iany bf e e; e4 + e? < ; lie 44. (=) [riv [riv`` thdt th th`` s`t ] if if prbl`s bs bafflabt`. (Hbat3 Miasbe`r 4 + p∈] p.) (5) ]hi ]hiw w thdt th th`` gdp g o`tw``a summ`ssbv` prbl` aulo`rs mda o` drobtrdrbny ndrg`.
∟
∟···
···
r i . l d b s . w w / / 3 p t t h ` ` s : t h g b r y p i m r i ` s a ` m b n L D B ] i t t m c ` o u s a i b t u o b r t s b e ` _ . ; 4 . 6 6 . 0 = ? . ; = 4 i t ; ? / 7 4 / 0 ; e ` e d i n a w i E
(Hbat3 Miasbe`r th` s`qu`am` if aulo`rs o`gbaabag wbth (g + 4)! + ?.) (0) Ba th` miat`x miat`xtt if ods` 0 lieundr lieundr drbt drbthl`t hl`tbm, bm, e`liastr e`liastrdt` dt` thdt th` `qudt `qudtbia bia ?
bat`g`r sinutbias. xwi
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