Lecture Notes o Graph Algorithms: Shortest Path Given a weighted

!"$#&%' (*),+"-%-./(0%213/%4
57698;:=<>@?A:=69BDCFEG:IHKJLHF6NMO:IPQEG:IHBRMS>ITUCWVYX[Z]\-^&_`SJL?*69EGCa?b:=6NBRCcEedgfU<hPQEG69iR<jYk4_[l m,no>pTUTq69<UB
:IHFBD:IrEOiaMG:I>psutv8D>psNfU:IHw?b:=6NBRCcESrQJAEOCU:?b:=6NBRCcEipdx>Ty>pEOCwz{X |~}pI^S};€S^=!==^S}D‚„ƒ6…rEGCU:r]fUn TFd/EGCU:
?b:!69BRCcESrbiId-6NESrb:IHcBR:Ir&k
‚ ‡9‹ ‡
†
‰
‡
ˆ
j/Z9zh`X € jŠZ~} € ^Œ} `Œ
Ž CU:xrCUiDMOEG:IrE]tvTh>IEOCdgMGiRnYoEOi}6…r
‘ Zv0^S}U`’X ” •“ n–£ 69^<h—Dj/Z9z˜`!™šœ › }Užc^{6ŸiRd EGCU>:=ThMO?*>pEG6…C@r: dgMGiRnYoEOi}:=¡c6¢r]ESr
  › }–HF:=<qiREO:„r*>Th>IEOCz¤dgMGiRnYoEOi¥}h¦
§
rCUiDMOEO:„rE7Th>pEGCaZ¨ª©$`-dgMGiRnYoEOi}6…r«>I<F¬Th>pEGC­zo?*69EGCo?b:=6NBRCcE7jŠZNz˜`X ‘ Z~0^S}y`Œ¦
®’' a "-% ¯¥%°+"-% .Š(0%21
Š% 1±4"²!.Š' (
³ ¨;6N<UBRsN:Qt´riRfqMSPQ:¨c©r*TUMOiDµUs9:=n¶k’·L69<hH¨c©rbdgMOiRn¸><UicHc:¹­Z~riDfUMSPQ:D`EOi>ps9sºiREGCU:=M7<UicHF:Ir&¦
³ EG¨;696NiR<U<ºBR`SsN:Q¦ t´HF:IrEG69<h>IEO6NiR<a¨c©reTUMOiDµUs9:!nk7·’69<hH¨ª©4r«dgMGiRn»>Is9s-<qicHF:IreEOi>­BD698;:=<<UicHc:­¼xZvHF:„rEO6N<h>„t
³ ¨;6N<UBRsN:Qt~Th>p6NMe¨c©½TUMGiRµUsN:=nk ·’69<hHo>I<¨c©adgMOiRn¸@EOi}hJF?*CU:=MG:x>p<yH}¥>pMO:xBR6N8;:=<–<UicHF:„rQ¦
³ §«sNsŸt~Th>p6NMSr*¨c©r¾TUMGiRµUsN:=nk ·’69<hHo>I<¿¨ª©½dgMOiRn¸@EOi}dgiRM7:=8;:=MO¬¥Th>I69M*ipd <UicHF:„r«>p<hH}˜¦
À
!"! #%$'&)(*+,-.0/+12%-,/.%+$34,5-.6.$7%-,/.8$39,5-.:<;4=>$?@
FJK$L
!$M.?N(IOP13/.
/
.?,E>FQ1/.
A $CBD-+$E>FGE.$"!8$EHB.!I.?IRTSU.?IEUV)"-W-,YXZ[SY\]V2Z@^F_$% `
/ `cb
` !$U+,5-N/+1 13/.
a
dQ-$?eF ` fgh%-,/.%8$3iI-U1/.
/ `cb
V
X Z]\-^&_` z@‡ X
|~}pI^S‡ };€S^=‡ =!=^S}R‚=ƒ
}I }D‚ ‡
z } }
‡z
‡} } z X
|~} ^Œ} €&^!===^S} ƒ
jTkDlil,mn ZgµF¬¤PQiR<cEOMŒ>RHF6…PQEO6NiR<KJ` o«:„PQiRn–Thi r:eTh>IEOCz@6N<FEGi
‡
› prq }  L› q s } ` › s5t }D‚p^
p
}


?b:/Ch>=8;:/j/Z9zh`bX jŠZNz  ‡ i` uwj/Z9z ‡ ‡ ` i` u jŠZNz ‡ ` ‚ `S¦
§erOr]fUn–:eEGCU:=MG:Š6…r*>Th>pEGC–z v w ` dgMOiDnY} EGi–} ` rfyP CoEOCy>pE7j/Z9#z ‡v ` ` \ j/Z9z ‡ ` `S¦ Ž Cq:=<
}p › prq } ‡ L› xq s } ` › s5t }D‚p^
6¢r*>Th>IEOC?*Cqi;r:/?A:=69BDCFE
j/Z9zc ‡ `iu jŠZNz ‡v ` `iuwj/Z9z ` ‚p`
6¢rbsN:IrOrbEGCh>p<¶jŠZNz˜`Œ¦ Ž CU6…r*PQiR<cEOMŒ>RHF6…PQESrbEGCU:x>RrGrfUn–TqEO69iD<oEOCh>IEbz–6…r*>p<¨ª©½dgMOiRn¸}p*EOi}R‚I¦
y
z|{}J~YI€J{ƒ‚„{€8}9…†@{ˆ‡U}{ˆ‰~‹ŠG‡ŒŠ4{}J~YI€J{ƒ‚„{€8}9…†[QŽ4i+{ˆ‰
o< TUMS>DPQEO6…P=>psºTUMOiDµUs9:=n¤rQJF<q:=B;>IEO698;:&tv?A:=69BDCFEA:IHFBR:„r*>pMO:x>Is9s9iI?A:IH–69<riDn–:/r6NEOfh>IEO6NiR<hr&¦
 deEGCU:=MG:@6…r­><U:=B >pEG698;:Qt~?b:=6NBRCcE¥PQ¬qPQs9:–EGCh>pE¥69<yPQs9fhHc:Ir>¿<qicHF:iD< riRn:–Th>pEGC dgMOiDn  EOi}˜JEOCq:=<
‘ Z~0^S}y`’
X ‘ £ ]¦ “ :n¤>„¬ P&iR<hr]6¢HF:=M>TUMOiDµUs9:=n69<yrES>I<hPQ:¶PQiR<cES>I69<U6N<UB >p<c¬a<U:=B >pEG698;:PQ¬qPQs9:–<qiR<cE
?b:!s9sŸt´HF!: ”y<U:IHK¦
• :=MGipt~?b:=6NBRCcEbP&¬UPQsN:Ir-P=>I<–µh:/HF:=sN:=EO:„H?*CU:!<oHF:=EG:IPQEO:„HKi¦ “:/>DrOr]fUn–:7EOCh>IEbEOCq:=MO:76¢r<Ui X tv?A:=69BDCFE$PQ¬UP&s9:
69<>I<¨c©L¦

–
+ #I¯­!.K(L" $. +o1œ(*)».e.Š(L.Н¥%#I¯ +o1œ(
§erGricPQ6…>pEO:/:I>DP Co<UicHc:}­?*6NEOC>g”h:=s…H
}~¦
o6¢r*} rTUMO:IHc:IPQ:IrGriRM
} hX ” •“ 0^ ^
}¥
Ch>Rr*<UiTUMG:IHF:IPQ:„rOriDM
S
Ž CU:xriDs9fUEG69iD<ipd Th>IEOChr7P=>p<µh:ŠMG:=TUMG:Ir:=<cEO:„H­µF¬¤>TUMG:IHF:IP&:IrOr]iRM*BRMŒ>pTUC V X Z\q^Q_c`SJU?*CU:!MO:
\ X —D} ™ } \-^ } X S ž —D¹Rž
_ X{—cZ } ~^S}U`=™ Z } v^G}y` _ ^S!
} "\ T‘ —D¹Rž;žc
».Š!$
#/%4#!"­¯w) % $. A¯&# % (
. ';"*
)a,¯ +#I¯ +o1œ(
- .} vk >p<>pEGEOMG69µUfUEG:/>RrGricPQ6…>pEG:IH–?*69EOCo<UicHc:}h¦  EA6¢r*>p<¨c©w:IrEG69no>IEO:;JD?*CU6¢PSC–6¢rA>p<ofUTUTh:=M*µhiDfU<hH@iD<
EOCq:Š?A:=69BDCFEAipdA>I<¨c©adgMOiDnY¹EOi¥}h¦
/ k>l10.2 3kD54?IK 76D$8 ( ?BK$8(*/.3"!$ ZVx^S¹R`
—
} \ ipdAV 2ˆl
mlk :I>RPSC–<UicHF: !
—
- .} 0kŸX :
£
9 .} KkuX S
ž
- ;¹ LkŸX X 9
ž
/ k>l10.2 3kD5< $% &= Zv0^S}h^Sj ` >?xiRTh:=MŒ>pEO6NiR<odgiRM >MO:=s…>p¡q>pEO6NiR<rEG:=ToiR<:IHFBR:¥Z~ ^Œ}y` ?>
—
@ m - .} 1AB- C #uwj/Zv0^S}U` DFE G
—
- .} 0kŸH
X - C #u j/Zv0^S}U` 9 } ºkŸX 
ž
ž
I
.,.ŠII' ¯ "+° !"b#&%4' c¯.e/%4#¥.,.Š#!­­% .$+./( .
I!"œ. +
Z]Vx^SjŠ^Œ¹ `
—
4?IK 76D$8 ( ?BK$ 8 (*/.,3"!$ Z]V^Œ¹ ` 9
mlkS X À D l ™ \™D‘ À 2l
mlk :I>RPSC–:IHFBD:¥Zv0^S}U
` _ 2ˆl <U$% &= Zv0^S}h^Sj ` 9
mlk :I>RPSC–:IHFBD:¥Zv0^S}U
` _ 2ˆl >?IPSCU:IPdgiRM ><U:=B;>IEO6N8;:P&¬UPQsN:$? >
@ m - }.1AB- C #uwj/Zv0^S}U` DFE G@kD D 3ˆk G ·h>Is¢r: 9 >?x<U:=B;>IEO6N8;:/PQ¬qPQs9:e:!¡F6…rESr?>
k> D 3ˆk GaŽ MGfU: >?x<Ui¥<U:=B;>IEO6N8;:/PQ¬qPQs9: ? >
/ k>l10.2 3kDU$%_ .?"8 /.3E
ž
·’69BDfUMO: À r CUiI?7r*CUiI? EGCU:b:=s9sNno>p<ct]·UiRMŒH­>Is9BDiRMO6NEOCUnY?AiRMqrbiR<¶>p<:=¡U>In–TUsN:;¦
@ 50.l / @ Dk
³  <U6NEO6¢>Is96I>pEG69iR< kxZ&™ \o™ `Œ¦
³ ·’6NMSr]EbdgiRMt~s9i;iDT0kZ&™ \o™ `69EO:!MS>pEG69iD<hrQJU:I>DP CES> ;:„r!Z&™ _™ `AEO69n:;¦ Ž CFfhr&J"xZQ™ \o™IU™ _–™ `-EO6Nn–:;¦
³ ¨;:„PQiR<hH­dgiDMt~s9i;iRT k!xZQ™ _™ `EG69n–:;¦
³ Ž iREŒ>psLEO69n:;k#Z&™ \™IU™ _–™ `S¦
$,"4. b%¯.Š(-(¸" ' %4. ,.ŠII' ¯ %" + !"b#Q%'
& :=E ‘ Zv¹;^ V `AHF:=<UiDEO:/EOCq:Š?A:=69BDCFEAipdAr]CUiRMGEO:Ir]E«Ty>pEOCodgMOiDnY¹ŠEGi V ¦('s9:„>pMOsN¬;J
‘ Z~¹ª^ V ` X ” •“ n–X ^ 6N<c‚*ˆ) ` — ‘ Zv¹;^ ^ `Quwj/Z ^ ^ V `Ožc^{6Ÿ6Ÿdd VV XX ¹¹
& :=E -,+ V HF:=<UiDEO:EOCq:­?A:=69BDCFEeiId7r]CUiRMGEO:Ir]ETh>IEOCdgMOiRn¹EOi V fyr69<qB<Uion–iDMO:EGCh>p<.- :IHcBR:Ir&¦
Ž CU:b:=sNs9no>I<Ft]·UiDMSH¥>ps9BDiRMG69EOCqn[r]iRsN8;:Ir4EGCU:xr69<qBRs9:&t]r]iRfUMŒPQ:¨c©r¾TUMGiRµUsN:=n µc¬oPQiDn–TUfUEG69<UB
- € V KX ” •“ Xj/^ Zv¹;^ `S^{6u6udd V XX ¹¹
‹ € ˆ V ‹ V€
- + V ˜X n–69<h— - + V v^On‚) 69< — - + ^ uwj/Z ^ ^ V `Gž;ž
`
/
5
6
-3
8
-2
6
-3
8
0
0
7
-4
8
-2
6
8
8
5
7
-4
2
2
7
(a)
(b)
9
9
5
6
5
4
-2
6
2
-3
8
0
0
7
-4
4
-2
6
-3
8
8
7
8
8
7
7
-4
2
2
7
7
7
(c)
2
7
(d)
9
2
9
5
2
4
-2
6
-3
8
0
7
-4
2
7
7
(e)
-2
·’6NBRfUMG: À k Ž CU:e:!¡F:IP&fUEO6NiR<ipdLEGCU: b:=s9sNno>p<Ft´·UiRMŒH>psNBRiDMO69EGCUn¦¨;Ch>RHc:IH:„HFBR:IrA>pMG:«EGCU:«iD<U:IrAr:=sN:IPQEO:„HK¦
9
b ¬ & :!n–no> À JUEOCU:/>Is9BDiRMO6NEOCUn 6¢rbP&iRMOMG:IPQEA6ŸdLEOCq:=MO:e6…r4<Ui<q:=B;>IEO698;: PQ¬UP&s9:769<V r69<yPQ:«<qi¥rCUiDMOEO:„rE
Th>IEOC{Ch>Dr@n–iDMO:EOCh>I< ™ \™‘ À :IHFBD:IrQ¦ 's9:I>IMOsN¬;J«EGCU6¢ro>psNBRiDMO69EGCUn fhr]:IroHF¬F<y>pn–6…P@TUMGiRBRMŒ>pn–n69<UB
EO:„P CU<q6 ;fU:;¦
]~Y,Q…4€3Š4} Š4{ˆ}~Y,€ J{ QŽ4i+{
§¾dgEG:=M™ \–™D‘ À 69EO:!MS>pEG69iD<hrbipd EOCU:U”hMŒrE*dgiRM]tvsNi;iRT0J
‹€
- } º:
X - }.X £
6Ÿd }6¢rbMO:„>RPSCh>pµUsN:edgMOiRn¸¹ª¦
¨;fUTUThi r:EGCU:=MG:Š6…r*><q:=B;>IEO698;: PQ¬UP&s9:
Z~}pI^S};€S^=!==^S}D‚„`
MO:„>RPSCh>pµUsN:«dgMOiDn2¹ªJc?*CU:=MG:}D‚«X }p!¦ Ž CU:=<0JUEGCU:=MG:Š:!¡F6…rESr S J XZ[SY\@^ JhrfyP CoEOCy>pE
‹€ ‡
‹€ ‡ ‡ ‡
} a € 0‚ A } Iu jŠZ~} ^Œ} a €
‚ `Œ
Ž CU6…r6¢rеh:IP!>pfhr]:@6NE6¢r/69n–Thi;rGr6NµUs9:dgiRM¥> PQ¬qPQs9:¥EOiCh>=8;:–:!8;:=MO¬<UicHc:@iId - 8R>Is9fU:6¢rrn¤>ps9sN:=MeEGCh>p<
iRM¥: ;fh>ps*EGia>Is9s*EGCU: -8D>ps9fq:IriId/>Is9s*6NESrTUMO:IHc:IPQ:IrGriRMŒr–Z 6N< > PQ6NMSPQfUs…>pM?$>„¬ Jb>Is9s*iREGCU:=M¥<UicHF:Ir¥>pMG:
69EŒr–TUMO:IHc:IPQ:IrGriRMŒr `Œ¦ Ž CU: b:=s9sNno>p<Ft´·UiRMŒH >psNBRiDMO69EGCUn P=>I<{HF:!EO:IPQEEOCU6…rd~>RPQE>I<hH MG:=EOfqMO< S·h>ps…r: cJ
69<yHF6¢P=>IEO6N<UBEOCh>IE*EOCU:/TUMGiRµUsN:=n 6N<hrEŒ>p<hPQ:/6…rb<UiRE7?b:=sNs0HF: ”h<U:IHK¦