CS计算机代考程序代写 algorithm CSE 565

CSE 565
,

Lecture 10
. 09/27/2021

Network Flow
Pet 1Mtwork) : A tuple

– directed graph G- = IV , E)
– capacity cle) 70 , He c- E.
– sources c-V and sink 1- c-V.

1indegree of s=o) lout- degree oft =D .

Example :

11-1–7
+ ¥¥÷%%

Def ls-t_fw of a network) : f : E- → Rt . satisfying
i. 1 capacity constraint) : 0£ fly a- Cle) Feet
2. ( conservation constraint) :

Ie c-Ivy -114
= Ieeow, -114

. the c- Nss ,t}

IN = { lu.DE E} : in – edges of ✓
OH = { iv. w) c- E) : out- edges of a .

Def-lvalueofans-t-twt.lt/–Ieeocg-tK1 .

Fonti If / = Jeep till = Ie c-Iit, fll) –

Proof : Jeezy fly = Zeeow, fieV-uc-VIHBFEVYS.tl¥-1M t ‘ =¥vys,tj ⇐ 01b$
left ¥v Ieez,,,tH

Jeezy, th

Ee
c-1¥”

= ⇐Etll) – o
– Ieezitfk

right = ¥, Igoµ,
tie – Eas,t'”

Eat,t'”

=

E-Efle)
– Ifl – 0

let = right

⇒ Ifl = Ie eat, till ☒ .

Probtem Imax – flow )
Input : network G- = ME) . cle) 70 He c-E, sit c-V.

output : an s – t flow f- sit . If / is maximized .

5- { s , a. b}

c 15,1-7=10
.

*:;¥÷÷¥¥÷÷÷÷:* ..
CCS, -11=4 -1121-31-8=27

Det 1st cut of a network ) , partition of V
CS

, 1- = V15) , sit . SES . and TET .

Note µt¥ElS,T)={uw)EE / UES. 0£31
•→• ETTTS) ={ lu.DE/uc-T.vc-S}
” ”

Els
, 5) = { Lu, DEE/ u . V c- S}

E- ( T, T) = { luis EE/ UN ET } .

Det-ccapa-i-yofans-tcutj.cl
ST) = Iee Els , -1)

d ‘

Probtem 1min – cut ) .

Inpu a network G- = IV, El, el e)70 V-ec-E.si/-c-V .

output : an s- t cut 1¥ sit . els , -1) is minimized .

Claim: Let f- beaty s -t flow , and let 1ST) be


-t cut

, of the same network . Then

Ifl I cls, -1) .

using Isil

⑤'”→⑦ 11-1–7 |s-+os¥%% “Sitio

feel – -2Proof: HI = ¥0 ,, fie,
= -2

£ “” → 0
tie) .

EEE 1ST) e c-FITS)

a- Iee-tls.pe/e)-0–clS,T) ☒,
Max-flow
I

→É⇒ IR
1ft I 4s , -1)

min- cut .

Font : 1£ 11-1=45,1-1 , then f- is a max -flow
and 1ST) is a min – cut .

Question : Do they# meet ?
Ante : Yes !
proof: constructive proof .

Algorithmi-ord-T-ulksoni.find an s-t flow f- and an s-t cut 1ST)
sit . Ifl = Cls , -1) .

⇒ f- is a max-flow and Csi is a min – out .

⇒ max-flow = min – cut . on every network.

Idea:_ iterative improvement .