Một nghiên cứu về mạng không tắc nghẽn

Khi nhu ciu ket noi mang thip s5 lugng it, viec thilt kl cac mang chuyen mach ket hgp nhieu thiet bi chuyen mach CO so vai nhau de dam bao n6 co the chuyen dugc bat ky mot goi tin nao ti

DAI HOC QUOC GIA H A NOI TRirdNG DAI HOC KHOA HOC TlT NHIEN

* * * * * * * * *

DETAI MOT NGHIEN ClTU VE MANG KHONG T A C N G H E N

M A SO: QT-09-03

CHU TRt DE TAI: Ts Le Trong VTnh CAC CAN BO THAM GIA: - Cn Hoang Qu6c Hung

- Ths Hoang Tuan Anh

DAI HOC QUOC GIA HA NO!

TRUNG TAM THONG TIN THlJ VIEN

oomoooo ^

H A NOI - 2010

MUC LUC

BAO CAO TOM TAT 4

PHAN CHINH BAO CAO 9

I Lai ma diu 9

II Kien true cua mang chuyen mach 10

II 1 Cac thanh phan va kien true cua mang chuyen mach 11

II.2 Mot so mang chuyen mach dac biet 13

III Truyen thong va Su khong tac nghen 16

111.1 Cac kieu truyen thong 16

111.2 Su tic nghen 17

111.3 Sir khong tac nghen 17

111.4 Cac vande ve mang khong tac nghen 18

IV Dieu kien can va dii de mpt mang la khong tac nghen hoan toan 18

IV 1 Mot s5 thuat ngu 18

IV.2 Cac ky hieu va menh de 20

IV.3 Dieu kien can va dii de mang khong tac nghen 23

V Ket luan vade xuat 30

VI Tai lieu tham khao 32

PHU LUC 33

PHAN CHJNH BAO CAO

I Lai md' dSu

Trong vai thap ky qua, anh huong ciia "mang" ngay cang ro ret trong viec to chirc he thing may tinh Mang may tinh la mpt he thing nhirng may tinh dpc lap dugc ket noi vai nhau nhim dap ung cong viec chung ciia t l chuc Mang dem lai nhimg thuan tien trong cupc sing nhu: cung cip phuang tien lien lac, chia se nhimg tai nguyen san co, cai tien su tin cay cua dich vu, giam thilu chi phi Trung tim ciia viec dam bao ket noi mang thanh cong hay that bai la cac thilt bi chuyin mach (switch) Mpt thilt bi chuyen mach (ca so) c6 nhiem vu chuyin mpt goi tin a mpt cong vao den mpt (hoac nhieu) cong ra Khi nhu ciu ket noi mang thip (s5 lugng it), viec thilt kl cac mang chuyen mach (ket hgp nhieu thiet bi chuyen mach

CO so vai nhau) de dam bao n6 co the chuyen dugc bat ky mot goi tin nao tir bat ky mot cong vao chua diing den bit ky cong ra chua diing khac ma khong co giy ra hien tugng tac nghen (cac goi tin dugc dinh tuyen sir dung cac lien ket khac nhau) la tuang doi de dang Tuy nhien,

su biing no cua viec sir dung cac san phim va cong nghe lien quan den mang may tinh dac biet la Internet ngay nay lam cho kich thuac ciia cac mang chuyen mach tang len rat nhieu Nhu ciu ket noi mang cang lan thi kich thuoc ciia cac mang chuyen mach cang tang Mac dii

vay, kich thuac ciia cac mang chuyen mach khong the tang vo han dugc, do do thiet ke vd xay

dung cdc mang chuyen mach khdng tdc nghen vai kich thuac be Id mot nhu cdu cap thiet de

dap img nhu ciu ket noi mang ngay cang tang hien nay Han niia, nhu ciu ket noi mang ngay nay khong chi tang ve so lugng ma con tang ve chat lugng nhu: bang thong lan hon, toe dp cao han, it loi hon vi vay, cac mang chuyen mach dien (electronic) la khong phii hgp Thay vao do, cac mang chuyen mach quang (optical) da va dang dugc sir dung de thay thi cac mang chuyen mach dien

Mat khac, "sir khong tac nghen" phu thupc vao nhieu yeu to nhu: cac dang khong tk

nghen, khuon mau ciia truyen thong, cac thuat toan sir dung de dinh tuyen, sir phu thupc vao cac thiet bi vat ly nen viec thiet ke cac mang chuyen mach noi chung va chuyen mach

quang noi rieng khong tac nghen cang tra nen kho khan Vi vay, vin dl khong tk nghen

trong mang chuyen mach dugc quan tim bai rat nhieu cac nha khoa hpc, cac cong ty, tip doan trong nuac va ngoai nuac dau tu nghien cim Do do, muc tieu cua dl tai nay la tim hilu

ve cac bai toan trong viec phan tich, thiet ke va xiy dung mang chuyen mach quang khong tac nghen

II Ki§n true cua mang chuyen mach

Ly do diu tien de ra dai mang chuyen mach la tir nhu ciu ket noi giiia cac cap may dien thoai Ban diu, vai mot so lugng khong nhieu ciia cac may dien thoai, mot diy (dien thoai) dugc noi true tiep giira hai may Tuy nhien, khi so lugng ciia cac may dien thoai tang len va gia truyen thong tren day (dien thoai) nay se tro nen qua tai va khai niem "switching" (chuyen mach) da dugc ra dai Thay vi ket noi true tiep moi cap hai may vai nhau, tit ca cac may trong mot khu vuc nao do deu dugc ket noi den mpt trung tam chuyen mach (SC-switching center) Cac may (dien thoai), day dan va bg chuyen mach tir nhieu mien khac nhau ket noi voi nhau hinh thanh mpt mang (network) va dugc ggi la mang chuyen mach (SW -switching network) - nhu trong hinh 1 duoi diy Ngay nay, khai mang chuyen mach dugc ma rpng trong viec ket noi cac may tinh trong mang may tinh, ket noi cac CPU, bp nho trong may tinh song song, v.v

network) Mac dii vay, cac ung dung va ly thuyet cho mang chuyen mach deu dugc nghien

cuu cho mang hai phia Trong mang hai phia : diu vao x ket noi vai dau ra y thi se khong giong nhu khi y ket noi vai x, trong khi chiing la giong nhau doi vai mang mpt phia Noi

each khac, mang mpt phia chi la truang hgp dac biet ciia mang hai phia Do vay, de tai nay chi tip trung nghien cuu ve mang hai phia va de cho ngan ggn, tu day ve sau, chung toi su dung "mang chuyen mach" dong nghia vai "mang chuyen mach hai phia" neu khong c6 mpt sir chi ro nao Phin tiep se trinh bay cac thanh phin va kien tnic ciia cac mang hai phia

11.1 Cac thanh p h i n va k i l n true cua mang chuyen mach

Phin nay trinh bay cac thanh phin chinh, cac khai niem va kien tnic mot mang chuyen mach [1]

• Crossbar (hay switch element)

Crossbar hay switch element (SE) la thanh phin co ban nhit ciia mpt mang chuyen mach Crossbar la mpt nut trong mang chuyen manh Mot crossbar gom co cac loi vao (in-

let) hay diu vao (input) va cac loi ra (out-let) hay diu ra (output) Mot crossbar vai n loi vao

va m loi ra dugc ki hieu la X^m ggi la no co ca « x m

Hinh 2 Vi du ve mot crossbar 3x4

Ro rang, trong mpt crossbar, vai bat ky mpt loi vao roi (chua sir dung) nao diu c6 thi ket noi tai bit ky mpt loi ra roi (chua sir dung) boi cac ket noi chua sir dung Hai kit nli chung mot lien ket se giy ra sir ngan chan (block) lan nhau (cai nay dung thi cai kia khong

diing dugc nua va ngugc lai) va dugc ggi la tdc nghen (blocking) Hai kit nli sir dung cac lien ket khac nhau se khong ngan chan lan nhau va dugc gpi la khdng tdc nghen (non-

bloking) Mpt mang co the thiet lip dugc moi ket noi sao cho khong co hien tugng blocking

xay ra dugc gpi la mang khdng tdc nghen Nhu vay, crossbar la khong tic nghen

• Mang chuyen mach

De phuc vu cho N cap ket noi, chiing ta mpt crossbar vai A^ l6i vao va N 16i ra Neu A^

dii Ian thi viec thiet ke mpt crossbar nhu vay trong thuc te la khong kha thi bai gia thanh

11

ciing nhu do phuc tap cua no De tranh phii thiet ke cac crossbar vai kich thuac lan nhu vay, nguai ta thuong thiet ke mot mang chuyen mach bang each ghep nhieu crossbar co kich thuac nho vai nhau theo mot kien true nao do, vi du nhu trong hinh 3 duoi diy

Hinh 3: Vi du ve mot mang chuyen mach gom cac crossbar noi vai nhau

ta luon co: r,x m, = r/.ix /7,.i(bao toan duang links).Voi / = l ^-l

Cac s-stage c6 the xep chong len nhau va ggi la cac stack Mpt mang chuyin mach ting quit dugc xem nhieu stack cua mpt kieu s-stage nao do Vi du trong hinh 3, mang gim 2

stacks cua 3-stage

• Cach xay dirng mang theo kien true 5-stage

So lugng cac crossbar trong cac ting ciia mot s-stage phai thoa man dieu kien: ting so lli

ra ciia cac crossbar a ting / phai bang so tong so loi vao cua ting thu /+! vai / tir 1 din s-l

Do vay ta co the xiy dung cac s-stage nhu sau

i) Gia su A'ab^^cd la cac crossbar de xiy dung 2-stage, khi do:

Ting stage-\ dung c crossbar ca

X^b-Tang stage-2 diing b crossbar ca Xcd

ii) Gia sir Xab, Xcd, Xbe la cac crossbar de xiy dung 3-stage, khi do;

Ting stage-l dung c crossbar co X^^

Ting stage-2 diing b crossbar caXcd

Ting stage-3 dimg d crossbar ca Xbe

iii) Tong quit, de xiy dung s-stage diing cac crossbar :

Ting stage-l diing is crossbar co ^/^/^

Ting stage-2 dimg 12 crossbar ca ^ ^

Tang stage-3 dimg i^ crossbar ca X

Ting stage-4 dimg i^ crossbar ca X^

Tang stage-} diing z" ,, crossbar X

Ting stages diing /^^, crossbar X- • (t bit ky)

s't

11.2 Mot 56 mang chuyen mach dac biet

Phin nay se trinh bay nhimg mang chuyen mach thuong dugc diing trong thuc tl va nghien cuu ly thuyet [2]

• Mang Clos

Trong linh vuc vien thong, mang Clos la mpt loai mang chuyen mach nhieu ting (stage) dugc dua ra bai Charles Clos nam 1953, dung de trinh bay ly thuyet ve he thong chuyen mach dien thoai nhieu ting Mang Clos dugc diing khi cac yeu ciu mach chuyen doi vat ly qua khi nang thong qua lan nhit cua mpt crossbar don LTu diem chinh cua mang Clos

la dimg cac crossbar kich thuoc be (gia thap) de thuc hien chuyen mach tit ca cac yeu ciu thay cho viec diing mot crossbar kich thuac lan (gia thanh cao)

13

crossbdis crossbars

Hinh 4: Mang Clos ba tang

Ciu tnic chuyen mach Clos gom ba ting, dugc the hien bai 3 so nguyen n, m, r Tang diu tien c6 r phin tir chuyen mach (n x m) va tang thu 3 c6 r phin tir chuyen mach {m x n) Ting thu hai (tang giira) c6 m phin tir chuyen mach (r x r) Moi phin tir trong so r phin tir chuyen mach cua ting diu tien dugc ket noi vdi tit ca cac phin tir chuyen mach cua ting thir hai (mdi phin tir chuyen mach ciia ting diu tien co m loi ra Moi loi ra dugc ket noi vai mpt

loi vao ciia timg phan tir chuyen mach khac nhau cua ting thu hai) Tuong tu nhu the, moi

phin tir trong so r phin tir chuyen mach ciia ting thu ba dugc ket noi vai tat ca cac phan tir

chuyen mach ciia ting thu hai

• Mang Benes

Xuit phat tir mang Clos c6 nhieu hon 3 tang vai m^n^2 ,y tuang chinh ciia mang

Benes la xiy dung mpt mang lan dua tren cac thanh phin chuyen mach ( SE) vai thupc tinh

CO kha nang sap xep khong tac nghen

1

,p\y°-^^^/^ I

Hinh 5: Vi du ve mang Benes

So dau vao va dau ra la N=rxn=2r Vai nhiing mang c6 2\og2N — 1 ting, moi ting chua N/2 crossbar 2x2, va su dung tong so Mog^A^ — N / 2 crossbar 2x2 Vi du, mang Benes 8x8 ( A^ = 8), co 21og^5 — 1 = 5, moi tang chira N/2=4 crossbar 2x2, va no sir dung ting si N\og2^ - N / 2 "=20 crossbar 2x2 Ba ting trung tam chua 2 mang Benes 4x4 nho han, trong khi do a ting trung tam, moi crossbar 2x2 co the coi chinh no li mpt mang Benes

La mpt trong nhimg kieu mang pho bien ciia mang ket noi da ting {Multistage

Interconnection Network- MIN) Mang Banyan dugc xiy dimg tir su lien ket ciia cac ting ciia

cac SE [3] Mot SE 2x2 co ban co the dinh tuyen mpt khoi (cell) vao theo mot bit dieu khien (dia chi dau ra) Neu bit dieu khien la 0, cell dugc dinh tuyen la dia chi cong cao hon, ngugc lai thi no dugc dinh tuyen dia chi cong thip hon

Liy vi du mang Banyan SxS.Sn lien ket cua 2 ting cua SE 2 x 2 co thi dugc lam bai

viec sir dung bit dau tien ciia dia chi ra de ky hieu cai SE dinh tuyen, va sau do su dung bit

cuoi ciing de xac dinh cong ra Mang Banyan 8 x 8 co the xay dung mot each de quy bing viec sir dung bit dau tien de dinh tuyen cell qua tang diu tien, hoac vao mang 4x4 thip han hoac cao hon, va khi do sir dung 2 bit cuoi de dinh tuyen cell qua mang 4x4 dk phii hgp vai

cong diu ra

15

Hinh 6: Mang Banyan 3-stages

Noi chung, de xay dung mang Banyan N x N, tang thu n sir dung bit thu n ciia dia chi diu ra de dinh tuyen cell Cho N = 2", mang Banyan se chua n = log2 N ting, moi ting chua N/2 SE Mot MIN dugc gpi la tu dinh tuyen dugc khi dia chi diu ra hoan toan xac dinh

duang di qua mang

Cong nghe mang Banyan la pho bien vi sir chuyen mach dugc thuc hien bai cac SE don

gian, cac cell dugc dinh tuyen song song, tat ca cac SE hoat dpng o ciing toe dp(nhu vay

khong han che kich thuoc ciia N va toe dp cong vao) va co the xay dung cau tnic chuyen mach vai dung lugng lan mpt each de dang

III T r u y I n t h o n g va SLF khong tac nghen

III.1 Cac k i l u truydn thong

Trong mpt mang chuyen mach, mpt yeu ciu truyen thong se thupc mot trong cac dang sau day :

• Unicast; Yeu cau truyen thong la truyen tin hieu tir mot loi vao den mot loi ra

• Broadcast: Yeu ciu truyen thong la truyen tin hieu tir mot loi vao din tit ca cac lli ra

• /-cast (hay multicast) : Yeu ciu truyen thong la truyen tin hieu tir mpt loi vao dln/16i ra

Ro rang, day la kieu truyen thong tong quit nhat, vi unicast va broadcast chi la hai trucmg hgp dac biet k h i / = 1 v a / = A^(vai A^ so loi ra cua mpt mang) tuang img

111.2 SIP tac nghen

De truyen tin hieu tir mot loi vao den mot loi ra, chung ra cin thiet lap mpt duong dan tir loi vao den loi ra thong qua cac lien ket (link) giiia cac crossbar va cac crossbar dugc diing

de xiy dung mang Cac crossbar la cac diem trung gian tren cac duang truyen tin hieu nen chiing ggi la mot crosspoint

Khi cac duang truyen tin hieu co mot lien ket triing nhau thi no dugc ggi la link-block Khi cac duang truyen tin hieu c6 mpt crosspoint triing nhau thi no dugc gpi la node-block Khi truyen tin gap cac truang hgp nay nguoi ta gpi la xay ra hien tugng blocking

Mpt mang co the thiet lip dugc mot duang truyen tin hien tir bat ky mpt loi vao chua sir dung (roi) den bit ky mpt loi ra chua sir dung (roi) sao cho khong xay ra hien tugng blocking

dugc gpi la mang khdng tdc nghen va bai toan cho mang khong tac nghen dugc phat bieu nhu

sau: "Vai mpt kieu truyen thong cho truoc va kich thuac mang, hay thiet mpt mang khong tac nghen sao cho so crossbar phai sir dung la it nhit"

111.3 SiF khong tdc nghen

Viec truyen tin tren mot mang deu lien quan den kieu truyen truyen thong, cau true cua cac crossbar, va each xac lap cic duang di tir loi vao den (cac) loi ra Do do yeu ciu vl su khong tac nghen se phu thupc vao cac yeu to nay Phin duoi diy se trinh bay cac dang (yeu ciu) khong tac nghen thuong dugc nghien cuu

• Strictly Non-Blocking (SNB)

Vai dang khong tac nghen nay, mang phai cho phep thiet lip dugc mpt duong truyin tir bit ky loi vao nao chua dugc sir dung vai bit ky loi ra nao cung chua dugc sir dung ma khong cin quan tim den trang thai ciia cac ket noi truac do trong mang

• Wide-sense Non-Blocking (WSNB)

Vai dang khong tac nghen nay, mang phai cho phep thiet lap dugc mpt duang truyin tir bit ky loi vao nao chua dugc sir dung vai bit ky loi ra nio ciing chua dugc sir dung ma khong cin phai dinh tuyen lai cac ket noi dang tin tai; mang chuyin mach dang nay su dung cac thuat toan dinh tuyen de dinh tuyin cho cac kit nli hien co sao cho dam bao khong xay ra nghen cho cac ket noi tiep sau do

cho phep dinh tuyen lai cho cac ket noi da ton tai Viec dinh tuyen lai cac ket noi co the hoac khong the dugc chap nhin con tuy thuoc vao ung dung vi chac chan cac ket noi se bi ngat trong mot khoang thai gian nao do khi chung dugc chuyen mach sang duang din khac III.4 Cac v i n d§ vd mang khong t i c nghen

Nhu da trinh bay a tren, cac crossbar la khong tac nghen nhung khong the san suit dugc mgi crossbar vai cac kich ca khac nhau trong khi do kich thuac cac mang chuyen thi thuc te la hoan toan khac nhau Hon nua, thuc te la khong the sin xuit dugc cic crossbar co kich thuac bang dung kich thuoc cua mang chuyen mach Vi vay, nguoi ta phai xiy dung cac

mang chuyen mach dua tren cac crossbar co san bang each xiy dung nhieu stages va nhieu

stacks Mat khac, nhiing kien tnic huu dung cho cac mang trong thuc te da dugc chi ra, tuc la

each thiet ke cac mang da dugc chi ra Do vay, ve mat ly thuyet, su khong tac nghen thuong dugc nghien cim nhu sau : Dua ra mpt kieu truyen thong, mpt kien true mang (clos, benes, banyan ), kich thuoc ciia mpt crossbar, kich thuoc mang va dang tac nghen Hay tim so crossbar cin diing nho nhat de mang khong tac nghen Noi each khac chiing ta phai tim so stages va stacks cin phai sir dung Mac dii vay, bai toan nay vin la rat kho, nen nguoi ta

thuong CO dinh so stages truac va chi tim so stacks

Do thai gian c6 han, nen trong de tai nay, chiing toi chi tim hieu dieu kien cin va du (so stacks toi thieu) de mot mang Banyan ngugc la khong tac nghen hoan toan cho kieu truyen thong tong quit/-cast

IV Dieu kien c i n va du de mot mang la khong tac nghen hoan toan

IV.1 Mpt s6 thuat ngu'

De cho de trinh bay noi dung chinh, chiing toi trinh bay truoc cac khii niem lien quan den mang khong tac nghen hoan toan:

• Mang \og^(N,0,m)

De cin bang giiia chi phi va phin cung, phin lan cac mang chuyen mach dugc thilt ke sir dung kien tnic da ting (multistages) va phin lan trong thuc the kieu kien tnic dugc sir dung la Banyan ngugc Vi vay, trong de tai nay chiing toi chi quan tim den kiln true da ting

kieu Banyan ngugc nhu dugc mo ta trong hinh 7 Moi Banyan ngugc dugc ggi la moX plane

Hinh 7: Kien tnic Banyan ngugc BY" (5)

Hon nira, chung toi nghien ciiru kien true mang chuyen mach tong quit multi-log d-dx^ voi nhieu stacks ciia cac planes theo chieu dpc nhu dugc mo ta trong hinh ve duoi diy

HinhS: Mang chuyen mach log3('27,0,2^

19

Thiet ke nay dugc sir dung trong thuc te bai vi dp sau nho (dp phuc tap cua viec tim

duang di tir mpt cong vao den mpt cong ra) - 0{logdN) vai mpt mang chuyen mach c6 A'^ cong vao va A^ cong ra xay dung tir SE co kich thuac \k d x d Do vay, chiing toi diing

\og^(N,0,m)de chi mot mang chuyen mach multi-log J-ary vai m stacks

• Kha nang Fanout (kha nang phan dau ra)

De ho trg multicast (^idiy f-cast), mot mang \og^(N,0,m)cdin co kha nang fanout Mpt plane hoac SE (i x J co sa dugc ggi la co kha nang fanout neu bit ky mot anh xa 1-

nhieu tir mpt cong vao tai cic cong ra ciia no deu ton tai Kha nang fanout cua mpt mang

\og^(N,0,m) se dugc xem xet trong cac truang hgp

- Chi a ting dau tien duge cung cip kha nang fanout

- Chi cac ting giira dugc cung cip kha nang fanout

- Tit ci cic ting cung cip kha nang fanout

Chu y: Neu nhu ciu fanout co a nhieu ting thi cang lam cho viec thuc hien no tra nen phuc tap va ton han Vi vay kha nang fanout can dugc xac dinh can than trong viec thiet ke mpt mang chuyen mach

• Tac nghen (block)

Tac nghen thuong dugc xem a mot trong hai dang duoi day:

o Tdc nghen lien ket (Link-block ) : chi co mpt yeu ciu dugc phep sir dung mot

link a mot thai diem (trong chuyen mach dien)

o Tdc nghen niit (Node-block): chi cho phep mot yeu ciu dugc sir dung mpt SE

a mpt thai diem

IV.2 Cac ky hieu va menh d§

Cho bit ky so nguyen duang I,deo:

+[l]: ky hieu tip {1, ,/}

+ Z^ : ky hieu tip {0, ,d-\} va cung dugc xem la cac ky hieu t^-ary

+ Z^ : ky hieu tap tat ca cac chuoi d-axy dp dai /

+ ^ : ky hieu chuoi / ky tu Z> e Z ^

+|s|: ky hieu dp dai ciia chuoi s d-ary (|31|= 2)

+ s- ky hieu day con s- -ciia mot chuoi s = Sj Si G Z^ Khi i > j quy uoc S- la

I J I J i.-j

day rong

Cho N = d" chung ta xem xet mang \og^(N,0,m) la mang gom m stacks ciia d-ary

BY" (n) {Banyan ngugc) vai A^ dau ra va N diu vio

Danh so cho dau vao va diu ra ciia BY~ (n) la cac chuoi d-ary co dp dai n Dac biet,

moi diu vao x^Tf^ va diu ra J^ GZ^ co dang x = x^ x^,y = yj y^ vox x^^y.^Tj^,

\/i G [n]

Tuong tu viec danh so trong cac SE c/xc/trong moi ting ciia n ting ciia BY" (n) voi

chuoi £^-ary dp dai n-I De thiy rang mpt diu vao x (mot diu ra y) la dugc ket noi voi SE co

nhan x^ ^_j 6 ting diu tien (y^ ^_j a ting cuoi)

Chung ta co the de dang thay rang tien to cua y^ ^_^ la dam nhin tien to cua x^ ^_j

tren duang di tir x tai y Tong quit, duang di duy nhit tir mot diu vao x tuy y tai mot diu ra y

tiiy y chinh xac la nhu sau :

Bay gia ta xem xet hai yeu ciu unicast {a, b) va {x,y) Trong truang hgp tac nghen mit, hai yeu cau nay khong the di qua ciing mpt stack ciia BY'^(n) khi va chi khi hai duang di

tuong img giao nhau a vai SE a ting giira (neu chiing dugc dinh tuyen qua ciing mot stack ) Chinh xac hon, (a,b) va (x,y) dugc gpi la tac nghen mit vai nhau khi va chi khi co

j G [n] sao cho by ^_j = y^ j_j va 3.j ^_j = x^ ^ ^ Trong truang hgp nay, hai duang di giao

nhau a ting j ciia SE Nen chii y rang hai duong di ciia hai yeu ciu co the giao nhau tai nhieu hon mot SE

Trong mot mang \og^(N,0,m), hai yeu ciu tac nghen mit voi nhau phai dugc dinh tuyen qua cac ban copy B Y " (n)^ac nhau Cho hai chudi bit ky u,v G Z^

- PRE{u,v) ky hieu tien to chung dai nhit cua u va v

- SUF{u,v) ky hieu hau to chung dai nhat ciia u va v

• Menh d e l :

Cho (a,b),(x,y) la 2yeu cdu unicast trong mot mang log^(N,0,m) Khi do 2 yeu cdu goi la tdc nghen niit khi vd chi khi \ SUF(ay „_y,X/ ^_J|+|PRE(b^ ^-z^Y/ n^/)\^^- ^ • Chitng minh:

Hai yeu ciu la tac nghen mit khi va chi khi chiing dimg chung mot crossbar

Xet 2 yeu ciu (a,b) va (x,y)

Hai yeu ciu nay tac nghen niit khi va chi khi co mpt vai so j e [n] ma b^ _^ = y^ ,

va a ^^j =x ^_j (Trong truang hgp nay hai duang dan giao nhau a stage-j SE)

Hay | S U F ( a , ^ _ y , x , ^ _ J | + | P R E ( b , „ _ , , y , , , _ y ) | > ^ - 7 (dpcm)

• Menh dl 2;

Cho (a,b),(x,y) la 2yeu cdu unicast trong mot mang \og^(N,0,m) Khi do 2 yeu cdu goi la tdc nghen lien kit khi vd chi khi | SUF(a/^,_/,Xy ^,_/)|+|PRE(by ^,_y,y/^_y)| > ^ Chung minh:

Hai yeu ciu li tic nghen lien ket khi va chi khi chiing diing chung mpt lien ket trong BY'Y^y'Cnlu chiing dugc dinh tuyen qua ciing mot ban copy )

Tuang tu chimg minh menh de 1, xet hai yeu ciu (a,b) va (x,y)

Hai yeu cau nay tac nghen lien ket khi va chi khi c6 mot vai so j G [Ả] ma b/ ,w = y/ ,-/, ^j „-i = ^j,.n-i ^ ^Lj = YL.J Vi ậ^y ^_y - x^^y ^_^

Hay tuang duong vai dieu kien by = y^ va a ^_y = x ^_ỵ

Hay I SUF(ay „_y,Xy ,_y)|+|PRE(by „_y,yy ^_y)| > n (dpcm)

IV.3 Didu kien c i n va du de mang khong tSc nghen

Phin nay se chi ra cic dieu kien cin vi dii dl mang \og^(N,0,m) la khong tic

nghen hoan toan [3]

Truang hap 1: Chi co ting giiia c6 kha nang fanout

Chicng minh:

Gia su mang da thiet lip duang di cho mot si tip 5? cac yeu ciu R = (x,Y) la mot

multicast mai yeu ciu tuong thich vai tinh trang mang hien tai, khi x la mot vai diu vao va

Y la mot tap con ciia / diu ra, / < / De mang la khong tic nghen hoan toan, chiing ta phii

tim dugc mpt bin copy ciia B Y " (n) dl dinh tuyIn R Ky hieu b(*}{) la si cac yeu ciu trong y{ ma tac nghen mit R (vi du truang hgp khong the dugc dinh tuyIn qua ciing ban copy cua BY" (n)voi R) Khi do R co kha nang dinh tuyIn khi va chi Wx m> b(y{) + 1

(*) Diu tien, quan tim toi truang hgp khi / < d"'^ — d"~^ Chiing ta tim giai han tren cho b^yi) Gioi han tren B se phu thuoc vao ?? va R Khi do, m>B + l la dilu kien dii dl mang logj(N,0,m)\a hoan toan khong tac nghen

Bai vi ting diu tien khong co kha nang fanout, mpt yeu ciu {x,Y)Xac nghen mit R khi va

chi khi CO mpt so y G F m a yeu ciu unicast (x,y) tac nghen mit R Vi vay, khong mat tinh tong quit, chiing ta gia sir rang 9? chiia toan bg cac yeu ciu unicast

Viet / = { y , ,y } De cho mpt yeu ciu (x,y) tac nghen mit R, (x,y) phai tac

(p)

nghen niit yeu ciu (^x, y ) cho mpt so 1 < p<l Nx vay chiing ta se thio luan chi tiet tren do {x,y) c6 the tac nghen mit ( x, y J

Cho moi / G {0, ,n - 7} , ky hieu X^ la tip cac dau vao x khic x ma Xy „_; va

X/ n~i CO SO hau to giong nhau voi dp dai chinh xic la i Hay co the dinh nghia nhu sau:

X, := {x G Z : - {x} I SUFrx,,,,_y,x/, -;; = /}

Tuong tu, cho mSi j e {0, ,n - 1} ,ky hieu 7 la tap cic diu ra khac y cai ma

CO SO tien to giong nhau voi y co dp dai chinh xac liy Cu the la:

Y^/^.- {y G z : - {y''} I PREry;,.,._y,7'^, _J = j}

Theo menh dl 1, {x,y) tic nghen niit (x,y ) khi va chi khi (x,y) G Xp(.Y^j^^vox i,

; : i + j > n - l

Vi vay ma {x,y) tic nghen mit voi (x, Y) khi va chi khi (x,y) G X,x7|^''cho

i,JG{0, ,n-I} va I < p<l cho i + j>n-1

Cho I = 7f^ ky hieu tip dau vao

O = 7f^ ky hieu tip diu ra

Xiy dung do thi hai phia Gj^=( I KJ0,E ) la hgp ciia tit ca cac do thj hai phia X-xYj ma i-\-j>n — l Khi do (x,y) tac nghen mit R khi va chi khi (x,y) la mpt canh ciia G^ Moi tip yeu ciu trong R tac nghen niit R phai la mpt cap ghep ciia Gy^, de cho ?J la mpt

tip yeu ciu gii tri Vi vay ma b(^) < v(Gn)vox v(Gf^)ky hieu la so cap ghep cue dai trong Gf^

m> max v(Gj^)-\-l {\)

R an 1-cast rcques/,I<f

la dieu kien cin de mang \og^(N,0,m) la/-cast hoan toan khong tac nghen

x(Gn) la kich thuoc toi thieu bao phii dinh trong G^ , khi do T(GJ^) = v(Gj^) theo

n-r -1

dinh ly Konig-Egevary [4] Cho j = [ j

Chii y ring trong truang hgp nay thi r <n- 2 Chinh vi vay ma j> 1

De dang xic dinh rang tip bao phii cac dinh ciia Gj^ la:

Vi I X, IH Y/^^ \= d"~' - d"~'~' cho mpi /, chiing ta co :

\C\<(d'^' -d"-'-' + + d^ -d') + l(d^ -d^-^ + + d^ -d')

25

<(d"-^-l+f(d^-l)

= mjn,f,d)-l

N\v^yv(GJ = x(G,)<\C\<mJn,f,d)-l

Theo (1) bo de dugc chiing minh

(**) Thii hai ta quan tam toi truang hgp / > d"~' — d"~ + 1

Nlu / < J " " ^ - ^ " " ^ k h i d 6 bC^) chi CO thi lan nhit la mjn,2"'^-2"~\d)-l bai phan tich truac va thuc te rang m^^(n, f ,d) li mpt ham khong giam trong/

NIU / > (^""^ - (^""^ -h 7, thi si diu ra khong su dung lan nhit la

d"-l<d" -d"-^ +d"-' -I

V\vayb(dl)<d"-d"-^ ^d"-'-l = mJn^d"-^ -d"-\2)-l

Dinh ly dugc chimg minh

Truang hop 2: Ci ting diu tien va ting giua co kha nang fanout

• D i n h l y 3 :

Cho r =[iog^ fj.Diiu kien cdn vd du de mang \og^{N,0,m)ld f-cast hoan todn

khdng tdc nghen trong truang hap tdc nghen nut la m> m^^^{n,f,d)

• Dinh ly 4 :

Cho r =• Wog^ f^.Diiu kien cdn vd dii de mang log^(N,0,m)ld f-cast hoan todn

khdng tdc nghen trong truang hap tdc nghen lien ket la m>m^f^(n,f,d)

Chirn^ minh:

Gia sir mang da thilt lap duong di cho mot so tip 9^ cac yeu ciu R = (x,Y)\a moxf

cast mai yeu ciu tuong thich voi tinh trang mang hien tai, khi x li mpt vai diu vao va

Y = {y , ,y ^lampt tip con ciia hiu h e t / d i u ra Cho moi / G/"//, phan (x,y )c6

thi dugc dinh tuyIn phu thupc vao tit ca (x,y ) khic m i j ^ / v a j G [I] Boi vi ca

hai tang deu co kha nang fanout Vi vay khong mit tinh tong quit ta gii sir rang Ychx chua

mpt diu ra y vi du 7? = (^x,y^

Khong giong nhu chimg mirih dinh ly 1 chiing ta khong the gii sir rang tip yeu ciu

trong 9{ la cic tip unicast, bai vi moi yeu ciu R trong 5R co the dugc dinh tuyen qua vai bin copy khic nhau BY~ (n) Tuy nhien chiing ta co the bo qua cic nhinh ciia R ma tai do

khong tie nghen nut 7?

Nhu vay cho moi yeu ciu 7? = (^x,{y , ,y JJ^^l chiing ta co the gia sir rang

(x,y ) tie nghen mit 7? cho mpi i<l

Tuong tu chung minh dinh ly 1 cho moi 0<i<n- 1 dinh nghia :

X, := {X e Z^ - {xj | SUF^x, „_„x/, -yj = i}

vi cho moi 0 < j <n — 1 :

r^ ;= {y e Z : - { y ; I P R E r y , „_py,„„_;; = 77

Chu y ring tip X^ li tip roi phu thupc lan nhau

Y la tip rai phu thupc lan nhau

Yionnn?i,\X,\=\Y,\^d"~'-d" yi

—(0) —(I)

Theo menh dl 2 cho mpi yeu ciu R = (x,{y , ,y })e9{ va cho mpi p^flj,

no phii la truang hop (x,y ^ y* ^ ^ , x Yj cho moi iJ mai + j>n-l

Gpi Gn la dl thi hai phia la hgp ciia tit ca X^ x Y^ vox i^- j>n- 1 Khi do mpi yeu

ciu 7? G ?{ phii hgp duy nhit vai mpt tip ciia cic canh/lien thupc vai ciing dinh vao

Tren phia diu ra, khong co dinh ra nao xuat hien hon mpt lin trong 9i Cho bC^i) la

si canh trong G^ cii ma ?? phii hgp voi Khi do, m > maxZ?(^9^^ -h 7 la hiln nhien dii dl

dinh tuyIn 9? Trong truong hgp xau nhat, moi nhanh (duong) ciia mdi yeu ciu trong 9^ duoc dinh tuyIn qua mpt ban copy khic ciia B Y " V ^ /

27

Biy gia ta ky hieu Vf(Gj^)\a kich thuac toi da cua mot tip con S cua cac canh ciia

Gj^ thoa man dieu kien sau:

a) m6i diu vao la lien thupc vai hiu hit/canh trong S

b) moi dau ra la lien thupc voi het cic canh 1 trong S

(chu y rang V^^f^Gy^^la so cap ghep ciia v(G)) Khi do tir phin tich tren ta co Z?(^9?^< v.f^Gy; J Vi vay m>m?iXV JGr.)+ 1 la dilu kien cin dl mang \ogJN,0,m)

J R ^

la hoan toan khong tic nghen trong truang hgp tic nghen mit

Xiy dung mpt luong mang Dj^ nhu sau:

Mang CO mpt nguon s, mpt dich /, va tip dinh X ^JY vox

Co mpt canh tir dinh nguon s tai mpi dinh x G X vai khi nang thong q u a /

Co mot canh (x,y) voi khi nang thong qua la 1 tir A' tai 7 khi va chi khi ( x , y ) G X , x Y j , z + y > ^ - 7

Culi ciing CO mot canh tir moi y G Y voi kha nang thong qua li 1 tai dinh dich /

Vi vay ''phdn giira " ciia luong mang chinh xac la Gj^ vox s,t phu hgp hai ben

Cho mpi tip S chiia cic canh cua G^ thoa man dieu kien (a),(b) a tren, de thay rang c6

su tuong ling mpt lulng nguyen vai gia tri chinh xic ca ciia S Chinh vi vay, 151 la gii tri lulng cue dai trong Dj^,ky hieu la MAXFLOWf^Dy^^, chinh bing gii tri kha nang thong qua cue tilu ciia lit cit s,t trong Dj^, ky hieu la MINCUTf^T:)/^; Vi vay

V (G ) < MINCUT(Z)y^) Trong do v^(G/^)la kha nang thong qua ciia lit cit s,t bit ky trong Dj^

Cho i = f / " ^ • De y den lit cat s, t trong Dj^ nhu minh hoa trong hinh

^ 2

i^O t=j+/ i^n-j-l i^O

Khi nang thong qua ciia lit cat nay chinh xac la

n-I n-I

Z f\X.\+T\Y.\-rnJn,f,d)-l

Vi vay m > m^^(n,f,d) la dieu kien cin

Truang hop 3: Chi co tang diu tien co kha nang fanout

• Dinh ly 5:

Cho r = [logoff Dieu kien cdn vd dii de mang \og^( N ,0,m) Id feast hoan todn khdng tdc nghen trong truang hap tdc nghen nut Id :

d'^-d''-^ +1 m>d" khi f >