89 90 98 00 Computing and Communications -- Ntwork Coding Dr. Zhiyong Chn Institut of Wirlss Communications Tchnology Shanghai Jiao Tong Univrsity China Lctur 5- Nov. 05 0
Classical Information Thory Sourc has ntropy rat H bits/sampl. Channl has capacity C bits/sampl. Rliabl communication is possibl iff H < C. Information is lik fluid passing through a pip. How about for ntworks? A Mathmatical Thory of Communication 98
Gnral Problm Each sourc is obsrvd by som nods and nds to b snt to othr nods Qustion: Undr what conditions can th sourcs b rliably snt to thir intndd nods?
Motivation Givn th wirlss ntwork as blow whr two nods A and C ar too far away to communicat dirctly. If transmitting on packt costs tim unit how many tim units do w nd to transmit on packt from A to C and on packt from C to A? A B C Traditionally intrmdiat nods in ntworks just forward data. Ntwork coding dviats from this paradigm in th sns that intrmdiat nods ar allowd to procss data bfor forwarding!
Ntwork Coding Savs Transmissions [Christina Fragouli EPFL] 5
Th Classic Exampl Givn two sourcs ach with a GB fil and two rcivrs. Each dirctd (wir-lin!) link can forward MB/s. How long dos it tak until both rcivrs hav rcivd both fils?
Without Ntwork Coding? Wll th naïv solution would first dlivr th first fil to both rcivrs thn th scond. Th total tim ndd is 000s. Can w do bttr (without ntwork coding)? First it sms that thr is a bttr forwarding-only solution. Th pictur shows that w can dlivr a total of MB/s. Howvr this is not tru. Indd crossing traffic must go through th bottlnck link A-B; to dlivr th GB information through this MB/s link w nd 000s What about with ntwork coding?
Max-Flow Min-Cut Thorm (From Wiki) Th max-flow min-cut thorm is a statmnt in optimization thory about maximal flows in flow ntworks Th maximal amount of flow is qual to th capacity of a minimal cut. In layman trms th maximum flow in a ntwork is dictatd by its bottlnck. Graph G(VE): consists of a st V of vrticsand a st E of dgs. V consists of sourcs sinks A and othr B nods A mmbr (uv) of E has a to snd information S capacity c(uv) T from u to v D C 8
Max-Flow Min-Cut Thorm Cuts: Partition of vrtics into two sts Siz of a Cut = Total Capacity Crossing th Cut Min-Cut: Minimum siz of Cuts = 5 Max-Flows from S to T Min-Cut = Max-Flow S A A B S D D C B T C T 9
Multicast Problm Buttrfly Ntworks: Each dg s capacity is. Max-Flow from A to D = Max-Flow from A to E = Multicast Max-Flow from A to D and E =.5 Max-Flow for ach individual connction is not achivd. B A F G C D E 0
With Ntwork Coding With ntwork coding w can indd dlivr all th data in 000s. Simply lt th bottlnck link transmit th XOR of th two packts (or bits) and rconstruct vrything at th rcivrs. Ntwork coding savs a factor! In this xampl this is optimal. In gnral? BTW: Sam xampl with on sourc only is known bttr: [Yunnan Wu]
Ntwork coding Nw rsarch ara (Ahlswd t. al. 000) Bnfits many aras Ntworking Communication Distributd computing Uss tools from Information Thory Algbra Combinatorics Has a potntial for improving: throughput robustnss rliability and scurity of ntworking and distributd systms.
Ntwork coding Introduc nods that do mor than forwarding + duplication. Each outgoing packts is a function of incoming packts. m F (m m m ) m F (m m m ) m ncoding
Ntwork coding Linar Coding ovr finit filds Exampls m m m +m +m m m m +m +m m m
Applications: Ntwork Bottlncks Nod B in th ntwork blow is a bottlnck bcaus it will nd to forward traffic for two flows (A to C and D to E). Howvr thanks to ovrharing it is nough if B transmits th XOR. In this xampl all nods hav th sam amount of traffic.
Applications: Scurity [Christina Fragouli EPFL] Without ntwork coding an avsdroppr may gt half of th information. With ntwork coding gtting usful information is hardr.
Digital Fountain A digital fountain strams data continuously and consumrs gt th full contnt aftr a fixd numbr of rcivd packts. Fil Transmission Clint Clint +
Digital Fountain Discussion With th right cods arbitrary n + o(n) out of n packts ar sufficint to rconstruct th complt fil. Th digital fountain ida is slightly oldr than th othr ntwork coding applications and may b sn as th original work on ntwork coding. Howvr in both th digital fountain and th scurity xampls intrmdiat nods simply forward th data without modification. As such it may b outsid th nw scop of ntwork coding. Digital fountains may also b usd to mak data mor availabl. Indd in pr-to-pr ntworks thanks to coding data may b availabl long aftr th sourc did.
Physical Layr Ntwork Coding Rmind -station xampl: Instad nod B may just rpat th rcivd physical signal saving on mor slot: [Christina Fragouli EPFL]
Linar Ntwork Coding Random Procsss in a Linar Ntwork Sourc Input: Info. Along Edgs: Sink Output: X Rlationship procsss among gnratd at thm v Y ( ) l X ( v l) ( v l) x0( v l) x ( v l)... Wightd Combination of Wightd Combination of Y( ) y ( ) ( )... Th 0 y indx procsss is a from adjacnt tim indx dgs of Z( v l) z ( ) ( )... 0 v l z v l Wightd Combination from all incoming ( v) dgs coms out of v Y ( l??? had (? tail( ) Z( v j)? jy(?? had(? v 0
Transfr Matrix Lt X X x ( X ( v) X ( v) X ( v)) z ( v ) ( v ) X ( v ) ( Z( v) Z ( v) vz ( v z x M )) 5 v 5 M A 5 v 0 5 5 5 AB Y ( ) X ( v) ( ) ( X v X v Y ( ) X ( v) ( ) ( X v X v Y ( ) X ( v) ( ) ( X v X v Z( v ) Y ( ) ( ) ( Y Y v Z ( v ) Y ( 5 ) Y ( ) Y ( 5 5 Z( v ) B Y ( ) Y ( ) Y ( Y ( ) Y ( ) Y ( Z( v) Y ( 5 ) Y ( ) Y ( 5 Z( v) Y ( 5 ) Y ( ) Y ( 5 Z( v) Y ( 5 ) Y ( ) Y ( 5 ) ) ) ) ) ) ) ) ) )
Ntwork Coding Solution W want Choos to b an idntity matrix. Choos B to b th invrs of 5 5 5 B A B A M 5 5 0 M x z z x A 5 5 0 NETWORK CODING SOLUTION EXISTS IF DETERMINANT OF M IS NON-ZERO
Connction btwn an Algbraic Quantity and A Graph Thortic Tool Kottr and Mdard (00): Lt a linar ntwork b givn with sourc nod sink nod and a dsird connction c of rat. Th following thr statmnts ar quivalnt.. Th connction c is possibl.. Th Min-Cut Max-Flow bound is satisfid. Th dtrminant of th ( c) R( c) transfr matrix is non-zro ovr th Ring F... l...?...? j R(c) R M...
Random Ntwork Coding Ho Kottr Mdard Kargr Effros (00/0) Random cofficints for linar ntwork coding Can dcod w.p. providd that th bas fild is sufficintly larg Enabls ntwork coding in unknown ntwork topologis Subspac coding: Kottr and Kschischang (00/08)
zhiyongchn@sjtu.du.cn http://iwct.sjtu.du.cn/prsonal/zychn/ 5