Orientation. Connections between network coding and stochastic network theory. Outline. Bruce Hajek. Multicast with lost packets

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1 Connecions beween nework coding and sochasic nework heory Bruce Hajek Orienaion On Thursday, Ralf Koeer discussed nework coding: coding wihin he nework Absrac: Randomly generaed coded informaion blocks form he basis of novel coding echniques used in communicaion neworks. The mos sudied case involves linear coding, in which coded blocks are generaed as linear combinaions of daa blocks, wih randomly chosen coefficiens. Digial founain codes, including Luby s LT codes, and Shokrollahi s Rapor codes, involve coding a he source, while nework coding involves coding wihin he nework. Recenly Maneva and Shokrollahi found a connecion beween he analysis of digial founain codes, and fluid and diffusion limi mehods, such as in he work of Darling and Norris. A simplified descripion of he connecion is presened, wih implicaions for code design. (Background reading can be found a hp://courses.ece.uiuc.edu/ece559/spring06bh/ ) This alk will be limied o he opic of coding a source nodes, which is much furher owards real world use. (See Sochasic nework heory can be helpful in learning how o apply such codes in neworks (e.g. radeoff beween buffer size and erasure proecion) and in designing such codes. This alk focuses on he laer. 2 Ouline Mulicas, and linear codes for erasures Luby s LT codes Markov performance analysis for Poisson model Fluid approximaion and an applicaion Diffusion approximaion and an applicaion Mulicas wih los packes source message: k symbols: (fixed lengh binary srings) Symbols are repeaedly broadcas in random order o many receivers, bu can be los. Each receiver ries o collec a complee se, for example: S S S4 S5 Sk-2 Sk Sk 3 4

2 Mulicas wih coding a source, and los packes Source message: k symbols: (fixed lengh binary srings) Source forms m coded symbols: S Sm Symbols are repeaedly broadcas in random order o many receivers, bu can be los. Each receiver ries o collec enough disinc coded packes: S S S4 S5 Sm-2 Sm Sk Linear coding and greedy decoding Source symbols: S 00 0 S3 0 S Received payloads: For a good code, only k or a few more coded packes are enough. If m > > k, hen problem of duplicaes a receiver is reduced. 5 6 Source symbols: Greedy decoding can ge suck: S S3 LT codes (M. Luby) random, raeless, linear codes Given he number of source symbols k, and a probabiliy disribuion on {,2,..., k}, a coded symbol is generaed as follows: Received payloads: Neverheless, we will sick o using greedy decoding. 7 Sep one: selec a degree d wih he given probabiliy disribuion. Sep wo: selec a subse of {,..., k} of size d. Sep hree: form he sum of he source symbols wih indices in he random se e.g., d=3. e.g., {3,5,6} code vecor The resuling coded symbol can be ransmied along wih is code vecor. e.g., k=8 e.g., form S3+S5+S6 8 2

3 Ideal solion disribuion for degree (Luby) Ideally we could recover all k source symbols from k coded symbols. Analysis A coded symbol wih reduced degree one (which has no been processed ye) is said o be in he gross ripple. Le X v denoe he number of symbols in he gross ripple afer v symbols have been decoded. Decoding successful if and only if ripple is nonempy hrough sep k-. Sudy Poisson case: --The ideal solion disribuion doesn work well due o sochasic flucuaions. Luby inroduced a robus variaion (see LT paper) 9 --Rapor codes use precoding plus LT coding o help a he end. 0 Examine arrival process for he gross ripple nex symbol decoded v v+ specific symbol o be decoded laer k Evoluion of gross ripple j-2 j + Degree j coded packe x 0 2 3

4 Inermediae Performance An applicaion of fluid limi (S. Sanghavi s poser, his conference) Sujay s bounds Le K = # inpu symbols. If # received coded symbols is rk, for r<, hen he number of inpus recovered is K where upper bound upper and lower bounds Example: for r S. Sanghavi invesigaed maximizing wih respec o he degree disribuion, for r fixed. 3 r 4 r Nex: diffusion analysis Recall: Diffusion limi of gross ripple x 0 Le 5 6 4

5 Design based on diffusion limi The diffusion limi resul suggess he represenaion: which in urn suggess guidelines for he degree disribuion: The ne ripple (used by Luby in LT paper) The ne ripple is he se of inpu symbols represened in he gross ripple. Le Y v denoe he number of symbols in he gross ripple afer v symbols have been decoded. Decoding successful if and only if he ne ripple is nonempy hrough sep k The free ripple (undersanding fluid approximaion) Suppose ha inpu symbols are revealed one a a ime by a genie, independenly of he packes received. The degrees of coded symbols are decreased as he genie reveals symbols. The free ripple is he se of coded symbols of reduced degree one in such sysem

6 Conclusions Coding a he source, nework coding, and peer-o-peer neworking (gossip ype algorihms) provide a rich source of nework design and analysis problems. Ieraing design and analysis can lead o ineresing racable problems (as in LT code design) Thanks!