1 Inroducion A (1 + ") loss-resilien code encodes essages consising of sybols ino an encoding consising of c sybols, c > 1, such ha he essage sybols c

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1 A Lower Bound for a Class of Grah Based Loss-Resilien Codes Johannes Bloer Bea Trachsler 1 May 5, 1998 Absrac. Recenly, Luby e al. consruced inforaion-heoreically alos oial loss-resilien codes. The consrucion is based on a sequence of biarie grahs. Using a robabilisic consrucion for he individual biarie grahs, hey obain loss-resilien codes ha have very ecien encoding and decoding algorihs. They lef oen he quesion wheher here are oher biarie grahs leading o codes wih even ore ecien encoding and decoding algorihs. In his aer we show ha if one follows he basic consrucion used by Luby e al., hen heir choice of biarie grahs leads o codes wih asyoically oial encoding and decoding algorihs. Auhors' addresses Johannes Bloer and Bea Trachsler Insiue for Theoreical Couer Science ETH Zenru CH-8092 Zurich Swizerland Technical Reor #298, Deareen Inforaik, ETH Zurich. Elecronically available fro f//f.inf.ehz.ch/ub/ublicaions/ech-reors/ 1 Work of boh auhors has been suored by Gran fro Schweizer Naionalfonds (SNF). 1

2 1 Inroducion A (1 + ") loss-resilien code encodes essages consising of sybols ino an encoding consising of c sybols, c > 1, such ha he essage sybols can be reconsruced fro any subse of (1 + ") sybols of he encoding. If " = 0 hen hese codes are usually called MDS codes (Maxiu Disance Searable). If he reconsrucion is guaraneed o be successful only wih high robabiliy, hen he code is called a robabilisic (1 + ") loss-resilien code. Here he robabiliy is wih resec o rando subses of (1 + ") sybols of he encoding. Loss-resilien codes have alicaions for exale in neworking. In realie alicaions, where reransission of los ackes is no feasible, hey can be used o roec agains he eecs of acke losses. For exale, any MPEG video layers canno deal wih disrued MPEG sreas. Even if hey can, video qualiy usually suers signicanly fro acke losses. In [2] Luby e al. inroduced a faily of robabilisic (1 + ") loss-resilien codes wih very ecien encoding and decoding algorihs. The codes are based on a sequence of biarie grahs. The individual biarie grahs are obained by a robabilisic consrucion using a very secial disribuion on biarie grahs. This lef oen he quesion, wheher choosing dieren biarie grahs can lead o codes wih ore ecien encoding and decoding algorihs. In his aer we show ha no aer how he individual biarie grahs are chosen, codes based on a sequence of biarie grahs as roosed in [2] lead o loss-resilien codes for which he encoding and decoding requires ie ( ln(1=")). To be ore recise, if we wan ha wih non-consan robabiliy a rando subse of (1 + ") sybols of he encoding allows o reconsruc he enire essage, hen he average degree of he biarie grahs has o be (ln(1=")). This lower bound on he average degree iediaely leads o he lower bounds for he encoding and decoding ies. The aer is organized as follows. In Secion 2 we review he codes of Luby e al. in ore deail. In Secions 3 and 4 we sae and rove our ain resuls. In Secion 5 we indicae how o rene he roofs of Secion 4 in order o obain lower bounds for he encoding and decoding ies ha alos ach he uer bounds given in [2]. 2 An Ecien Loss-Resilien Code In his secion we inroduce he coding schee by Luby e al. The goal is o give a shor overview of he consrucion. Proofs can be found in [2, 1]. For an inroducion o coding heory we refer o [3]. In he following we denoe essage sybols by bis alhough in racice one acually uses bi srings. I is sraighforward o generalize he consrucion of Luby e al. o bi srings. 2

3 The ain ar of he schee is a rando direced acyclic grah whose nodes are ariioned ino ses S i, 0 i `, such ha js i+1 j = js i j, where 0 < < 1. The nodes in S 0 corresond o he essage bis of he code. The nuber of essage bis is denoed by. The nodes in S i, 1 i `, corresond o he check bis. The edges of he grah go only fro S i o S i+1, 0 i ` 1. The consrucion allows for a very sile and ecien encoding algorih which 2 ` S` loss-resilien code C S 2 S 1 S 0 Figure 2.1 Coding schee by Luby e al. coues he check bis in he ses S 2 ; ; S` fro lef o righ by ieraing he following oeraion. Encoding. The value of a node in S i+1 is coued as he XOR of he values of is neighbors in S i. The running ie of his ar of he encoding algorih is roorional o he nuber of edges in he grah. The decoding is done fro righ o lef. I urns ou o be ore robleaic alhough he underlying oeraion is eleenary as well. Decoding. Assue ha he value of a node c in S i+1 and he values of all bu one of is neighbors in S i are known. Then he value of he issing neighbor in S i is coued by he XOR of he value of c and he values of is known neighbors. For his rocedure o recover all essage bis, one has o guaranee ha as long as here are nodes in S i, whose values are unknown, here is a node c in S i+1 o which he basic decoding oeraion can be alied. In [2] his guaranee is obained using rando biarie grahs wih he following roeries o connec S i and S i+1 3

4 1. Wih high robabiliy he bis in S i can be reconsruced by he decoding oeraion fro above if all he bis in S i+1 are known and only a (1 ") ar of he bis in S i is los. 2. Aroxiaely, he average ou-degree of he nodes in S i is ln(1="). To sar his rocedure wih levels S` and S`1 one has o guaranee ha all of he bis a he las level S` are known. The idea is o choose ` in such a way ha an ecien sandard loss-resilien code C wih rae 1 can encode and decode S` in linear ie wih resec o. Since in racice he os ecien loss-resilien codes have quadraic ie encoding and decoding algorihs, we choose ` such ha js`j 2 Hence for his ar of he encoding and decoding can be done in ie linear in. Since he encoding and he decoding in he grah ake ie roorional o he nuber of edges, he running ie of he overall encoding and decoding algorihs is O( ln(1=")). Fro he discussion above i also follows ha he code is a (1 + ") lossresilien code. Finally, we clai ha he rae of he code is 1. This follows fro he fac ha he encoding lengh is given by `X j j=0 {z } # nodes in S 0 [ [ S` 3 The Lower Bound + `+1 1 {z } # check bis of C = 1 For he following arguens we only need he srucure of he biarie grah ha connecs he nodes in S 0 o he nodes in S 1. Firs, we ake he following inforaion-heoreical observaion. I is he key arguen for he resuls below. Proosiion 3.1. Denoe he se of nodes corresonding o he los essage bis by T. Then he se (T ) of nodes conneced o T has o be a leas as large as T if he decoding algorih erinaes successfully. Proof. If j(t )j < jt j, he check bis do no conain enough inforaion o recover all he essage bis. Fro his we can derive a lower bound for he average degree ` of he nodes in S 0. Lea 3.2. Assue ha a node in S 0 is conained in T uniforly and indeendenly wih robabiliy (1 "), where 0 < " < 1 is consan. If he average 4

5 degree ` of he nodes in S 0 saises ` < 1 4 (1 ) ln 1 " ; (3.1) hen he robabiliy ha he se (T ) is saller han T is (1). The roof of his lea will be given in he nex secion. Reark 3.1. Noice ha is he inforaion-heoreically oial erasure robabiliy for he essage bis. A coding schee ha guaranees a successful decoding a his erasure robabiliy leads o an oial MDS code. Proosiion 3.1 and Lea 3.2 ily he following lower bound for he running ie of he encoding and decoding algorih by Luby e al. Theore 3.3. Assue ha he essage bis are los uniforly and indeendenly wih robabiliy (1 "), where 0 < " < 1 is consan. If he robabiliy ha he decoding fails is required o be o(1), hen he encoding and he decoding akes ie ( ln(1=")). 4 The Proof of Lea 3.2 For he roof we use he following sraegy. To show ha for ` < 1 4 (1 ) ln 1 " he robabiliy Pr[j(T )j < jt j] is (1), for all values of jt j in soe inerval I, we esiae he execed size of he se (T ) of neighbors of T. Then we aly Markov's inequaliy o obain a consan lower bound on Pr[j(T )j < jt j j jt j = ]. Choosing I such ha Pr[jT j 2 I] is consan, nishes he roof. Since every node in S 0 lies in T wih robabiliy = (1"), he execaion and he sandard deviaion are Dene I by E[jT j] = ; [jt j] = (1 ) I = [ in ; ax ] where in def = (1 2") def ax = Noice ha he execaion of jt j lies in he iddle of I. inequaliy Pr [jt j =2 I] = Pr jt j E[jT j] > " jt = Pr j E[jT j] " > [jt j] (1) By Chebyshev's k("; ) 1 where k("; ) = (1) " 2 2 5

6 Hence Pr [jt j 2 I] 1 k("; ) 1 jt j = We wan o show ha for any 2 I he robabiliy Pr j(t )j < jt j is (1). For his we coue E j(t )j jt j =. Fix an arbirary 2 I. Condiioned on jt j = he robabiliy ha a xed d node w 2 S 1 is no conneced o a rando se T of size is w =, where d w denoes he in-degree of w. Hence Pr w 2 (T ) dw jt j = = 1 Since js 1 j =, E j(t )j X dw jt j = = w2s 1 Lea 4.1. If he average in-degree r in S 1 saises r, hen X 1 r w2s 1 dw x Proof. Since is a convex funcion in x, X dw w2s 1 r Now he roof is coleed by he following calculaion r = ( r)( r 1) ( r + 1) ( 1) ( + 1) = 1 r 1 1 r 1 r 1 r + 1 Since r = ` and since ` is bounded by a consan, we can assue ha r 1 2 ( ) for large enough. By Lea 4.1 and using 1 x e2x ; 0 x 1=2, E j(t )j jt j = 1 1 r 1 e 2 r! 6

7 Since ax ax = 1, E j(t )j jt j = 1 e 2 1 r (4.1) for an arbirary 2 I. Now we derive a suiable inequaliy for he condiional execaion of j(t )j. Lea 4.2. If ` < 1 4 (1 ) ln 1 ", hen for any 2 I. E j(t )j jt j = < 1 " (4.2) Proof. Assue ha E j(t )j jt j = 1 " Then wih (4.1) " e 2 1 r Hence r ln 1 " and nally ` = r 1 4 (1 ) ln 1 " This conradics he condiion of he lea. Now we can nish he roof of Lea 3.2. Pr j(t )j jt j jt j = Pr j(t )j in jt j = for an arbirary 2 I. Since in Lea 3.2 we assue ` < 1 4 (1 ) ln 1 ", inequaliy (4.2) ilies in = (1 2") = 1 2" 1 " (1 ") for any 2 I. By Markov's inequaliy Pr j(t )j jt j > 1 2" 1 " E j(t )j 1 " jt j = 1 2" 7 jt j =

8 Thus for an arbirary 2 I. Hence Pr j(t )j < jt j Therefore Pr j(t )j < jt j " 2" jt j = 1 2" X jt j 2 I = Pr j(t )j < jt j jt j = Pr [jt j = ] 2I " 2" 1 2" " 2" = 1 2" X 2I Pr [jt j = ] Pr [jt j 2 I] Pr [j(t )j < jt j] Pr j(t )j < jt j " 2" (Pr [jt j 2 I])2 1 2" " 2" 1 2k("; ) 1 1 2" This colees he roof. jt j 2 I Pr [jt j 2 I] = (1) 5 A Degree Bound wih Beer Consans In his secion we wan o irove he consans in Lea 3.2. Corollary 5.1. Assue ha a node in S 0 is conained in T uniforly and indeendenly wih robabiliy (1 "), where 0 < " < 1 is consan. If he average degree ` of he nodes in S 0 saises ` < (1 )(1 ") ln 1 " C (5.1) where C = (1 )(1 ") ln 2, hen he robabiliy ha he se (T ) is saller han T is (1). Reark 5.1. In he consrucion by Luby e al. he average degree of he nodes in S 0 is a leas ln 1 (see [2]). This is for sall " u o a consan facor (1 ) " he sae as he righ hand side of (5.1). The roof of his corollary is alos he sae as he roof of Lea 3.2. Consider inequaliy (4.1). I can be relaced by E j(t )j jt j = 1 e c(1) r for an arbirary 2 I where 0 < c < 1. Insead of (4.2) we obain (5.2) 8

9 Lea 5.2. If ` < (1 )(1 ") ln 1 2", hen for any 2 I. Proof. Assue ha E j(t )j E j(t )j jt j = < 1 (2") 1" jt j = 1 (2") 1" (5.3) Then by (5.2) wih c = 1 " (2") 1" e r 1"(1) Hence r 1 (1 ") ln 1 2" and nally ` = r (1 )(1 ") ln 1 2" This conradics he condiion of he lea. The res of he roof follows along he sae lines as he roof in Secion 4. References [1] Michael G. Luby, Michael Mizenacher, and M. Ain Shokrollahi. Analysis of rando rocesses via and-or ree evaluaion. In Proceedings of he Ninh Annual ACM-SIAM Syosiu on Discree Algorihs, ages 364{ 373, San Francisco, California, 25{27 January [2] Michael G. Luby, Michael Mizenacher, M. Ain Shokrollahi, Daniel A. Sielan, and Volker Seann. Pracical loss-resilien codes. In Proceedings of he Tweny-Ninh Annual ACM Syosiu on Theory of Couing, ages 150{159, El Paso, Texas, 4{6 May [3] J. H. van Lin. Inroducion o Coding Theory. Sringer-Verlag,