Degree Distribution Optimization in Raptor Network Coding

Transcription

1 2 IEEE Internatonal Symosum on Informaton Theory Proceedngs Degree Dstrbuton Otmzaton n Rator Network Codng Nkolaos Thomos and Pascal Frossard Sgnal Processng Laboratory (LTS4) Swss Federal Insttute of Technology, Lausanne (EPFL), Lausanne, Swtzerland {nkolaos.thomos,ascal.frossard}@efl.ch Abstract We consder a mult-source delvery system, where Rator codng at sources and lnear network codng n overlay nodes work n concert for effcent data delvery n networks wth dversty. Such a combnaton ermts to ncrease throughut and loss reslency n multcast scenaros wth ossbly multle sources. The network codng oeratons however change the degree dstrbuton n the set of ackets that reach the recevers, so that the low comlexty decodng benefts of Rator codes are unfortunately dmnshed. We roose n ths aer to change the degree dstrbuton at encoder, n such a way that the degree dstrbuton after network codng oeratons recovers a form that leads to low comlexty decodng. We frst analyze how the degree dstrbuton of the encoded symbols s altered by network codng oeratons and losses n a regular network. Then we formulate a geometrc otmzaton roblem n order to comute the best degree dstrbuton for encodng at sources, such that the decodng comlexty s low and close to Rator decoders erformance. Smulatons show that t s ossble to mantan the low comlexty decodng erformance of Rator codes even after lnear network codng oeratons, as long as the codng at sources s adated to the network characterstcs. Index Terms Network codng, Rator codes, degree dstrbuton, geometrc rogrammng. I. INTRODUCTION Mesh networks are very nterestng for data dstrbuton as they tycally offer several transmsson aths between servers and clents. These aths mght share common lnks and nodes, whch motvates the desgn of arorate dstrbuton and codng strateges that can roerly exlot the network dversty. Network codng () [] takes advantage of network nodes comutaton caabltes to ncrease network throughut and enhance transmsson robustness n networks wth dversty. The network nodes tycally erform codng oeratons wth the receved data before forwardng them to next ho nodes. When combned effcently wth codng at sources, network codng leads to dstrbuted data delvery solutons wthout the need for reconclaton among nodes nor dfferental treatment of data ackets wth dfferent mortance. Ths ermts to beneft from the network dversty for both ncreased throughut and mroved reslence to losses. In ths aer, we consder the transmsson framework reresented n Fg.. We buld on the algorthm that has been roosed n [2] n the secfc context of vdeo streamng n overlay mesh networks. We consder that the source symbols Ths work has been suorted by the Swss Natonal Scence Foundaton, under grant PZP Rator Sender Rator Sender 2 Lossy network Stage Stage 2 2 : Network codng node : Erasure channel Recever Fg.. Illustraton of a transmsson framework wth network codng of Rator symbols. are frst rocessed by Rator encoders [3] wth a degree dstrbuton. Ratorcodesarechosenduetotherrateless erformance as well as ther lnear encodng and decodng tmes. The encoded symbols are then sent on a regular mesh network, where ntermedate nodes mlement lnear network codng. In case of acket loss, the network nodes generate substtute ackets by combnng ars of ackets, whch are selected n order to maxmze the symbol dversty [2]. Due to losses and network codng n successve stages, the degree dstrbuton of the Rator encoders at sources s altered; t mght fnally dffer sgnfcantly from the degree dstrbuton of tycal Rator codes that are characterzed by low-comlexty decodng. In artcular, the generator matrx becomes denser due to combnatons of Rator symbols by network codng, whch leads to ncreased decodng comlexty. The comutatonal comlexty of the decodng oeratons at recevers are clearly drven by the degree dstrbuton of the receved symbols. We thus roose to re-desgn the degree dstrbuton of the encoders at sources such that the actual degree dstrbuton of the recever ermts low-comlexty decodng. We roose here a generc method for comutng an arorate source degree dstrbuton for any target degree dstrbuton functon after lnear network codng n regular toologes. The otmzaton roblem s formulated as ageometrcrogrammng(gp)[4]roblem.gpstycally characterzed by objectve and constrant functons of secal forms. Furthermore, t s well adated for solvng large-scale otmzaton roblems. We assume that the servers are aware of the network statstcs (network losses and toology), and we desgn effcent source degree dstrbutons so that the r //$26. 2 IEEE 2736

2 lnear decodng tme roerty s reserved. We show through smulatons that the resultng codes only suffer from a mnmal erformance loss comared to otmal degree dstrbutons at decoder and stll rovde effectve reslency to losses and network dynamcs. A related roblem has been studed n [5] where the desgn of LT codes degree dstrbuton for smle relay toologes has been nvestgated. The robust solton dstrbuton () [6] s decomosed nto to two comonent dstrbutons ror to deconvoluton. It reserves the skes of that guarantee the success of the decodng rocess. Although the algorthm n [5] ensures that clents receve symbols whose degree dstrbuton s close to, t moses rather comlcated encodng rules. Furthermore, the extenson of ths method to comlex network toologes s non trval. II. ANALYSIS OF LOW-COMPLEXITY RAPTOR NETWORK CODES We analyze here n detals the Rator network codng scheme roosed n [2] for regular networks that have the same number of nodes er codng stage as shown n Fg. and each node s connected wth all nodes n the revous codng stage. By Rator codng at the senders, the data symbols are frst recoded and then fed nto a LT encoder. The transmtted symbols are roduced by samlng a degree dstrbuton functon (x) = x, whch determnes the number of orgnal data symbols that has been combned for generatng the Rator encoded symbols. denotes the robablty of generatng a symbol wth degree. Snce s are degree dstrbuton functon coeffcents the followng constrants should hold: = (), [,...,L] As the ackets travel through the network, they are combned to comensate for acket losses due to erasures and network varatons. The network coded symbols follow a new denser degree dstrbuton functon that s dfferent from the orgnal functon used at the servers. The degree dstrbuton at the k th codng stage s denoted as k (x) = k x where k,=,...l, are the coeffcents of the degree dstrbuton functon at nodes n the k th codng stage. The degree dstrbuton evolves as the symbols traverse the dfferent encodng stages, and eventually lead to the degree dstrbuton r observed at the clent (decoder). We now analyze n more detals the evoluton of the degree dstrbuton n regular networks where all network lnks have equal caacty and dentcal loss rate. Frst, sngle ho transmsson s examned and then the fndngs are extended for larger networks. At the frst network codng stage, the reencoded ackets corresond to a degree dstrbuton functon (x) gven by (x) = x =( ) + Ψ x = x + (2) { ( ) + Ψ } x where Ψ (x) s the convoluton ( ) (x) snce the symbols that are combned follow the degree dstrbuton (x). Itcan be wrtten as µ ν Ψ = µ+ν=,=,...,l (3) µ ν or equvalently Ψ = µ+ν= j j,=,...,l (4) l j l j l= The Ψ values are normalzed by A = L l l= j l j snce some symbols combnatons are not elgble as the degree of the re-encoded symbols can not exceed L, whchsequal to the number of source symbols. Obvously, the dstrbuton (x) can be seen as the weghted sum of (x) and Ψ (x) wth weght arameter (.e., the acket loss rate). For the degree dstrbutons of tycal Rator codes such as [7] and, the denomnator of Eq. (4) s aroxmately wrtten as l L/2 j l j = 2 +2 l= The term l=l/2+ l= ( L ) 2 = l= L l=,l L l l=l/2+ 2 L/2 tends to zero as l= l L l = 2 = = for the and dstrbutons that are concentrated nto small degree values. Thus, the codng coeffcents of Eq. (2) can be wrtten as =( ) + j j, for =,...,L (5) Unfortunately Rator and degree dstrbutons contan skes that do not ermt drect deconvoluton. Therefore, the resultng equaton system cannot be solved as some of the constrants of Eq. () are volated. In order to exlan ths lmtaton, we rewrte Eq. (5) as 2737

3 = { j j }, for =,...,L (6) In the case of the, t holds that 2 > and + <,,...,M,M +,...L where M = K/S.The K and S are resectvely the number of source symbols and aarameterthatcontrolstheszeoftherleneveryste of the decodng rocedure. The rle contans the symbols that have been recovered but not rocessed yet. We have the followng lemma, whose roof s gven n the Aendx. Lemma : If (x) has a ske at M then t should be: { } c> max j [2,] and c > max j [2,] 2 M, {, 2 M j M j M j M j+ for M >. c, c determne the magntude of the ske and t holds M = c M and M = c M+. Therefore, t s not ossble to fnd a functon (x) that reserves the ske at the M th oston and satsfes the condtons of Eq. (), snce s become negatve when c and c take large ostve value. Smlar conclusons can be drawn for other degree dstrbutons wth skes. Therefore, we can only desgn subotmal degree dstrbutons (x) that lead to fnal degree dstrbuton r (x) that are close but not equal to the target degree dstrbuton functons such as or. III. DEGREE DISTRIBUTION OPTIMIZATION We roose now to otmze the degree dstrbuton at the source by an teratve algorthm. If there s only one codng stage, the otmzaton roblem can be formulated as mn (7) { } ( ) + j j where (x) s the fnal dstrbuton. The otmzaton condtons are gven n Eq. (). We cast ths otmzaton roblem nto a Geometrc Programmng (GP) roblem. The general form of a GP roblem s mn f (W ) subject to f (W ) for =,...,m g (W )= for =,...,n where g, =,...,n and f, =,...,m are resectvely monomals and generalzed osynomals [8]. W = (W,...W n, ) s a vector wth n real varables. To transform Eq. (7) nto a vald GP form, we use the geometrc nequalty L L L } Thus, the objectve functon s wrtten as mn { } ( ) / / ( j j ) / whle the otmzaton constrants are L L /L = Snce we are aware of the exected loss rate, we add two new constrants ( + ) ( ) These constrants lmt the search sace and allow fast convergence. The resultng roblem can then be solved wth hel of software ackages such as [9], []. When we have two codng stages, we consder the otmzaton roblem as a cascade of two deendent GP roblems. Snce we know the desred degree dstrbuton, we frst solve mn { } 2 ( ) / ( )/ then after determnng the solve mn { } ( ) / / ( j / j) coeffcents, we successvely ( j j ) / to fnd the orgnal degree dstrbuton. The same rocess s reeated teratvely for larger networks. IV. SIMULATION RESULTS We analyze the erformance of the above desgn algorthm n a Rator network codng system. We study the erformance of a system where we target a -lke degree dstrbuton functon at decoder, wth the desgn rocedure descrbed above. Ths dstrbuton s used along wth a re-coder at the encoder. We comare ths system wth a varant that combnes the same re-coder as codes and LT codes wth dstrbuton (denoted as ). The erformance s comared n terms of decodng robablty and decodng comlexty. For the sake of comleteness, we also study the erformance of classcal degree dstrbuton (denoted as ), whch are heurstc dstrbutons commonly used n wreless broadcastng systems. We frst look at the erformance of these codes for dfferent network szes. We consder regular toologes wth varous number of network codng stages between servers and clents wth three nodes er codng stage. The source ackets are rotected by (23, 249) Rator codes. For all the lnks, the 2738

4 Probablty of decodng falure Probablty of decodng falure Probablty of decodng falure... Number of receved symbols Number of receved symbols Number of receved symbols (a) (b) (c) Probablty of symbol wth degree x Probablty of symbol wth degree x Probablty of symbol wth degree x (d) (e) (f) Fg. 2. Performance evaluaton of (23, 249) Rator codes for regular network toologes wth three ntermedate nodes er stage for codes (), codes modfed wth dstrbuton () and the roosed dstrbuton. The Rator decodng falure robabltes at clents wth resect to the number of receved symbols are resented for (a) three, (b) sx and (c) nne codng stages. Cumulatve dstrbuton of symbols degree receved by each clent are shown for networks wth (d) three, (e) sx and (f) nne codng stages. Probablty of decodng falure Probablty of decodng falure.. Number of receved symbols Number of receved symbols (a) (b) Fg. 3. Performance comarson of the network codng schemes emloyngwth, andtherooseddegreedstrbuton functons at encoder, for regular networks wth sx stages and (a) three and (b) sx nodes er codng stage. symbol loss rato s set to 5%, whlethelnkbandwdthb ermts the transmsson of 83 symbols er lnk er tme nterval. The results for varous network toologes are demonstrated n Fg. 2. The Rator decodng robabltes wth resect to the number of receved ackets for three-, sx- and nnestage network toologes are shown n Fgs. 2(a), (b), and (c) resectvely. For three-stage network toologes all methods erform equally well. As the number of stages ncreases, the roosed dstrbuton erforms slghtly worst n terms of decodng robablty, but t remans close to the erformance of the other schemes. Ths erformance degradaton s due to the neffcency of the desgn method n reservng the exact value of the skes n the degree dstrbuton functon. Ths makes the roosed codes less effcent for large sze toologes as the ske value becomes smaller than the deal value. However, when the methods are comared n terms of the cumulatve degree dstrbuton functon (cdf) as dected n Fgs. 2(d), (e), and (f), t s clear that the roosed codes outerform the varant wth n terms of decodng comlexty. The scheme results n very dense generator matrces, whle the desgned code that targets a -lke degree dstrbuton at the decoder results n sarser matrces and hence lower decodng comlexty. Ths confrms that the roosed desgn algorthm succeeds n achevng low comlexty decodng even after symbols have been combned by network codng. Fnally, note that both schemes have nferor erformance to the codes n terms of decodng robablty. In addton, the codes offers sarser generator matrces. Ths s exected as codes otmze the re-coder n order to 2739

5 use sarser LT generator matrces. However, we can observe n Fgs. 2(d), (e), and (f) that the codes result n denser matrces and become therefore less favorable to lowcomlexty decodng when the sze of the network ncreases. The codes have very sarse matrces, but network codng affects the sarsty of the generator matrces as the number of codng stages ncreases. For larger networks the cdf of the codes roosed n ths aer becomes smlar to the one of codes for low degree values. They lead to smlar decodng comlexty as the codes, whch are among the most effectve codes n ractce. Ths shows that shang the source (Rator codes) degree dstrbuton n order to guarantee low-comlexty decodng s ncreasngly mortant for large network szes. We also study the erformance of the dfferent codng strateges for dfferent number of nodes er codng stage. We consder network toologes wth sx codng stages, but resectvely 3 and 6 network codng nodes er stage (see Fgs. 3 (a) and (b)). We have set the lnk caactes n order to delver 249 symbols to each recever when transmsson s error free. All lnks face 5% symbol loss rate. When network dversty s lmted, the desgned codes have nferor erformance n terms of decodng robablty, comared to that of the other two schemes. Ths erformance dfference s however not sgnfcant. All schemes take advantage of the mroved network dversty as shown n Fg. 3(b). The erformance ga between the desgned codes and the varant, however, decreases when the number of nodes er codng stage ncreases. In ths case, the multle receton robablty gets lower. Ths can be exlaned by a larger network dversty as more aths connect the clents wth the servers, whch favorably comensates for the larger robablty of low degree symbols to be re-combned n network nodes. The otmzed degree dstrbuton ermts to take advantage from the network dversty and leads to decodng erformance that s comettve wth the other dstrbutons. At the same tme, t guarantees a low decodng comlexty, whle the other source degree dstrbutons lead to ncreased erformance enalty when the network dversty augments. V. COLUSIONS In ths aer, we resented a novel method for desgnng adegreedstrbutonforratorlkeencodngatsenders,so that the degree dstrbuton of the receved symbols could be controlled n network codng data delvery system. A generc otmzaton roblem has been roosed for determnng through geometrc rogrammng the arorate source degree dstrbutons n regular network toologes. The smulaton results show that these roosed codes erform close to tycal 3GP P Rator codes n terms of decodng robablty. At the same tme, they ermt to mantan a lnear decodng tme smlar to deal degree dstrbuton functons such as. The multle receton robablty s the robablty that anodereceve multle tmes the same symbol, whch s non-zero as symbols that have been combned can be combned agan. ACKNOWLEDGMENTS The authors would lke to thank Dr Amn Shokrollah for ntal dscussons on the degree otmzaton roblem. APPENDIX Proof of lemma : From eq. (6) for the ske at M we have M = c M and thus we derve M = c M ( ) M j M j c j M j In order to determne the desgnng constrants for c, we examne when M takes negatve values. Therefore, f M < t s c M < ( ) c j M j + + M ( ) j M j snce M > and the nequalty s true ff {c 2 M } + j {c M j M j } < + j=2 We want all terms n the left sde to be negatve. Thus, c< 2 M and c< M j M j,j=2,...,m 2. As we have seen due to ske at M we have M = c M+.Smlartotheaboveanalysstcanbeseenthat M < when c < ( ) 2 and < M j c M j+,j=2,...,m. REFEREES [] R. Ahlswede, N. Ca, S.-Y. R. L, and R. W. Yeung, Network Informaton Flow, IEEE Trans. Informaton Theory, vol. 46, no. 4, , Jul. 2. [2] N. Thomos and P. Frossard, Network Codng of Rateless Vdeo n Streamng overlays, IEEE Trans. Crcuts and Systems for Vdeo Technology,vol.2,no.2, ,Dec.2. [3] A. Shokrollah, Rator codes, IEEE Trans. Informaton Theory, vol. 52, no. 6, , June 26. [4] M. Chang, Geometrc Programmng for Communcatons Systems. Boston, MA: Now Publsers Inc, 25. [5] S. Puducher, J. Klewer, and T. E. Fuja, The Desgn and Performance of Dstrbuted LT codes, IEEE Trans. Informaton Theory, vol. 53, no., , Oct. 27. [6] M. Luby, LT codes, n Proc. of the 43rd Annual IEEE Symosum on Foundatons of Comuter Scence (FOCS 2), Vancouver,Canada, Nov. 22, [7] TS V7.., Techncal Secfcaton Grou Servces and System Asects; Multmeda Broadcast/Multcast Servce; Protocols and Codecs, June 25. [8] S. Boyd and L. Vandenberghe, Convex Otmzaton. New York: Cambrdge Unversty Press, 24. [9] J. Lofberg, Yalm : A toolbox for modelng and otmzaton n MATLAB, n Proceedngs of the CACSD Conference, Tae,Tawan, 24. [Onlne]. Avalable: htt://control.ee.ethz.ch/ joloef/yalm.h [] The mosek otmzaton tools verson 5.. user s manual and reference, 27, [Onlne]. Avalable: htt:// 274