Evolutionary Approaches To Minimizing Network Coding Resources

Transcription

1 This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2007 proceedings. Evolutionry Approches To Minimizing Network Coding Resources Minkyu Kim, Muriel Médrd, Vrun Aggrwl, Un-My O Reilly, Wonsik Kim, Chng Wook Ahn, nd Michelle Effros Astrct We wish to minimize the resources used for network coding while chieving the desired throughput in multicst scenrio. We employ evolutionry pproches, sed on genetic lgorithm, tht void the computtionl complexity tht mkes the prolem NP-hrd. Our experiments show gret improvements over the su-optiml solutions of prior methods. Our new lgorithms improve over our previously proposed lgorithm in three wys. First, wheres the previous lgorithm cn e pplied only to cyclic networks, our new method works lso with networks with cycles. Second, we enrich the set of components used in the genetic lgorithm, which improves the performnce. Third, we develop novel distriuted frmework. Comining distriuted rndom network coding with our distriuted optimiztion yields network coding protocol where the resources used for coding re optimized in the setup phse y running our evolutionry lgorithm t ech node of the network. We demonstrte the effectiveness of our pproch y crrying out simultions on numer of different sets of network topologies. I. INTRODUCTION It is now well known tht network throughput cn e significntly incresed y employing the novel technique of network coding, where the intermedite nodes re llowed to comine dt received from different links [1], [2]. While most network coding solutions employ coding t ll possile nodes, it is often possile to chieve the network coding dvntge y coding only t suset of nodes. Exmple 1: In the cnonicl exmple of network B (Fig. 1()) [1], only node z needs to comine its two inputs while ll other nodes perform routing only. If we suppose tht link (z,w) in network B hs cpcity 2, which we represent y two prllel unit-cpcity links in network B (Fig. 1()), multicst of rte 2 is possile without network coding. In network C (Fig. 1(c)), where node s wishes to trnsmit dt t rte 2 to the 3 lef nodes, network coding is required t either node or node, ut not oth. Exmple 1 leds us to the following question: At which nodes does network coding need to occur to chieve the multicst cpcity? If network coding is hndled t the ppliction lyer, we cn minimize the cost of network coding M. Kim, M. Médrd, nd W. Kim re with the Lortory of Informtion nd Decision Systems, Msschusetts Institute of Technology, Cmridge, MA 02139, USA ({minkyu, medrd, wskim14}@mit.edu). V. Aggrwl nd U.-M. O Reilly re with the Computer Science nd Artificil Intelligence Lortory, Msschusetts Institute of Technology, Cmridge, MA 02139, USA ({vrun g@, unmy@csil.}mit.edu). C. W. Ahn is with the Deprtment of Informtion nd Communictions, Gwngju Institute of Science nd Technology, Gwngju , Kore (cwn@evolution.re.kr). M. Effros is with the Dt Compression Lortory, Cliforni Institute of Technology, Psden, CA 91125, USA (effros@cltech.edu). x s z w y () Network B x s z w y () Network B x s y c d Fig. 1. Smple Networks for Exmple 1 (c) Network C y identifying the nodes where ccess up to the ppliction lyer is not necessry. If network coding is integrted in the uffer mngement of router, it is importnt to understnd where nd how mny such specil routers must e deployed to stisfy the communiction demnds. Determining miniml set of nodes where coding is required is difficult. The prolem of deciding whether given multicst rte is chievle without coding, i.e., whether the minimum numer of required coding nodes is zero or not, reduces to multiple Steiner sugrph prolem, which is NP-hrd [3]. Hence, the optimiztion prolem to find the miniml numer of required coding nodes is NP-hrd. Even pproximting the miniml numer of coding nodes within ny multiplictive fctor or within n dditive fctor of V 1 ɛ is NP-hrd [4]. In [5], we introduce n evolutionry pproch for finding prcticl multicst protocol tht provides the full enefit of network coding with reduced numer of coding nodes. The proposed pproch uses Genetic Algorithm (GA) tht opertes on set of cndidte solutions which it improves sequentilly vi mechnisms inspired y iologicl evolution (e.g., recomintion/muttion of genes nd survivl of the fittest). The lgorithm proposed in [5] reduces the numer of coding links/nodes reltive to prior pproches nd pplies to vriety of generlized scenrios. The new lgorithm proposed here improves on the one in [5] in three key wys. First, wheres the lgorithm in [5] cn e pplied to only cyclic networks, we devise modified method tht works lso with networks with cycles. Second, we introduce new set of GA components tht in our experiments significntly outperforms the one used in [5]. Third, we develop novel frmework where most time consuming computtions of the evolutionry lgorithm re distriuted over the network. This new frmework, comined z X/07/$ IEEE 1991 Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloded on Octoer 13, 2008 t 15:22 from IEEE Xplore. Restrictions pply.

2 This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2007 proceedings. with the distriuted rndom network coding scheme of [6], cn mke distriuted network coding protocol where the resources used for coding re optimized in the setup phse s our proposed lgorithm running t ech node of the network. The rest of the pper is orgnized s follows. Section II presents the prolem formultion nd summrizes relted work. Section III descries the improvements of the lgorithm using centrlized frmework. Section IV extends this pproch to distriuted frmework. Section V presents experimentl results. Section VI concludes with topics for future reserch. II. PROBLEM FORMULATION AND RELATED WORK A. Prolem Formultion We ssume tht the network is given y directed multigrph G = (V,E), where ech link hs unit cpcity. Connections with lrger cpcities re represented y multiple links. Only integer flows re llowed, hence there is either no flow or unit rte of flow on ech link. We consider the single multicst scenrio in which single source s V wishes to trnsmit dt t rte R tosett V of sink nodes, where T = d. RteR is sid to e chievle if there exists trnsmission scheme tht enles ll d sinks to receive ll of the informtion sent. We consider only liner coding, where node s output on n outgoing link is liner comintion of the inputs from its incoming links. Liner coding is sufficient for multicst [2]. Given the trget rte R, which we ssume is chievle if coding is llowed t ll nodes, we wish to determine miniml set of nodes where coding is required in order to chieve this rte. Coding is necessry t node v V if coding is necessry on t lest one of node v s outgoing links. As pointed out lso in [4], the numer of coding links is more ccurte estimtor of the mount of computtion incurred y coding. We ssume herefter tht our ojective is to minimize the numer of coding links rther thn nodes. Note, however, tht s demonstrted in [5], it is strightforwrd to generlize the proposed lgorithm to the cse of minimizing the numer of coding nodes. Furthermore, [5] shows tht, with pproprite chnges, the lgorithm cn e redily pplied to more generlized optimiztion scenrios, e.g., where different links/nodes hve different costs for coding. It is cler tht no coding is required t node with only single input since it hs nothing to comine with. We refer to node with multiple incoming links s merging node. If the linerly coded output on prticulr outgoing link of prticulr merging node weights ll ut one incoming messge y zero, then no coding occurs on tht link. (Even if the only nonzero coefficient is not identity, there is nother coding scheme tht replces the coefficient y identity [4].) Thus, to determine whether coding is necessry on n outgoing link of merging node, we need to verify whether we cn constrin the output on the link to depend on single input without destroying the chievility of the given rte. Consider merging node with k( 2) incoming links nd l( 1) outgoing links. For ech i {1,..., k} nd ech j {1,..., l}, weset ij =1if the input from incoming link i contriutes to the linerly coded output on outgoing link j, nd ij =0otherwise; we cll these the ctive nd inctive sttes, respectively. Network coding is required over link j only if two or more link sttes re ctive. Thus, it is useful to think of j =( ij ) i {1,...,k} s lock of length k (see Fig. 2 for n exmple). x x x y v y () Merging node v x x x y v lock for y x x x v y = = lock for y () Two locks for outgoing links Fig. 2. Node v with 3 incoming nd 2 outgoing links hs inputs descried y vectors 1 =( 11, 21, 31 ) nd 2 =( 12, 22, 32 ). As in network C of Exmple 1, whether node v must code over link y j vries depending on which other nodes re coding. Thus deciding which nodes should code in generl involves selection out of exponentilly mny possile choices. We employ GA-sed serch method to efficiently ddress the lrge nd exponentilly scling size of the spce. B. Relted Work Frgouli et l. [7] show tht coding is required t no more thn (d 1) nodes in cyclic networks with 2 unit-rte sources nd d sinks. This result, however, is not esily generlized to more thn 2 sources. They lso present n lgorithm to construct miniml sutree grph. For trget rte R, they first select sugrph consisting of R link-disjoint pths to ech of d sinks nd then construct the corresponding leled line grph in which they sequentilly remove the links whose removl does not ffect the chievle rte. Lngerg et l. [4] derive n upper ound on the numer of required coding nodes for oth cyclic nd cyclic networks. They give n lgorithm to construct network code tht chieves the ounds, where the network is first trnsformed such tht ech node hs degree t most 3 nd ech of the links is sequentilly exmined nd removed if the trget rte is still chievle without it. Both of the ove pproches remove links sequentilly in greedy fshion, ssuming tht network coding is done t ll nodes with multiple incoming links in the remining grph. Note tht, unless good order of the link trversl is found, the qulity of the solution cnnot e much improved s illustrted in [5]. Bhttd et l. [8] give liner progrmming formultions for the prolems of optimizing over vrious resources used for network coding, sed on model llowing continuous flows. Their optiml formultions, however, involve numer of vriles nd constrints tht grows exponentilly with the 1992 Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloded on Octoer 13, 2008 t 15:22 from IEEE Xplore. Restrictions pply.

3 This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2007 proceedings. numer of sinks, which mkes it hrd to pply the formultions to the cse of lrge numer of sinks, even t the price of scrificed optimlity. We conclude this section with rief introduction to GA. C. A Brief Introduction to GA GAs [9] operte on set of cndidte solutions, clled popultion. Ech solution is typiclly represented y it string, clled chromosome. Ech chromosome is ssigned fitness vlue tht mesures how well the chromosome solves the prolem t hnd, compred with other chromosomes in the popultion. From the current popultion, new popultion is generted typiclly using three genetic opertors: selection, crossover nd muttion. Chromosomes for the new popultion re selected rndomly (with replcement) in such wy tht fitter chromosomes re selected with higher proility. For crossover, survived chromosomes re rndomly pired, nd then two chromosomes in ech pir exchnge suset of their it strings to crete two offspring. Chromosomes re then suject to muttion, which refers to rndom flips of the its pplied individully to ech of the new chromosomes. The process of evlution, selection, crossover nd muttion forms one genertion in the execution of GA. The ove process is iterted with the newly generted popultion successively replcing the current one. The GA termintes when certin stopping criterion is reched, e.g., fter predefined numer of genertions. GAs hve een pplied to lrge numer of scientific nd engineering prolems, including mny comintoril optimiztion prolems in networks (e.g., [10], [11]). There re severl spects of our prolem suggesting tht GA-sed method my e promising cndidte: GA hs proven to work well if the spce to e serched is lrge, ut known not to e perfectly smooth or unimodl, or even if the spce is not well understood [9] (which mkes trditionl optimiztion methods difficult to pply). Note tht the serch spce of our prolem is pprently not smooth or unimodl with respect to the numer of coding links nd the structure of the spce consisting of the fesile inry vectors is not well understood. Since the prolem is NP-hrd, it is not criticl tht the clculted solution my not e glol optimum. Note lso tht, while it is hrd to chrcterize the structure of the serch spce, once provided with solution we cn verify its fesiility nd count the numer of coding links therein in polynomil time. Thus, if the use of genetic opertions cn suitly limit the size of the spce to e ctully serched, solution cn e otined firly efficiently. III. CENTRALIZED APPROACH We first present the centrlized version of the lgorithm, whose overll structure, sed on simple GA [9], is shown in Fig. 3. Sections III-A nd III-B present two different methods for mpping the network coding prolem to GA frmework (procedure [C1] in Fig. 3) nd evluting the chromosomes ([C3, C8] in Fig. 3). Either of the two methods cn e comined with the computtionl prt of the lgorithm (remining procedures in Fig. 3) which is descried in Section III-C. [C1] preliminry processing; [C2] initilize popultion; [C3] evlute popultion; [C4] while termintion criterion not reched { [C5] select solutions for next popultion; [C6] perform crossover; [C7] perform muttion; [C8] evlute popultion; } [C9] perform greedy sweep; Fig. 3. A. Algeric Method Flow of Centrlized Algorithm We first descrie the lgeric method y which choice of coding links is mpped to GA prolem nd given cndidte solution (chromosome) is evluted. Owing to spce limittions, here we present only the min concepts; the reder is referred to [5] for detils. This lgeric method will lso e used lter in the distriuted version of the lgorithm. This method pplies only to cyclic networks; for cyclic networks, it cn e very inefficient s discussed in Section III-B. Given n cyclic grph G = (V,E), we first construct the corresponding leled line grph G = (V,E ) [12], where ech node in V represents link in E nd ech link (v,w ) E implies tht the links e, f E corresponding to nodes v,w V, respectively, re connected in G vi some node u V such tht u = hed(e) =til(f). To construct network code, we ssign coefficient to ech link in G s in [12]. Note tht there is one-to-one correspondence etween the inry vriles ij introduced in Section II-A nd the coefficients ssigned to the incoming links to node with multiple incoming links in G. Thus, for ech inry vrile ij we cn consider the ssocited coefficient. If there re m such coefficients in G, chromosome is represented y vector consisting of m inry vriles; if we denote y d v in nd d v out the in-degree nd the out-degree of node v V, m is given y m = v V dv in dv out, where V V is the set of ll merging nodes in G. To evlute given chromosome, we first verify its fesiility. If ij =0in the chromosome, then input x i is inctive with respect to output y j nd we set ssocited coefficient to zero. If ij =1, then we let the ssocited coefficient e n indeterminte nonzero vlue. To determine whether the trget rte R is chievle, we rely on rndom liner coding; i.e., to ech of the remining coefficients we ssign rndom element from finite field nd check whether the system mtrix is nonsingulr. Note tht this fesiility test entils ounded error, which is shown in [5] not to e criticl since the error is one-sided, i.e., fesile chromosome my mistkenly e declred infesile ut not vice vers, nd we cn lower the ound on the error proility s much s we desire t n dditionl cost of computtion Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloded on Octoer 13, 2008 t 15:22 from IEEE Xplore. Restrictions pply.

4 This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2007 proceedings. We then define the fitness vlue F of chromosome z s { numer of coding links, if z is fesile, F (z) = (1), if z is infesile, where the numer of coding links cn e esily clculted y counting the numer of locks in the chromosome with t lest two 1 s. It is not hrd to verify tht the computtionl complexity required to evlute single chromosome is O(d ( E R 3 )). B. Grph Decomposition Method Note tht the ove lgeric method dels explicitly with the sclr coefficients tht pper in the system mtrix ssuming tht the network opertes with zero dely (nd thus the network is cycle-free). In the presence of cycles, dely must e tken into ccount, hence the system mtrix ecomes mtrix over the polynomil ring with coefficients tht re rtionl functions in the dely vrile D [12]. In this cse, the mtrix computtion involves clculting the coefficient for ech power of the dely vrile D, which in generl renders the fesiility test prohiitively inefficient. In this susection, we show tht, with n pproprite grph decomposition, the evlution cn e done y clculting the mx-flows etween the source nd the sinks. Note tht the minimum of those mx-flows equls the mximum chievle multicst rte regrdless of the existence of cycles in the network [1]. Unlike the lgeric one, this modified method opertes on the ctul grph G rther thn the leled line grph. In the first stge of the lgorithm ([C1] in Fig. 3), we decompose ech merging node tht is not sink s follows. (For sink node with nonzero out-degree, introduce virtul sink connected vi R links nd decompose the originl sink.) Consider merging node v with d in ( 2) incoming links nd d out outgoing links (see Fig. 4). We introduce d in nodes u 1,..., u din, which we cll incoming uxiliry nodes, nd redirect the i-th (1 i d in ) incoming link of node v to node u i. Similrly, we crete d out nodes w 1,..., w dout, which we cll outgoing uxiliry nodes, nd let the j-th (1 j d out ) outgoing link of node v e the only outgoing link of node w j. We then insert link (u i,w j ) etween ech pir of nodes u i nd w j (1 i d in, 1 j d out ). v () Before decomposition u u u w w () After decomposition Fig. 4. Decomposition of node with d in =3nd d out =2. With ech of the newly introduced links etween the uxiliry nodes, we ssocite inry vrile in chromosome. Since ech of those new links corresponds to pir of connected incoming nd outgoing links in the originl grph, we cn see one-to-one correspondence etween the inry vriles here nd those introduced in the lgeric method. To verify the fesiility of given chromosome z, we first delete ll the links ssocited with 0 in z nd then compute the mx-flow etween the source nd ech of the d sinks. If n outgoing uxiliry node hs only one incoming link, we cn replce the incoming link, together with its outgoing link, y single link without chnging ny of the mxflows. If n uxiliry outgoing node hs no incoming link, we simply delete the node. If ll d mx-flows re t lest R, rte R is chievle with network coding only t the outgoing uxiliry nodes with two or more incoming links. The numer of remining outgoing uxiliry nodes equls the numer of coding links in the originl grph. Hence, y counting the numer of such outgoing uxiliry nodes, we cn clculte the fitness vlue F of chromosome z, defined s in (1). The mx-flow sed evlution of single chromosome requires O(d E 2 V ) time [13], where E nd V re the numer of links nd nodes, respectively, in the decomposed grph. Note tht, unlike the lgeric method, this fesiility test incurs no error. Since it works oth with nd without cycles, the grph decomposition method my e preferle to the lgeric method when centrlized opertion is fesile. Centrl opertion requires tht the topology of the whole network e known to centrl computing node nd my e slow. This pproch my e pproprite, for exmple, in the plnning stge of network. However, the lgeric method plys crucil role in the distriuted version of the lgorithm, s shown in Section IV. C. Computtionl Prt of GA In descriing the computtionl prt of the GA, we first discuss two novel elements designed specificlly for the prolem nd then descrie other typicl GA components. The discussion pplies for oth the lgeric nd grph decomposition methods. 1) Block-Wise Representtion nd Opertors: A lock of length k is defined to e suset of chromosome consisting of k inry vriles tht indicte the link sttes for the trnsmission onto prticulr outgoing link fed y k incoming links, i.e., the k components of vector j =( ij ) i {1,...,k} introduced in Section II-A. We my llow for length-k lock to tke ll possile 2 k strings s in [5], which we refer to s it-wise representtion 1. Note, however, tht once lock hs t lest two 1 s, replcing ll the remining 0 s with 1 hs no effect on whether coding is done nd tht sustituting 0 with 1, s opposed to sustituting 1 with 0, does not hurt the fesiility. Therefore, for fesile chromosome, ny lock with two or more 1 s cn e treted the sme s the lock with ll 1 s. 1 In the GA community, the method for representing cndidte solution s it string is clled genotype encoding; we void the use of this term to minimize confusion with the term encoding in the context of network coding Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloded on Octoer 13, 2008 t 15:22 from IEEE Xplore. Restrictions pply.

5 This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2007 proceedings. The ove oservtion leds to lock-wise representtion, where ech lock of length k is llowed to tke one of the following (k+2) strings: [ ] (1 string for no trnsmission stte), [ ], [ ], [ ],..., [ ] (k strings for uncoded trnsmission stte), [ ] (1 string for coded trnsmission stte). If we let w e the totl numer of locks (i.e., w = v V dv out) nd k i denote the length of the i- th lock (i = 1,..., w), the serch spce size is reduced to w i=1 (k w i +2), from 2 i=1 ki in the cse of it-wise representtion. To preserve the structure of lock-wise representtion, we need set of new genetic opertors. In [5], uniform crossover [9] nd inry muttion [9] re used, which for comprison we refer to s it-wise opertors 2. Let us now define lock-wise opertors s follows. For lock-wise uniform crossover, we let ech pir of chromosomes suject to crossover exchnge ech full lock, rther thn ech individul it, independently. For lock-wise muttion, we let ech lock of length k suject to muttion tke nother string chosen uniformly t rndom out of the other (k +1) llowed strings. It is interesting to note tht the enefit of the smller serch spce size in fct comes t the price of losing the informtion on the locks with prtilly ctive link sttes tht my serve s intermedite steps towrd n uncoded trnsmission stte. Also, wheres the verge numer of its flipped y lockwise muttion of length-k lock using muttion rte α 4k is 2 (k+1)(k+2) α, which is smller thn tht y the it-wise muttion (kα), the proility tht 2 or more its re flipped is often much lrger for lock-wise muttion; this my negtively ffect the GA s ility to improve the solution through fine rndom chnges. Hence, the overll effect of lock-wise representtion nd opertors on the lgorithm s performnce is not esy to predict theoreticlly. Section V includes n experimentl evlution of this question. 2) Greedy Sweep: We introduce nother novel opertor, referred to s greedy sweep, where we inspect the est chromosome otined t the end of the itertion nd switch ech of the remining 1 s to 0 if it cn e done without violting fesiility ([C8] in Fig. 3). This procedure cn only improve the solution, nd sometimes the improvement is sustntil. Moreover, if we denote y z the chromosome fter the greedy sweep, then z gives n upper ound on the numer of coding links the sme s in [4]. Lemm 1: The numer of coding links ssocited with z is upper ounded y R 3 d 2 for n cyclic network nd (2B + 1)R 3 d 2 for cyclic network, where B is the minimum numer of links tht must e removed from the network in order to eliminte cycles. Proof : Let us consider the grph decomposition method. Here, switching 1 to 0 in chromosome implies tht we delete the ssocited link in the decomposed grph. In the decomposed grph ssocited with z, there is no link etween uxiliry 2 For uniform crossover, pir of chromosomes exchnges ech it independently with given proility, nd for inry muttion, ech it of chromosome is flipped independently with given proility. links tht cn e removed without violting the chievility. One cn esily verify tht there is one-to-one correspondence etween the links etween uxiliry nodes in our decomposed grph (see Fig. 4) nd the set of the pths within the gdget Γ v (see Fig. 2 in [4]). Now we cn replce ll non-merging nodes with degree lrger thn 3 y the gdgets nd greedily remove links in the sme wy s ove, which, however, is irrelevnt of the numer of coding links. Therefore, from z we cn construct simple instnce, s defined in [4], nd it gives the desired upper ounds on the numer of coding links (Lemm 14 in [4]). Lemm 1 provides our lgorithm with gurntee on its performnce which is t lest no worse thn tht of the lgorithm in [4]. 3) Typicl GA Components: When initilizing the popultion ([C2] in Fig. 3), we rndomly generte ech lock nd insert n ll-one vector, whose effect s fesile strting point is crucil s discussed in [5]. The itertion is terminted if the genertion numer reches the predefined limit ([C4] in Fig. 3). For selection ([C5] in Fig. 3), we employ tournment selection [9], where we repet tournment etween predefined numer of rndomly selected chromosomes out of which the est one is selected (with replcement) for the next genertion. IV. DISTRIBUTED APPROACH Noting tht the min dvntge of network coding sed multicst is tht n efficient cpcity-chieving code cn e constructed in distriuted fshion [6], [14], motivtion for decentrliztion of the lgorithm ecomes pprent. Tht is, given tht the ctul multicst cn proceed in decentrlized mnner, n lgorithm used for resource optimiztion should e more desirle if it does not require centrlized computtion. Moreover, s will e discussed elow, such decentrliztion enles the most time consuming tsk, fitness evlution, to e distriuted over the network such tht the computtionl complexity required t ech node depends only on locl prmeters. The size of the popultion often serves s n importnt fctor for the ility of GA to find good solution [15]. Though it is not n esy tsk to predict the ccurte popultion size required for specific prolem, it is lwys desirle to devise n evlution method with low complexity, which llows for flexiility in dopting lrgesized popultion when needed. In this section, we present novel distriuted frmework for our evolutionry lgorithm, in which the fesiility test is done loclly t ech sink while the intermedite nodes ctully construct rndom liner codes. With limited mount of feedck informtion from the sinks nd the merging nodes, fitness evlution cn e done with sustntilly lower complexity. Furthermore, the popultion cn e mnged in distriuted mnner such tht ech merging node loclly mnges suset of the popultion tht determines the locl opertions t tht node (see Fig. 5). Also, it will e shown tht, with some mount of coordintion, ll genetic opertions cn e done loclly t individul merging nodes Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloded on Octoer 13, 2008 t 15:22 from IEEE Xplore. Restrictions pply.

6 This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2007 proceedings. Ech lock indictes trnsmission stte of n outgoing link : : : : : : : : : : : : : : :... Ech set of locks determines locl opertions t node, thus cn e mnged loclly t tht node. Fig Structure of Popultion N Chromosomes In ddition to computtionl efficiency, this distriuted pproch hs n importnt enefit tht the coding resource optimiztion cn e done on the fly while network is opertionl, llowing for the following network coding protocol: As the source node sends n optimize signl, ll the nodes prticipting in the multicst go into the optimiztion mode, nd s the distriuted evolutionry lgorithm proceeds, the links/nodes where coding is not required re identified. When the source node sends trnsmit signl, the network strts to multicst dt sed on the est network code found, in which coding is done only t the required links/nodes. Since the distriuted version of the lgorithm is sed on the lgeric method descried in Section III-A, we need to tke different pproches depending on whether the network hs cycles or not. In the following, we egin to descrie the detils of the distriuted pproch ssuming tht the network is cyclic, nd lter in this section we extend the pproch to cyclic networks, highlighting the chnges to e mde. A. Assumptions nd Preliminries We ssume tht ech link cn trnsmit one pcket of fixed size per time unit in the given direction. Ech link is lso ssumed to e le to send some mount of feedck dt, typiclly much smller thn the pcket size, in the reverse direction. Also, we ssume tht ech interior node opertes in urst-oriented mode; i.e., for the forwrd (ckwrd) evlution phse, ech node strts updting its output only fter n updted input hs een received from ll incoming (outgoing) links. The overll structure of our distriuted lgorithm is shown in Fig. 6 with the loctions of ech procedure specified. We now proceed to descrie the detiled procedures of the lgorithm in the order of their occurrences. B. Detils of Algorithm 1) Preliminry Processing [D1]: The source initites the lgorithm y trnsmitting the optimize signl contining the following predetermined prmeters: trget multicst rte R, popultion size N, the size q of the finite field to e used, crossover proility, nd muttion rte. Ech prticipting node tht hs received the signl psses the signl to its downstrem nodes. [D1] preliminry processing; (ll nodes) [D2] initilize popultion; (merging nodes) [D3] run forwrd evlution phse; (ll nodes) [D4] run ckwrd evlution phse; (ll nodes) [D5] clculte fitness; (source) [D6] while termintion criterion not reched (source) { [D7] clculte coordintion vector; (source) [D8] run forwrd evlution phse; (ll nodes) [D9] perform selection, crossover, muttion; (merging nodes) [D10] run ckwrd evlution phse; (ll nodes) [D11] clculte fitness; (source) } [D12] perform greedy sweep; (ll nodes) Fig. 6. Flow of Distriuted Algorithm 2) Popultion Initiliztion [D2]: Let us consider merging node with d in ( 2) incoming links. For ech of its d out outgoing links, the node hs to mnge inry vector of length d in, which we refer to s coding vector, to indicte the link sttes for single chromosome. Hence, for the popultion of size N, the node must hve N d out coding vectors to determine the opertions t tht node. To initilize this suset of the popultion, the merging node rndomly genertes N d in d out inry numers nd set ll the components to 1 for the coding vectors tht correspond to the first of the N chromosomes. 3) Forwrd Evlution Phse [D3, D8]: For the fesiility test of chromosome, ech node trnsmits vector consisting of R components, which we refer to s pilot vector. Ech of its the components is from the finite field F q nd the i- th component represents the coefficient used to encode the i-th source dt. We ssume tht set of N pilot vectors is trnsmitted together y single pcket. The source initites the forwrd evlution phse y sending out on ech of its outgoing links set of N rndom pilot vectors. Ech non-merging node simply forwrds ll the pilot vectors received from its incoming link to ll its outgoing links. Ech merging node trnsmits on ech of its outgoing links rndom liner comintion of the received pilot vectors, computed sed on the node s coding vectors s follows. Let us consider prticulr outgoing link nd denote the ssocited d in coding vectors y v 1, v 2,..., v din.forthei-th (1 i N) output pilot vector u i, we denote the i-th input pilot vectors received form the incoming links y w 1, w 2,..., w din. Define the set J of indices s Then, J = {1 j d in the i-th component of v j is 1}. (2) u i = j J w j rnd(f q ), (3) where rnd(f q ) denotes rndom element from F q.iftheset J is empty, u i is ssumed to e zero. 4) Bckwrd Evlution Phse [D4, D10]: To clculte chromosome s fitness vlue, two kinds of informtion need to e gthered: 1) whether ech sink cn decode dt of rte R 1996 Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloded on Octoer 13, 2008 t 15:22 from IEEE Xplore. Restrictions pply.

7 This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2007 proceedings. nd 2) how mny links re used for coding t ech merging node. Ech sink cn determine whether dt of rte R is decodle for ech of the N chromosomes y computing the rnk of the collection of received pilot vectors. It is worth to point out tht this is the sme lgeric evlution method descried in Section III-A, with the difference tht, rther thn computing the system mtrix with rndomized elements centrlly, we now ctully construct rndom liner codes over the network in decentrlized fshion. Hence, this fesiility test lso ers the sme, ut uncriticl, possiility of errors s in the centrlized cse. Regrding the numer of coding links, ech merging node cn simply count the numer links where coding is required y inspecting its coding vectors used in the forwrd evlution phse. For the feedck of this informtion, ech node trnsmits vector consisting of N components, which is referred to s fitness vector. Ech of the components must e t lest log( E +2) its long since for ech chromosome the numer of coding links cn rnge from zero to E nd n dditionl symol (infinity) is needed to signify infesiility. The ckwrd evlution phse proceeds s follows: After the fesiility tests of the N chromosomes re done, ech sink genertes fitness vector whose i-th (1 i N) component is zero if the i-th chromosome is fesile t the sink, nd infinity otherwise. Ech sink then initites the ckwrd evlution phse y trnsmitting its fitness vector to ll of its prents. Ech interior node clcultes its own fitness vector whose i-th (1 i N) component is the numer of coding links t the node for the i-th chromosome plus the sum of ll the i-th components of the received fitness vectors. Ech node then trnsmits the clculted fitness vector to only one of its prents, nd n ll-zero fitness vector (for just signling) to the other prent nodes. Note tht, since the network is ssumed to e cyclic, ech coding link of chromosome contriutes exctly once to the corresponding component of the source node s fitness vector, nd thus the ove updte procedure provides the source with the correct totl numer of coding links. 5) Fitness Clcultion [D5, D11]: The source clcultes the fitness vlues of N chromosomes simply y performing component-wise summtion of the received fitness vectors. Note tht if n infinity were generted y ny of the sinks, it should dominte the summtions ll the wy up to the source, nd thus the source cn clculte the correct fitness vlue for the infesile chromosome. 6) Termintion Criterion [D6]: The source cn determine when to terminte the optimiztion y counting the numer of genertions iterted thus fr. 7) Coordintion Vector Clcultion [D7]: Since the popultion is divided into susets tht re mnged t the merging nodes, genetic opertions lso need to e done loclly t the merging nodes. However, some mount of coordintion is required for consistent genetic opertions throughout ll the merging nodes; more specificlly, for 1) consistent selection of chromosomes, 2) consistent pring of chromosomes for crossover, nd 3) consistent decision on whether ech pir is suject to crossover. This informtion is crried y coordintion vector, clculted t the source, consisting of the indices of selected chromosomes tht re rndomly pired nd 1-it dt for ech pir indicting whether the pir needs to e crossed over. The coordintion vector is trnsmitted together with the pilot vectors in the next forwrd evlution phse. 8) Genetic Opertions [D8]: Bsed on the received coordintion vector, ech merging node cn loclly perform genetic opertions nd renew its portion of the popultion s follows: For selection, ech node only retins the coding vectors tht correspond to the indices of selected chromosomes. For lock-wise crossover, ech node independently determines whether ech lock is crossed over. Since no lock is shred y multiple merging nodes, this cn e done independently t ech merging node. For lock-wise muttion, ech node independently determines whether ech lock is mutted without ny coordintion with other nodes. 9) Greedy Sweep [D12]: Greedy sweep requires n dditionl protocol where the source is notified of the merging nodes with t lest one coding link in the est solution otined t the end of the itertion. Then, for ech of such merging nodes, the source sends out pcket to test if uncoded trnsmission is possile on the link(s) where currently coding is required. Since this dditionl protocol requires more extensive coordintion etween nodes, we my leve this procedure optionl, whose detiled description is omitted owing to spce limittions. Note, however, tht in our experiments the solutions otined with the lock-wise representtion/opertions re lredy good enough so tht further improvement y greedy sweep hs never een oserved. Nevertheless, greedy sweep my e useful s sfegurd tht prevents the lgorithm s poor performnce due to misdjusted prmeters, e.g., too smll popultion size. C. Complexity For evlution of single chromosome, ech merging node v computes rndom liner comintions of inputs in the forwrd evlution phse, which requires O(d v in dv out R), nd ech non-merging node w simply forwrds the received dt, which requires O(d w out). Fesiility test t ech sink t is done y clculting the rnk of d t in R mtrix, where we ssume d t in R, hence it requires O(dt in2 R). In the ckwrd evlution phse, updte of fitness vector tkes O(d v in + dv out). Therefore, the computtionl complexity required for evlution of single chromosome is O( v V dv in dv outr + w V \V dw out + t T dt in2 R), which cn e sustntilly less thn tht for the centrlized version of the lgorithm. D. Networks with Cycles Cycles cn e delt with in two different wys s in other network coding prolems. First, we cn select sugrph tht does not contin directed cycle, sed on which we proceed 1997 Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloded on Octoer 13, 2008 t 15:22 from IEEE Xplore. Restrictions pply.

8 This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2007 proceedings. to code construction nd decoding in essentilly the sme mnner s in the cyclic cse [16], [17]. Alterntively, we my directly pply coding over cycles y comining informtion from possily different time periods t intermedite nodes nd deploying memory t the receivers for decoding [12], [18], [19], where the network code cn e considered essentilly convolutionl code. The former of the ove two scenrios llows for simple coding nd decoding, ut it my necessitte coding t the links/nodes where coding is not necessry if some link connections were not removed in the erlier stge [5]. On the other hnd, the ltter scenrio my llow us to explore the full-fledged trdeoff etween coding nd cpcity, ut oth specifying nd decoding the code re more complex thn in the former cse. Here we focus on the first scenrio nd descrie how our distriuted lgorithm cn e incorported in the whole frmework of such network coding schemes. Note tht, if the originl coding scheme is designed to operte on n cyclic sugrph selected eforehnd, which seems more prcticle, there is no reson to employ more complex network codes sed on the originl cyclic grph to minimize coding resources. However, we expect tht similr pproch cn e redily pplied to the convolutionl network coding scenrio with n pproprite cycle-voding mechnism for the trnsmission of the control messges such s the feedck informtion. To set up n cyclic set of connections on given network, we use the distriuted lgorithm in [17], where inry vrile is ssigned to ech pir (l, l ) of incident links indicting tht the connection from link l to link l is llowed or not. The vlue of ech inry vrile is determined such tht the trnsmission long directed cycle is prohiited. It is interesting to note tht those inry vriles used for sugrph selection re ssigned to the link coefficients the sme wy s in our lgorithm. Hence, our lgorithm cn e incorported into the whole frmework s follows: i) Use the lgorithm in [17] to select the set of link coefficients to e used for trnsmission. ii) Ech node then exchnges the inry vriles ssigned to its links with its neighors so tht ech node cn identify the llowed connections. iii) We then pply the our distriuted lgorithm ignoring the link coefficients tht correspond to the disllowed connections. Alterntively, if minimizing the link cost is our primry concern, we my use the lgorithms tht find the minimum cost sugrph in decentrlized fshion, such s the one in [14]. The resulting sugrph does not contin directed cycle if the link costs re ll positive. Hence, two-stge method is possile where the minimum cost sugrph selection is followed y our distriuted lgorithm. This two-stge method my perform very well in prctice, s will e demonstrted in the next section. V. EXPERIMENTAL RESULTS The prmeters used for the experiments re s follows: Popultion size is 150 nd the itertion termintes fter 1000 genertions. Tournment size (for selection) nd muttion rte re 100 nd 0.012, respectively, for the lock-wise cse, nd 10 nd for the it-wise cse. Crossover rte is fixed t 0.8. A. Effects of Block-Wise Representtion nd Opertors We evlute the performnce of our lgorithm with the lock-wise nd the it-wise representtions nd opertors, using the centrlized version of the lgorithm with the grph decomposition method. The experiments re sed on the two topologies generted y the lgorithm in [20] with the following prmeters: (50 nodes, 87 links, 10 sinks, rte 5) nd (75 nodes, 156 links, 15 sinks, rte 7). For comprison, we lso perform experiments with two existing greedy pproches y Frgouli et l. [7] ( Miniml 1 ) nd Lngerg et l. [4] ( Miniml 2 ). For oth pproches, link removl is done in rndom order. For Miniml 1, the sugrph is selected lso in greedy fshion y sequentilly removing links. Tle I shows the est nd the verge vlues, s well s stndrd vrition, otined in 30 trils. (50,87,10,5) (75,156,15,7) Best Avg. Std. Best Avg. Std. Block-wise Bit-wise Miniml Miniml TABLE I NUMBER OF REQUIRED CODING LINKS Tel I shows tht the lock-wise representtion nd opertors clerly outperform the it-wise counterprt in ll spects. We cn lso oserve tht the performnce of our lgorithm, with either of the two representtion nd opertors, is t lest s good nd often etter thn tht of oth Miniml 1 nd Miniml 2, except only in the est vlue of the itwise cse for the lrger network. More in-depth comprisons etween these lgorithms cn e found in our susequent pper [21], where our lgorithm with the lock-wise representtion nd opertors is found to exhiit fr greter performnce dvntge over the other three cses. B. Performnce of Distriuted Algorithm Since the distriuted lgorithm shres the sme computtionl prt of GA with the centrlized one, the two lgorithms show the sme performnce in terms of solution qulity. However, s shown in Section IV-C, the computtionl complexity required y the distriuted lgorithm depends on locl topologicl prmeters, nd this cn often led to significnt gin in terms of the running time. To compre the running times of the two pproches, we generte set of highly connected topologies such tht there exists link etween ech pir of nodes i nd j (i < j), where the source is node 1 nd the sinks re the lst 10 nodes. This test is pessimistic in the sense tht the distriuted 1998 Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloded on Octoer 13, 2008 t 15:22 from IEEE Xplore. Restrictions pply.

9 This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2007 proceedings. lgorithm is simulted on single mchine while ech node s function is performed y seprte thred, thus it cnnot enefit from the multi-processing gin wheres it only suffers from dditionl computtionl urdens for mnging numer of threds. Tle II shows tht, nevertheless, the distriuted lgorithm exhiits n dvntge in running time s the size of the network grows. Numer of nodes Centrlized Distriuted TABLE II ELAPSED TIME PER GENERATION (SECONDS) C. Effectiveness of Two-Stge Method We introduced in Section IV-D two-stge method where we first select minimum-cost sugrph ssuming network coding is done everywhere nd then pply our evolutionry lgorithm to the resulting sugrph. Though not optiml, this two-stge method cn e very useful when optimiztion over oth link cost nd coding cost is required; the minimum link cost is gurnteed, nd the resulting cyclic sugrph cn often e sustntilly smller thn the originl network. We test the two-stge method on ISP 1755 nd 3967 topologies from the Rocketfuel project [22], using the lgorithm in [14] to otin minimum cost sugrph. With 10 rndomly selected sinks nd trget rtes 2, 3, nd 4 on oth topologies, ech of 30 runs lwys ends up with zero coding links. These results my suggest tht, while ssuming network coding enles to clculte minimum-cost sugrph, there my e very few links/nodes where network coding is ctully required in the end. VI. CONCLUSIONS AND FUTURE WORK We hve presented evolutionry pproches to minimizing the resources used for network coding in single multicst scenrio. The proposed lgorithms hve een shown to hve dvntges over our previously proposed lgorithm [5], s well s other existing greedy lgorithms, in terms of the pplicility to generl topologies, the solution qulity, nd the prcticility in distriuted environment. For future reserch, we my further improve the distriuted lgorithm, y smrter mngement of popultion nd pcket trnsmissions, such tht it converges fster to etter solution nd works synchronously, providing roustness ginst dely, filure, or topologicl chnges in the network. Also, we could oserve trdeoff etween coding nd link usge in the sense tht in some networks [5], reducing link usge first y sugrph selection my increse coding in the remining sugrph, which is not the cse in our experiments in Section V-C. Hence, whether there exists such trdeoff cn e considered topologicl property of network nd thus the effectiveness of the two-stge method discussed in Section IV-D my depend on the network topology. We my further investigte this trdeoff using evolutionry lgorithms designed for multi-ojective optimiztion. ACKNOWLEDGMENT The first uthor would like to thnk Fng Zho for her help on the experiments. REFERENCES [1] R. Ahlswede, N. Ci, S.-Y. R. Li, nd R. W. Yeung, Network informtion flow, IEEE Trns. Inf. Theory, vol. 46, no. 4, pp , [2] S.-Y. R. Li, R. W. Yeung, nd N. Ci, Liner network coding, IEEE Trns. Inf. Theory, vol. 49, no. 2, pp , [3] M. B. Richey nd R. G. Prker, On multiple steiner sugrph prolems, Networks, vol. 16, no. 4, pp , [4] M. Lngerg, A. Sprintson, nd J. Bruck, The encoding complexity of network coding, IEEE Trns. Inf. Theory, vol. 52, no. 6, pp , [5] M. Kim, C. W. Ahn, M. Médrd, nd M. Effros, On minimizing network coding resources: An evolutionry pproch, in Proc. NetCod, [6] T. Ho, R. Koetter, M. Médrd, D. R. Krger, nd M. Effros, The enefits of coding over routing in rndomized setting, in Proc. IEEE ISIT, [7] C. Frgouli nd E. 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Kim, Genetic representtions for evolutionry optimiztion of network coding resources, in Applictions of Evolutionry Computing, [22] N. Spring, R. Mhjn, nd D. Wetherll, Mesuring ISP topologies with rocketfuel, in Proc. ACM/SIGCOMM 02, 2002, pp Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloded on Octoer 13, 2008 t 15:22 from IEEE Xplore. Restrictions pply.