An evolutionry pproch to inter-session network coding The MIT Fculty hs mde this rticle openly ville. Plese shre how this ccess enefits you. Your story mtters. Cittion As Pulished Pulisher Kim, M. et l. An Evolutionry Approch To Inter-Session Network Coding. INFOCOM 2009, IEEE. 2009. 450-458. 2009 IEEE http://dx.doi.org/10.1109/infcom.2009.5061950 Version Finl pulished version Accessed Fri Jn 26 23:47:07 EST 2018 Citle Link Terms of Use Detiled Terms http://hdl.hndle.net/1721.1/54694 Article is mde ville in ccordnce with the pulisher's policy nd my e suject to US copyright lw. Plese refer to the pulisher's site for terms of use.
This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2009 proceedings. An Evolutionry Approch To Inter-Session Network Coding Minkyu Kim, Muriel Médrd, Un-My O Reilly, Dnil Trskov Astrct Wheres the theory nd ppliction of optiml network coding re well studied for the single-session multicst scenrio, there is no known optiml network coding strtegy for more generl connection prolem where there re more thn one session nd receivers my demnd different sets of informtion. Though there hve een numer of recent studies tht demonstrte vrious utilities of network coding in the multisession scenrio, they rely on very restricted clsses of codes in terms of the coding opertions llowed nd/or the loction of decoding. In this pper, we propose novel inter-session network coding strtegy for generl connection prolem. Our coding strtegy llows firly generl rndom liner coding over lrge finite field, in which decoding is done t receivers nd the mixture of informtion t interior nodes is controlled y evolutionry mechnisms. We demonstrte how our coding strtegy my surpss existing end-to-end pirwise XOR coding schemes in terms of effectiveness nd prcticlity. I. INTRODUCTION In the multicst scenrio with single session, the theory of network coding is well founded on the fmous theorem y Ahlswede et l. [1] tht chrcterizes the cpcity region y the mx-flow (min-cut) ounds. Susequently, it is shown tht the optiml cpcity cn e chieved using only sclr liner network codes [2]. However, when we generlize the prolem such tht there re more thn one session nd receivers my demnd different sets of informtion, finding the optiml network coding strtegy is still n open question. First of ll, in such generlized prolem, chrcterizing the cpcity region ecomes prohiitively difficult nd even its inner/outer ounds cnnot e computed in prctice [3] [6]. Moreover, liner coding is shown to e insufficient for optiml coding in the multi-session cse [7]. A grph theoretic pproch is proposed s systemtic method for deciding solvility of given network with either liner or nonliner codes [8], whose sclility issue, however, is unresolved. Even within liner codes, the solvility decision prolem is shown to involve Gröner sis computtion [9], whose complexity my prohiit prcticl implementtions for lrge prolems. Recently, [10] suggests tht solving generl network coding prolem is equivlent in terms of complexity to solving set of polynomil equtions. M. Kim nd M. Médrd re with the Lortory of Informtion nd Decision Systems, Msschusetts Institute of Technology, Cmridge, MA 02139, USA ({minkyu, medrd}@mit.edu). U.-M. O Reilly is with the Computer Science nd Artificil Intelligence Lortory, Msschusetts Institute of Technology, Cmridge, MA 02139, USA (unmy@csil.mit.edu). D. Trskov is with the Institute for Communictions Engineering, Technicl University Munich, Munich, Germny (dnil.trskov@tum.de). In short, it is very unlikely tht network coding strtegy tht is theoreticlly optiml will soon emerge. Nevertheless, there hve een numer of studies tht demonstrte vrious utilities of network coding in the multi-session scenrio, sed on some restricted clsses of codes in terms of the coding opertions llowed nd/or the loction of decoding. Ktti et l. s opportunistic coding [11] leds to restricted, yet very prcticl coding scheme, tking dvntge of the rodcst nture of wireless medium. In their coding scheme, multiple, possily more thn two, pckets cn e comined using inry XOR, ut decoding needs to e done immeditely t the neighors of the coding node. If we wish to perform decoding t receivers, we my need to put n even stronger restriction: XOR opertions re llowed only etween two flows, thus clled pirwise XOR coding. The simplicity of coding nd decoding opertions of pirwise XOR codes llows the code construction prolem to e descried s flow formultions, which thus cn e solved jointly with vrious other network flow prolems. Trskov et l. [12] present liner nd integer optimiztion formultions for the clss of pirwise XOR coding for multiple unicst sessions. Ho et l. [13] develop ck pressure lgorithms for finding pproximtely throughput-optiml network codes within the clss of pirwise XOR coding. Eryilmz et l. [14] propose dynmic scheduling strtegy tht supports the chievle rtes for multiple unicst sessions given in [12]. Note, however, tht the enefit of pirwise XOR coding within tht frmework presented in [12], in terms of svings in link cost compred with trditionl routing, is found to e only modest depending on network topologies. It is not cler whether such modest gin is ecuse the simultions performed in [12] were restrictive, or it ctully indictes the limittions of the pirwise XOR coding. Even if we my wish to lift the restriction on the numer of sessions tht cn e coded together or the type of coding opertion, it is hrd to do so ecuse of the lck of pproprite tools for tht. The flow formultions in [12] or [13] my e generlized to represent more generl XOR coding scheme tht llows coding mong more thn two sessions. However, keeping trck of ll possile comintions of coded strems mong up to k sessions would require t lest O(m k E V ) vriles, where m is the totl numer of sessions, which leds to prohiitively lrge numer of constrints, hindering prcticl implementtions. Beyond the inry field, pthsed chrcteriztion is presented for the fesiility of the connection prolem with two unicst sessions [15], sed on which distriuted rte control lgorithm is proposed 978-1-4244-3513-5/09/$25.00 2009 IEEE 450 Authorized licensed use limited to: MIT Lirries. Downloded on April 09,2010 t 16:34:56 UTC from IEEE Xplore. Restrictions pply.
This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2009 proceedings. [16]. However, generlized chrcteriztion in the cse of coding mong more thn two sessions still seems difficult. For more generl coding schemes thn the pirwise XOR tht llow decoding t receivers, liner optimiztion prolem is proposed y Lun et l. [17], whose minimum cost is shown to e no greter thn the minimum cost of ny routing solution. However, its formultion is sed on the given set of the source processes tht cn e mixed on ech link, which still remins difficult to decide optimlly. In this pper, given tht there exists wide gp etween the presumly difficult quest for optiml inter-session network coding nd mny existing pproches tht re constrined within very restrictive clsses of coding strtegies, we investigte how the evolutionry pproches sed on Genetic Algorithm (GA), which in our previous works [18] [22] hve een used for the network coding resource optimiztion prolem, cn e utilized in n effort to fill the gp. In prticulr, we present novel rndomized liner coding scheme, in which decoding hppens t receivers while interior nodes perform rndom liner coding with selective mixture of informtion, controlled y evolutionry mechnisms. We then demonstrte, through simultions, how our coding strtegy my surpss existing end-to-end XOR coding schemes in terms of effectiveness nd prcticlity. The rest of the pper is orgnized s follows. Section II descries the prolem setup with relted ckground. Section III presents our coding strtegy with rief introduction to GA. Section IV descries how the computtionl components of our evolutionry pproch need to e designed, nd then Section V shows how our pproch cn e implemented in distriuted fshion over the network. Section VI exhiits numer of simultions for the evlution of the performnce of our coding strtegy. Section VII concludes the pper with some directions for future reserch. A. Prolem Setup II. PROBLEM SETUP AND BACKGROUND We ssume tht the network is given y n cyclic directed multigrph G =(V,E) where ech link hs unit cpcity nd links 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. Let us ssume tht there re R independent rndom processes of unit entropy rte, denoted y X ={X 1, X 2,..., X R }, originting t s( 1) source nodes. There re d receiver nodes, v 1, v 2,..., v d. For ech receiver v i (i =1,..., d), we denote the set of requested source processes y X i X. We ssume tht for ech source process in X i (i =1,..., d), there exists pth from its originting node to receiver v i ; otherwise, it is esy to check nd declre tht the prolem is not solvle. Other thn this connectivity condition, we do not put ny restriction on the source processes ech receiver node my request, i.e., X i cn e ny nonempty suset of X provided tht receiver v i is rechle from the originting nodes of the source processes in it. For this generlized scenrio, there is no simple chrcteriztion known for the fesiility of the connection prolem with network coding, even for the simpler cse of unicst connections, i.e., when ll X i s re disjoint [9], [23]. Though liner coding hs een shown to e suoptiml in generl [7], here we focus on the sclr liner coding in generl form, i.e., we do not restrict ourselves within either inry XOR opertions or pirwise mixing. Becuse the fesiility chrcteriztion still remins hrd even within liner coding [9], we ssume tht the cpcity constrints cn e relxed for some links, i.e., links cn e scled up, if necessry, to llow multiple trnsmissions. Note tht this my e the cse in mny prcticl networking scenrios; e.g., in wireless networks, nodes cn exploit more cpcities t the expense of more energy, nd in opticl networks, cpcity itself is rrely limited resource, though using more resources incurs n dditionl cost. With this ssumption, the fesiility prolem cn e resolved y scling links ppropritely. Then, we minly focus on finding cost-efficient trnsmission scheme using network coding. Hence, our primry ojective is to minimize the link cost required to stisfy the given communiction demnds. Lter, we consider nother ojective of minimizing the coding resources, i.e., numer of coding nodes/links. B. Bckground For the multicst cse, the solvility of connection prolem oils down to whether the mx-flows etween the source node nd ech of the receiver nodes ll exceed the desired multicst rte [1]. This cn e trnslted into the lgeric frmework [9] such tht, if we let ξ denote the vector consisting of ll the link coefficients s defined in [1], the prolem of finding fesile network code ecomes finding n ssignment of numers to vriles ξ such tht the product of the determinnt polynomils does not evlute to zero, which cn e done reltively esily with mny efficient rndomized lgorithms, e.g., s in [24]. As we go eyond the multicst cse, we now hve nother condition for solvility: within the trnsfer mtrix, the sumtrices relting the input processes tht re not requested to the corresponding output processes t the receiver nodes should evlute to zero (the first condition in Theorem 6 in [9]). Let us denote the entries of the sumtrices tht hve to evlute to zero y f 1 (ξ),f 2 (ξ),..., f K (ξ), which we refer to s interference polynomils. As in the multicst cse, we still hve the condition tht the sumtrices ssocited with the input nd output processes specified in the connection requests should e nonsingulr (the second condition in Theorem 6 in [9]). Let g 1 (ξ),g 2 (ξ),..., g L (ξ), referred to s determinnt polynomils, e the determinnts of the sumtrices tht should e nonsingulr. While finding n ssignment tht mkes determinnt polynomils nonzero cn e done esily, the difficult prt is to solve the set of determinnt polynomils to find n ssignment of numers to ξ tht mkes them ll zero. To determine whether Authorized licensed use limited to: MIT Lirries. Downloded on April 09,2010 t 16:34:56 UTC from IEEE Xplore. Restrictions pply. 451
This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2009 proceedings. such n ssignment is possile, [9] suggests Gröner sis method for determining the emptiness of vriety of the idel generted y the interference polynomils nd function of the determinnt polynomils. However, the worst cse complexity of the Bucherger lgorithm, which is used predominntly to compute Gröner sis, is douly exponentil [25] nd moreover, in our cse the input to the Bucherger lgorithm, i.e., the interference nd determinnt polynomils, my lredy hve n exponentil numer of terms, nd hence clculting Gröner sis my not e prcticl solution nywy. On the other hnd, s solving generl liner network coding prolem is no esier thn solving set of polynomil equtions [10], it is unlikely tht one cn develop specilized method tht possily provides more efficiency, utilizing some structurl properties of the interference nd determinnt polynomils. III. OUR CODING STRATEGY: SELECTIVE RANDOM LINEAR CODING In our coding strtegy, we control the mixture of informtion t ech node y deciding whether to include the input from prticulr incoming link when clculting the output on ech outgoing link. However, once the decisions re mde, we clculte the output y forming rndom liner comintion of the inputs llowed. Hence, the nme selective rndom liner coding. Intuitively, the strtegy we employ to del with the interference polynomils is 1) to mke some interference polynomils identiclly zero y zeroing out n enough numer of ssocited components of ξ nd 2) for the interference polynomils tht remin nonzero, to llow enough degrees of freedom t the receivers so tht the unwnted informtion cn e successfully cnceled out. For node v with d in incoming links nd d out outgoing links, we ssign inry vrile ij to ech pir of the i {1,..., d in }-th incoming link nd the j {1,..., d out }-th outgoing link. For the j-th (j =1,..., d out ) outgoing link, we refer to the ssocited inry vriles j =( ij ) i {1,...,din} s coding vector (see Fig. 1 for n exmple). The coding vectors re the vriles tht we need to decide in our coding strtegy. x1 x2 x3 y1 v y2 () Node v x1 x2 x3 v x1 x2 x3 v y1 y2 1= 1 0 1 2= 0 1 1 coding vector for y1 coding vector for y2 () Coding vectors for outgoing links Fig. 1. Node v with 3 incoming nd 2 outgoing links is ssocited with two coding vectors 1 =( 11, 21, 31 ) nd 2 =( 12, 22, 32 ). Once set of the coding vectors is given, we employ rndom liner coding t interior nodes, selectively using only those inputs ssocited with 1 s in the coding vectors. More specificlly, node v clcultes the output y j (j =1,..., d out ) on the j-th outgoing link s follows. Define the set I j of indices s I j = {1 i d in the i-th component of j is 1}. If we denote the input from the i-th (i =1,..., d in ) incoming link y x i, y j = i I j rnd(f q ) x i where rnd(f q ) denotes nonzero rndom element from F q. If the set I j is empty, y j is ssumed to e zero. Given set of coding vectors, the fesiility verifiction cn e done y first performing selective rndom liner coding t interior nodes s descried ove. Then, ech receiver node v i (i =1,..., d) performs Gussin elimintion to determine whether ll the desired input processes cn e recovered, with the interference prt cnceled out. Note tht this whole process cn e done in distriuted mnner, which will e utilized lter. This rndomized decision rule incurs n error when nonzero polynomils evlute to zero fter rndomly ssigning vlues to vriles; for zero polynomils, rndom ssignments in the evlution process do not ffect the finl result. Hence, the error proility is ounded y 1 (1 d/q) ν where q is the size of the finite field used for coding nd ν is the mximum numer of links in ny set of links constituting flow solution from the source to ny receiver [24]. Note tht this ound remins the sme even if we scle up some of the links s will e discussed lter. It remins to find n optiml ssignment of the coding vectors out of exponentilly scling numer of possile choices, which we ddress using GA sed serch method. Before proceeding, let us provide rief introduction to GA. A. A Brief Introduction to GA GAs re stochstic serch methods tht mimic genetic phenomen such s gene recomintion, muttion nd survivl of the fittest. Hving een pplied to lrge numer of scientific nd engineering prolems, GAs re especilly shown to e effective for the prolem of network coding resource optimiztion [18] [22]. The min control flow of the stndrd form of GA, clled simple GA, is shown in Fig. 2 [26]. Simple GA [26] opertes 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 chromosomes tht re more fit re selected with higher proility. For crossover, chromosomes re rndomly pired, nd then two chromosomes in ech pir exchnge suset Authorized licensed use limited to: MIT Lirries. Downloded on April 09,2010 t 16:34:56 UTC from IEEE Xplore. Restrictions pply. 452
This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2009 proceedings. initilize popultion; evlute popultion; while termintion criterion not reched { select solutions for next popultion; perform crossover; perform muttion; evlute popultion; } Fig. 2. Min control flow of simple GA. 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 simple GA. The ove process is iterted with the newly generted popultion successively replcing the current one. Simple GA termintes when certin stopping criterion is reched, e.g., fter predefined numer of genertions. For further detils of stndrd simple GA, the reder is referred to [18], [26]. IV. COMPUTATIONAL ASPECTS We first descrie how the computtionl components of our evolutionry pproch need to e designed, nd then in the next section, we present how our pproch cn e implemented in distriuted mnner over the network. A. Chromosome nd Fitness Function The decision vriles in our coding scheme re set of coding vectors s defined ove in Section III. Hence, ech chromosome (i.e., cndidte solution) consists of the collection of ll coding vectors. For given chromosome y,we must verify its fesiility first y performing selective rndom liner coding s descried in Section III. Once the fesiility test is done, given tht our primry ojective is to otin the cost-efficient trnsmission scheme vi network coding, the fitness function F is defined s { totl cost of link usge, if y is fesile, F (y) = (1), if y is infesile. Note tht if we wish to minimize the resources engged in network coding, we my replce the link cost y the numer of coding nodes/links for fesile chromosomes. Moreover, the two ojective vlues, i.e., the link nd coding costs, cn e jointly considered to investigte possile trdeoff etween those two ojectives [22], which leds to more informed decisions on whether or where to employ network coding. B. Initil Popultion Construction Typiclly the initil popultion of GA is composed of rndom chromosomes. However, s pointed out in [18], inserting some non-rndom chromosomes cn gretly improve the performnce of the lgorithm. When we dd numer of non-rndom solutions to the initil popultion, cre must e tken to insert only neutrl solutions in the sense tht they re not prticulrly close to some locl optimum. Otherwise, those inserted strting points tend to tke over the whole popultion in the erly stge of the evolution process so tht the lgorithm my end up converging just to neighorhood of the strting points. To crete neutrl strting point, we first scle up the links in G to mke the prolem multicst-like in the sense tht ech receiver node now receives ll source processes tht hve directed pth to it. Then, we employ s neutrl strting point the ll-one chromosome tht indictes mixing everything t ll interior nodes. Let us now show how we cn find n pproprite scling fctor. Before proceeding, for ech receiver node v i (i = 1,..., d) in G, welety i X e the set of the source processes from whose originting node, receiver v i is rechle (i.e., there exists directed pth from the source process s originting node to node v i ). Also, we let r i (i =1,..., d) denote the size of set Y i. Now let us crete n uxiliry network H from G y introducing virtul source node S nd dding unitcpcity link from node S to the source node t which ech source process X j X(j =1,..., R) origintes. Then, we let f i (i =1,..., d) e the vlue of the mximum flow from the virtul source node S to ech receiver v i (i =1,..., d). We define k s k = mx i {1,...,d} ( ri f i ). (2) Theorem 1: Let G e scled version of G with ech link replced y k multiple links, where k is defined s in (2). In network G, network code tht performs rndom liner coding using ll ville inputs t every interior node in sufficiently lrge finite field is fesile. Proof Outline: Let us crete nother uxiliry network H y scling up the links in H y fctor of k nd dding unit-cpcity links etween ech source process in X\Y i nd ech receiver node v i (i = 1,..., d). Then, we hve multicst prolem for which fesile network code cn e otined y employing rndom liner coding in sufficiently lrge finite field. Given such fesile network code C on network H, it cn e shown tht C remins fesile even if we remove the dded links to otin network G.(Afull proof cn e found in [27].) Note tht the ll-one chromosome shown ove is fesile ut hs the worst cost in terms of either the link cost or the coding cost nd thus it my not is the initil popultion towrd ny prticulr suoptiml solution. C. Genetic Opertors In conventionl crossover nd muttion opertors, which we refer to s it-wise opertors, the unit of interchnged or pertured sucomponents is ech it of the chromosomes. However, it is pointed out in [19], [20] tht, for the prolem of the coding resource optimiztion for multicst, significnt Authorized licensed use limited to: MIT Lirries. Downloded on April 09,2010 t 16:34:56 UTC from IEEE Xplore. Restrictions pply. 453
This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2009 proceedings. performnce gin cn e ttined y pplying the crossover nd muttion opertions on ech full coding vector. Tht is, for vector-wise crossover, we let two chromosomes suject to crossover exchnge ech full coding vector (rther thn ech it) independently with the given crossover proility. For vector-wise muttion, we rndomly regenerte ech coding vector (gin, rther thn ech it) independently with the given muttion proility. We refer to these two opertors s vector-wise opertors. Given tht enforcing coding vector-level modulrity in genetic opertions cn led to significnt performnce gin, we my develop nother set of genetic opertors tht further exploit, this time, the node-level modulrity. In node-wise opertors, we now interchnge or pertur ll coding vectors ssocited with the outgoing links of ech node with the crossover/muttion proility. The intuition ehind the nodewise genetic opertors is tht the coefficients of the links directly connected with ech other or within few hops wy re more likely to hve strong dependencies thn those ssocited with the links fr wy from ech other. Note, however, tht the node-wise genetic opertors enforce roder level of modulrity thn the vector-wise opertors, exchnging or perturing lrger numer of coding vectors t once. If this is the right level of modulrity, it would trnslte into fster convergence of the lgorithm to the sme or etter solution; otherwise, the lgorithm my tend to converge premturely to lower qulity solution. We will lter verify through simultions the effect of these different genetic opertors on the performnce of the lgorithm. V. DISTRIBUTED IMPLEMENTATION Recll tht in Section III we discussed how the fesiility test of single chromosome cn e done y employing rndom liner coding t interior nodes. Note tht in doing so, ech interior node only refers the relevnt portion of the chromosome, i.e., the coding vectors tht indicte the opertions t tht node. Hence, we cn divide up the popultion y letting ech node hndle only the coding vectors it needs from every chromosome in the popultion. If the coding vectors re stored t locl nodes, we only need to trnsmit the set of the coefficients tht indicte the overll effect of network coding reltive to the source dt, which is commonly referred to s glol encoding vector in the network coding literture (see e.g., [28]), for fesiility test. Hence, fesiility test cn e done in distriuted fshion y trnsmitting pckets contining such coefficients. Moreover, it cn e shown tht the whole fitness clcultion nd ll genetic opertions discussed in Section IV cn e done independently t locl interior nodes with some coordintion informtion emedded in dt pckets, ssuming tht the source nodes cn communicte with one nother [27]. Note tht when we consider network coding mong multiple sessions tht my originte from multiple source nodes, it must e ssumed in the first plce nywy tht the source nodes cn communicte with one nother to find out the totl numer of the source processes eing considered together for possile coding so tht the coefficients for liner coding cn e ligned consistently cross ll the source nodes involved. Hence, we ssume tht mong the prticipting source nodes, we cn designte one s the mster node which serves s the min controller of the lgorithm y gthering informtion from other source nodes nd sending the clculted coordintion informtion to other source nodes. With the ove ssumptions, the whole evolutionry lgorithm to serch for n optiml set of coding vectors cn operte in distriuted fshion [27], whose overll flow is shown in Fig. 3 with the loction of ech procedure specified. More detiled descriptions of the distriuted lgorithm re omitted for spce considertion (the reder is referred to [27, Chpter 7] for detils). Note tht the clcultion of the scling fctor k defined in (2) cn lso e done in distriuted mnner [27]. Computtionlly, the distriuted lgorithm performs the sme tsk s simple GA (Fig. 2) with the computtionl components descried in Section IV. [S1] initilize; (ll nodes) [S2] run forwrd evlution phse; (ll nodes) [S3] run ckwrd evlution phse; (ll nodes) [S4] send prtil fitness to mster node; (source nodes) [S5] clculte fitness; (mster node) [S6] while termintion criterion not reched (mster node) { [S7] clculte coordintion vector; (mster node) [S8] fetch coordintion vector from mster node; (source nodes) [S9] run forwrd evlution phse; (ll nodes) [S10] perform selection, crossover, muttion; (interior nodes) [S11] run ckwrd evlution phse; (ll nodes) [S12] [S13] Fig. 3. send prtil fitness to mster node; (source nodes) clculte fitness; (mster node) } Flow of distriuted lgorithm for selective rndom liner coding Within this distriuted setup, the fitness evlution of ech chromosome requires the computtionl complexity of O(d in d out R) t ech interior node nd O(d 2 in R) t ech receiver node [27]. The most importnt enefit of this distriuted structure is tht network code cn e constructed on the fly while the network is opertionl, llowing for the following network coding protocol: As the mster node sends pcket tht signifies the strt of the code construction, ll prticipting nodes go into the code construction mode, running the lgorithm descried ove. As the distriuted evolutionry lgorithm proceeds, ech interior node stores nd improves its relevnt network codes. At the end of the lgorithm, the mster node only needs to send the index of the est chromosome of the lst popultion nd ll prticipting nodes now strt to trnsmit dt sed on the loclly stored network code tht corresponds to the received est index. Authorized licensed use limited to: MIT Lirries. Downloded on April 09,2010 t 16:34:56 UTC from IEEE Xplore. Restrictions pply. 454
This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2009 proceedings. VI. PERFORMANCE EVALUATION In order to evlute the effect of different genetic opertors nd lso compre the performnce of our coding scheme with others, we first perform simultions for the multipleunicst scenrio, for which other network coding schemes re ville. Then, to demonstrte our lgorithm s ility to hndle more generl prolems, we further consider the cse of wireless networks with no restriction on the type of connection requests. A. Multiple Unicst Connections We performed numer of simultions for the multipleunicst cse, i.e., ll connection requests X i s (i =1,..., d) re disjoint. Our simultions re sed on the grid network introduced in [12] (network D in Fig. 4). In [12], the cost of ech link is ssigned rndomly, which llows only slim chnces tht the network coding dvntge exists. Note, however, tht network coding gin tkes effect only if there exists n expensive ottleneck link tht hs to e used y numer of flows nd lso lower cost detours round it, which hppens rrely when the connection requests nd link costs re rndomly chosen. Hence, in our experiments, we pick cost ssignment s depicted in Figure 4, where the links with cost higher thn 1 re highlighted y thicker rrows with the ctul cost shown y their side, to mke the network coding dvntge clerly exist t lest for some connection requests. For comprison, we tke Trskov et l. s pirwise XOR coding scheme [12]. We do not include the pproch y Wng et l. [15] here ecuse it cn e shown tht the gril structure considered therein does not led to link cost svings in our prolem setup (i.e., our primry ojective to minimize the link cost wheres the cpcity constrint my e relxed if needed); nevertheless, it does increse the cpcity region when the cpcity constrint is strictly enforced. Fig. 4. 10 u1 u2 u3 v1 10 10 100 x y v2 Grid network D for multiple-unicst simultions 1) Two-Connection Cse: First, we consider the cse of two connections, i.e., R =2. We repetedly rn our lgorithm, vrying the loction of the source processes nd connection requests such tht two source processes X 1 nd X 2 my originte t ny of nodes {u 1,u 2,u 3 } nd ech source process v3 10 10 my e requested t ny of receiver nodes {v 1,v 2,v 3 }.For comprison with the routing cse nd the pirwise XOR coding scheme, we used the multi-commodity formultion with integer constrints nd Trskov et l. s formultion [12], respectively. Among ll possile rrngements of the source processes nd connection requests, there is only one cse in which network coding sves the link cost: X 1 nd X 2 originte t nodes u 1 nd u 3, respectively, while X 1 nd X 2 re requested t nodes v 3 nd v 1, respectively. Fig. 5() shows the est solution otined y our selective rndom liner coding which requires the link cost of 131. Note tht, for this simple prolem, our evolutionry lgorithm, despite its stochstic nture, lwys yields the sme solution regrdless of the genetic opertors used. In comprison, the optiml cost chieved y routing is 212. In Fig. 5(), we use the symol to represent liner comintion with rndom coefficients from the designted finite field; i.e., x 1 x 2... x n = n i=1 rnd(f q) x i, where gin rnd(f q ) represents nonzero rndom element from the designted finite field F q. Interestingly, this exmple highlights the difference etween our coding scheme nd the pirwise XOR coding scheme, whose est solution, s depicted in Fig. 5(), offers n even lower cost of 129. In our selective rndom liner coding, coding opertion is performed only once t node x nd decoding is done t receiver nodes. On the other hnd, in the est pirwise XOR code, nother XOR opertion is done t node z, which, in fct, serves s decoding t n interior node nd consequently sves the link cost of 2 (y not sending β t node w long longer pth down to node u). Note tht the sme trnsmission strtegy would not work for our selective rndom liner coding ecuse nother coding opertion t node z would yield nother rndom liner comintion ( ) rther thn. From this exmple, we oserve tht, while oth coding schemes provide dvntges over trditionl routing, our selective rndom liner coding scheme my require some dditionl link cost compred with the pirwise XOR coding, which cn e considered the price of employing rndomized coding in finite field lrger thn the inry field. Note, however, tht s we increse the numer of connections, our coding scheme leds to much more prcticl solution despite the possile expense of such dditionl link cost, s will e discussed next. 2) Five-Connection Cse: Let us now increse the numer of source processes to 5 nd pick up n rrngement of the source processes nd connection requests tht llows the network coding dvntge: {(source process, source node, receiver node)} = {(X 1,u 1,v 3 ), (X 2,u 1,v 1 ), (X 3,u 3,v 1 ), (X 4,u 3,v 2 ), (X 5,u 3,v 2 )}. As opposed to the simpler prolem in the previous susection, the performnce of our lgorithm vries depending on the type of genetic opertors used. Tle I summrizes the performnce of our lgorithm with different genetic opertors. For ech type of genetic opertors, we performed 30 simultion runs nd t the end of ech run we pick the est coding solution out of the lst popultion. The first column shows the lowest cost of those 30 est coding solutions nd the next Authorized licensed use limited to: MIT Lirries. Downloded on April 09,2010 t 16:34:56 UTC from IEEE Xplore. Restrictions pply. 455
This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2009 proceedings. u1 u2 u3 u1 u2 u3 x x y w z y z w v1 v2 v3 v1 v2 v3 () Selective rndom liner coding () Pirwise XOR coding Fig. 5. Comprison of est coding solutions otined. two columns show the verge nd stndrd devition of the cost of those 30 solutions. For comprison, the est routing solution yields the optiml cost of 242. The fourth column of the tle displys the rtio of the lgorithm runs, out of the totl of 30, in which the cost of the est coding solution found is ctully lower thn the est routing solution. The fifth column clcultes the verge numer of genertions required for coding solution tht outperforms the routing solution, if found, to pper in the popultion. The lst column shows the rtio of the lgorithm runs in which coding solution outperforming the routing solution is found efore the 100- th genertion, reltive to the totl numer of lgorithm runs where such coding solution is ever found. TABLE I SUMMARY OF THE LINK COSTS OF THE CODING SOLUTIONS FOUND BY DIFFERENT GENETIC OPERATORS AND RELATED STATISTICS. THE BEST ROUTING SOLUTION REQUIRES THE LINK COST OF 242. Link Cost Outperform Routing Best Avg Std Rtio At <100 Bit-wise 181 204.2 10.52 1.00 296.7 0.07 Vector-wise 156 164.8 5.87 1.00 177.3 0.73 Node-wise 158 236.2 41.96 0.43 124.8 0.77 From Tle I, we first notice tht our lgorithm with the itwise or vector-wise opertors relily yields network coding solution tht outperforms the est routing solution. Between the two kinds of opertors, the vector-wise opertors led to much etter solutions, oth in terms of the men nd stndrd devition. Also, the numer of genertions required to find network coding etter thn the est routing solution is much smller on verge for the vector-wise opertors; most of the time it is found efore the 100-th genertion. Hence, we my conclude tht the vector-wise opertors llow the lgorithm to find much etter solutions much fster thn the it-wise opertors. For the cse of the node-wise opertors, our lgorithm finds network coding solution tht exceeds the routing solution only in 43% of the simultions. However, in those successful simultions, such network coding solution is found much fster thn the cse of the vector-wise opertors. Hence, we my conclude tht with the node-wise opertors the lgorithm tends to converge fster, ut premturely to lower qulity solution. This my e due to tht the node-level modulrity enforced y the node-wise opertors is too strong in the sense tht it chnges too mny coding vectors t once, which conceptully my correspond to setting the step too lrge in n itertive optimiztion scenrio. On the other hnd, the optimiztion formultions for the pirwise XOR coding [12] filed to converge within resonle mount of time (during full week of simultions) sed on the simultion environment used in [12]. Note tht the liner nd integer progrms in [12], even for this fiveconnection prolem, contin round 68700 nd 1400 vriles (including the slck vriles to hndle the mx opertor in the constrints) nd 67500 nd 1700 constrints, respectively. Though we my hve een le to otin converged results if we hd experimented it with much fster mchine, the point we would like to mke here is tht the optimiztion formultions considered in [12] my not provide prcticl sclility s the numer of connections increses. For comprison, our lgorithm tkes out 1.5 seconds for ech genertion in the sme simultion environment, where we simulted ech node s opertion sequentilly (one t ech time from upstrem to downstrem nodes) using MATLAB on single-processor mchine. In rel network, where ech node uses its own computtionl resources, we my expect much fster execution of our lgorithm. B. Generl Connections We performed nother set of experiments in fully generlized networking scenrios where the receiver nodes my request ny comintion of the ville source processes. For this, we generted 100 rndom wireless networks where 40 nodes re plced rndomly within 10 10 squre with rdius of connectivity 3. A unit-rte hyperlink is originted from ech node towrd the set of nodes tht re within the Authorized licensed use limited to: MIT Lirries. Downloded on April 09,2010 t 16:34:56 UTC from IEEE Xplore. Restrictions pply. 456
This full text pper ws peer reviewed t the direction of IEEE Communictions Society suject mtter experts for puliction in the IEEE INFOCOM 2009 proceedings. connectivity rnge nd hve higher horizontl coordinte. In ech rndom topology, 5 source processes were rndomly plced t 5 nodes chosen in the left hlf nd ech of 5 receiver nodes rndomly chosen in the right hlf demnds rndomly chosen nonempty suset of X = {X 1,..., X 5 }. For such generted rndom connection prolems, we pply two-stge method in which we first run our lgorithm with the fitness function defined s (1) to minimize the link cost without restricting network coding. Then, from the est coding solution found, we tke only the links tht re used for trnsmission to form the sugrph to e used in the second stge. In the second stge, we use our lgorithm to minimize the numer of coding nodes y chnging the fitness function to reflect the numer of nodes where network coding is performed. Tle II shows the distriution of the found minimum numer of coding nodes through the two-stge method. Note tht in most cses (out 80% of the rndom topologies tested), the clculted minimum cost cn e chieved without network coding t ll. Also, for the topologies where network coding is required, it is required only t smll suset of nodes, i.e., not ll nodes need to perform network coding, which ws lso the cse in mny multicst scenrios [27]. TABLE II DISTRIBUTION OF THE CALCULATED MINIMUM NUMBER OF CODING NODES IN 100 RANDOM TOPOLOGIES. # Coding Nodes 0 1 3 4 5 6 7 9 10 # Topologies 79 6 1 1 1 3 2 2 5 C. Discussion Aove simultions exhiit tht our evolutionry pproch offers coding strtegy in the generl connection prolem which is still somewht restricted ut with enhnced fetures in mny spects, compred with existing pirwise XOR coding. For the prolem of multiple unicst connections, we showed tht our evolutionry pproch pproch yields network coding solution tht offers n dvntge over trditionl routing in terms of link cost, wheres n existing pproch for pirwise XOR coding my fil to provide sclility within prcticl rnges s the numer of connections increses. Also, we demonstrted tht our evolutionry pproch cn tckle more generl prolems in which there is no restriction on the type of connection requests, tking the coding cost into ccount s well. Note tht, though only the two-stge method ws experimented for n illustrtive purpose in the second set of simultions, multi-ojective evolutionry pproch cn lso e utilized, similrly s in [22], to investigte the trdeoff etween the two ojectives. In ddition to the distriuted structure presented in Section V, nother distriuted structure, i.e., temporlly distriuted structure, cn lso e dopted for more efficient utiliztion of computtionl resources s well s more roust opertion ginst pcket losses [21]. A possile drwck of our evolutionry pproch is tht, s opposed to the cse to the coding resource optimiztion for multicst [19], it lcks performnce ound. However, given tht there re no prcticl lterntives tht tke into ccount coding mong more thn two flows, our pproch my serve s unique mens for exploring network coding dvntges in much more generlized setup thn the pirwise coding. Another possile limittion is regrding the sclility; i.e., s is typicl for GA, the popultion size my need to e incresed significntly for lrge prolems. However, with the distriuted structure descried in Section V, the popultion size cn e incresed rther flexily y incresing the size of the pcket used for fitness evlution, given tht the computtionl complexity t ech node scle linerly with the popultion size. Moreover, with the temporlly distriuted structure [21], we my further increse the effective popultion size y incresing the numer of (su)popultions. VII. CONCLUSION AND FUTURE WORK We hve proposed novel inter-session network coding strtegy for generl connection prolem eyond multicst, for which no optiml network coding scheme is known. Our coding strtegy llows rndom liner coding over lrge finite field, in which decoding is done t receivers nd the mixture of informtion t interior nodes is controlled y evolutionry mechnisms. We hve demonstrted how our coding strtegy my surpss existing end-to-end pirwise XOR coding schemes in terms of effectiveness nd prcticlity. There hve een mny recent developments in the field of evolutionry computtion tht significntly improve the sclility of the trditionl simple GA, on which our current pproch is sed. Such dvnced GA frmeworks cn e redily employed, without chnging the overll frmework presented here, to find nd exploit further linkge informtion, i.e., dependencies mong vriles, for n improved performnce. Also, one my develop method to construct solutions tht provide some useful ounds on the link or coding cost, which in turn cn e comined with our evolutionry pproch to generte etter initil popultion or to refine the finl solution (s in the coding resource optimiztion prolem for multicst [19]). In ddition, one my utilize the core structure of our proposed evolutionry pproch within other network prolems tht involve comintoril optimiztions with n unresolved scling issue. 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