Genetic Representations for Evolutionary Minimization of Network Coding Resources

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1 Genetic Representtions for Evolutionry Minimiztion of Network Coding Resources Minkyu Kim 1, Vrun Aggrwl 2, Un-My O Reilly 2, Muriel Médrd 1, nd Wonsik Kim 1 1 Lortory for Informtion nd Decision Systems 2 Computer Science nd Artificil Intelligence Lortory Msschusetts Institute of Technology, Cmridge, MA 02139, USA {minkyu@, vrun g@, unmy@csil., medrd@, wskim14@}mit.edu Astrct. We demonstrte how genetic lgorithm solves the prolem of minimizing the resources used for network coding, suject to throughput constrint, in multicst scenrio. A genetic lgorithm voids the computtionl complexity tht mkes the prolem NP-hrd nd, for our experiments, gretly improves on su-optiml solutions of estlished methods. We compre two different genotype encodings, which trdeoff serch spce size with fitness lndscpe, s well s the ssocited genetic opertors. Our finding fvors smller encoding despite its fewer intermedite solutions nd demonstrtes the impct of the modulrity enforced y genetic opertors on the performnce of the lgorithm. 1 Introduction Network coding is novel technique tht generlizes routing. In trditionl routing, ech interior network node, which is not source or sink node, simply forwrds the received dt or sends out multiple copies of it. In contrst, network coding llows interior network nodes to perform ritrry mthemticl opertions, e.g., summtion or sutrction, to comine the dt received from different links. It is well known tht network throughput cn e significntly incresed y network coding [1, 2, 3]. While network coding is ssumed to e done t ll possile nodes in most of the network coding literture, it is often the cse tht network coding is required only t suset of nodes to chieve the desired throughput. Consider Exmple 1: Exmple 1. In the cnonicl exmple of network B (Fig. 1()) [1], where ech link hs unit cpcity, source s cn send 2 units of dt simultneously to the sinks t 1 nd t 2, which is not possile with routing lone. But only node z needs to comine its two inputs while ll other nodes perform routing only. If we suppose tht link (z,w) innetworkb 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 is to trnsmit dt t rte 2 to the 3 lef nodes, network coding is required either t node or t node, ut not t oth. M. Gicoini et l. (Eds.): EvoWorkshops 2007, LNCS 4448, pp , c Springer-Verlg Berlin Heidelerg 2007

2 22 M. Kim et l. x s z y x s z y x s y z t1 w t2 t1 w t2 c t1 t2 d t3 () Network B () Network B (c) Network C Fig. 1. Smple networks for Exmple 1 Exmple 1 leds us to the following question: To chieve the desired throughput, t which nodes does network coding need to occur? This question s nswer is vlule ecuse eliminting unnecessry coding nodes will sve computtion t the ppliction lyer if tht is where network coding is hndled. Alterntively, if network coding is integrted in the uffer mngement of routers, it will reduce the numer of routers tht need to perform coding opertions without compromising communiction cpcity. For GA, the prolem cn e posed s the minimiztion of coding cost (in links or nodes) suject to the constrint of fesiility (chieving the desired throughput). The prolem of determining miniml set of nodes where coding is required is NP-hrd; its decision prolem, which decides whether the given multicst rte is chievle without coding, reduces to multiple Steiner sugrph prolem, which is NP-hrd [4]. It is shown tht even pproximting the minimum numer of coding nodes within ny multiplictive fctor or within n dditive fctor of V 1 ɛ is NP-hrd [5]. Note, however, tht once the set of coding nodes is identified, network code chieving the desired throughput cn e efficiently constructed for the multicst scenrio, either in deterministic [6] or rndomized fshion [7]. In the network reserch community, [8] nd [9] hve documented results tht demonstrte the enefit of the GA over other existing pproches in terms of reducing the numer of coding links or nodes nd its pplicility to vriety of generlized scenrios. These contriutions emphsized the computtionl howto spects of fesiility checking nd the trnsformtions of the network grph into secondry grphs tht express possile coding situtions uniformly since these re key to evluting the fitness function of the GA. In the course of investigting the fesiility checks, grph trnsformtions, nd vlue of using GA, we experimented with two different genotype encodings 1 nd ssocited opertors. For oth encodings, we use genotype composed of numer of locks, ech of which consists of set of vriles indicting the link sttes. For lock of length k, using n lphet of crdinlity 2, the Binry Link Stte (BLS) encoding represents ll possile 2 k sttes of k links. On the other hnd, for the Block Trnsmission Stte (BTS) encoding, we group 1 To minimize confusion, throughout the pper, the term encoding refers to genotype encoding only, while the term coding mens network coding.

3 Genetic Representtions for Evolutionry Minimiztion 23 those link sttes into (k + 2) trnsmission sttes. Despite the smller serch spce size of BTS encoding, it is not cler tht it should e superior to BLS encoding ecuse in grouping mny link sttes into one, less informtion tht would relte the fitness of solutions intermedite to the est solution is ville in contrst to BLS encoding which provides more informtion through its intermedite solutions. In this pper we focus on the two different encodings with ssocited genetic opertors nd conduct more comprehensive comprison etween them. Specificlly relevnt to the GA community, we consider into the GA encoding trdeoff issues relted to serch spce size nd fitness lndscpe. The rest of the pper is orgnized s follows. Section 2 presents the prolem formultion, nd Section 3 descries the network coding GA (NCGA) with the two different encodings nd ssocited opertors. Section 4 sets up set of experiments into reltive vlues of the encodings nd discusses the results. Section 5 presents summry of the results nd our conclusions. 2 Prolem Formultion We ssume tht network is given y directed multigrph G =(V,E) s in [10] where ech link hs unit cpcity whose unit cn e ritrrily chosen, e.g., k its per second for constnt k, or fixed size pcket per unit time, etc. 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. We consider the single multicst scenrio in which single source s V wishes to trnsmit dt t rte R to set T V of sink nodes. Rte R is sid to e chievle if there exists trnsmission scheme tht enles ll T sinks to receive ll of the informtion sent. We only consider 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 n chievle rte R, we wish to determine miniml set of nodes where coding is required in order to chieve this rte. However, whether coding is necessry t node is determined y whether coding is necessry t t lest one of the node s outgoing links nd thus, s pointed out lso in [5], the numer of coding links is in fct 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 [8], it is strightforwrd to generlize the proposed lgorithm to the cse of minimizing the numer of coding nodes. Furthermore, [8] shows tht, with pproprite chnges, the lgorithm cn e redily pplied to more generlized optimiztion scenrios, e.g., when links nd nodes hve different coding costs. It is cler tht no coding is required t node with only single input since these nodes hve nothing to comine with [8]. For node with multiple incoming links, which we refer to s merging node, if the linerly coded output to prticulr outgoing link weights ll ut one incoming messge y zero, effectively no coding occurs on tht link; even if the only nonzero coefficient is not

4 24 M. Kim et l. identity, there is nother coding scheme tht replces the coefficient y identity [5]. Thus, to determine whether coding is necessry t n outgoing link of merging node, we need to verify whether we cn constrin the output of the link to depend on single input without destroying the chievility of the given rte. As in network C of Exmple 1, the necessity of coding t link depends on which other links code nd thus the prolem of deciding where to perform network coding 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. 3 Network Coding GA (NCGA) Prior to using the NCGA, the given network grph G is trnsformed into secondry grph y either of the two methods presented in [8, 9] 2. Regrdless which method is used, mpping the network coding prolem to GA frmework is done s follows. Suppose merging node with k( 2) incoming links. To consider the trnsmission to ech of its outgoing links, we ssign inry vrile to ech of its k incoming links, which eing 1 indictes tht the link stte is ctive (the input from the ssocited incoming link is trnsmitted to the outgoing link) nd 0 indictes it is inctive. Given tht network coding is required for the trnsmission only if two or more link sttes re ctive, we my need to consider those k vriles together. We refer to the set of the k vriles s lock of length k (see Fig. 2 for n exmple). The wy how those inry vriles re ctully encoded s genotype will e descried lter in this section. x1 x2 x3 x1 x2 x3 x1 x2 x3 v v v y1 y2 y1 y lock for y1 lock for y2 () Merging node v () Two locks for outgoing links of v Fig. 2. Node v with 3 incoming nd 2 outgoing links results in 2 locks, ech with 3 vriles indicting the sttes of incoming links (x 1,x 2,x 3) onto the ssocited outgoing link Constrint nd Fitness Function. A genotype is clled fesile if there exists network coding scheme tht chieves the given rte R with the link sttes 2 An interested reder is referred to [8,9] for the detils of the two methods for grph trnsformtion nd fesiility testing.

5 Genetic Representtions for Evolutionry Minimiztion 25 determined y the genotype. To clculte the fitness of genotype y, its fesiility must e checked y either of the two methods in [8], [9] depending on the secondry grph chosen erlier, nd the fitness vlue F is ssigned s { numer of locks with two or more ctive links, if y is fesile, F (y) =, if y is infesile. The NCGA uses stndrd genertion-sed GA control loop with tournment selection. It termintes t some mximum numer of genertions. Afterwrd, the est solution of the run is optimized with greedy sweep: weswitch ech of the remining 1 s to 0 if it cn e done without violting fesiility. This procedure my only improve the solution, nd sometimes the improvement cn e sustntil. Reference [9] proves tht the NCGA with greedy sweep is gurnteed to perform no worse thn the existing lgorithm in [5]. Binry Link Stte (BLS) Encoding nd Opertors. This encoding llows lock of length k to tke ny of 2 k possile inry strings of length k. Ifwe denote y d v in nd dv out the in-degree nd the out-degree of node v,nodev hs dv out locks of length d v in, nd we hve totl of m = v V dv in dv out inry vriles, where V is the set of ll merging nodes. We must explore the m-dimensionl inry spce of 2 m cndidtes to find the desired miniml set of coding links. For BLS encoding we use uniform crossover, where ech pir of genotypes is selected for crossover with given proility (mixing rtio) nd the two genotypes in selected pir exchnge ech it independently with nother given proility (crossover proility). For muttion, we use simple inry muttion, where ech it in ech genotype is flipped independently with given proility (muttion rte). Since these opertors del with ech it seprtely, we refer to the opertors used for BLS encoding s it-wise genetic opertors. Block Trnsmission Stte (BTS) Encoding nd Opertors. As mentioned ove, once lock hs t lest two 1 s, replcing ll the remining 0 s with 1 s hs no effect on whether coding is done. Moreover, it cn e shown tht sustituting 0 with 1, s opposed to sustituting 1 with 0, does not hurt the fesiility. Therefore, for fesile genotype, ny lock with two or more 1 s cn e treted the sme s the lock with ll 1 s. Thus we could group ll the sttes with two or more ctive links into single stte, coded trnsmission. This stte is rounded out y k sttes for the uncoded trnsmissions of the input received from one of the k single incoming links nd one stte indicting no trnsmission. Thus BTS encoding emerges where ech lock of length k cn only tke one of the following (k + 2) strings: , , , ,..., , The net effect is reduction in the numer of possile sttes for lock to (k + 2) rther thn 2 k.ifweletw e the totl numer of locks (i.e., w = v V dv out )ndk i denote the length of the i-th lock (i =1,..., w), the serch spce size is w i=1 (k i + 2). However, the enefit of the smller spce size in fct comes t the price of losing the informtion on the prtilly ctive link sttes tht my serve s intermedite steps towrd n uncoded trnsmission stte. This trdeoff will e discussed more in depth in Section 4.

6 26 M. Kim et l. To preserve the BTS encoding structure throughout genetic opertions, we need to define new set of genetic opertors, which we refer to s lock-wise genetic opertors. For lock-wise uniform crossover, we let two genotypes suject to crossover exchnge ech lock, rther thn it, independently with the given crossover proility. For lock-wise muttion, we let ech lock under muttion tke nother string chosen uniformly t rndom out of (k + 1)otherstringsfor length-k lock. If muttion rte is α, the verge numer of chnged its in α, wheres it is kα for it-wise muttion. While the difference ecomes more pprent when k is lrge since the ltter is upper ounded y 4α, those vlues re still different for k =2,where the lock structure mkes no difference in the spce size. Though lock-wise muttion my led to much smller numer of flipped its, it is more likely to cuse sudden chnge in genotype. length-k lock is now clculted s 4k 2 (k+1)(k+2) 4 Experiments 4.1 Experiment Setup The two encodings not only differ in the size of serch spce, ut in the wy the genetic opertors re pplied. In BTS encoding with the lock-wise opertors, crossover is pplied t the lock oundries nd muttion is performed intrlock. However, in BLS encoding with the it-wise opertors, crossover nd muttion re rndomly pplied without respecting ny lock oundries. While evluting the effect of the serch spce size reduction, we lso wnt to investigte whether the exploittion of lock level modulrity y the lock-wise opertors gives ny significnt improvement in the lgorithm s performnce. We thus set up two experiments: Experiment I compres the effect of the two encodings comined with ssocited opertors on the performnce of the NCGA, while Experiment II tests the effect of the opertors lone y isolting the effect of the encodings tht led to different spce sizes. Experiment I: We use two cyclic networks, I-50 nd I-75, generted y the lgorithm in [11], whose detils re given in Tle 1. Note tht BTS encoding reduces the size of the serch spce y 30.3 nd 115 orders of mgnitude for networks I-50 nd I-75, respectively, compred with tht in the cse of BLS encoding. This experiment tests which encoding is etter given the trdeoff in the serch spce size nd ese of trversing the fitness lndscpe. Experiment II: We construct set of synthetic networks with only locks of length 2. Note tht for lock of length 2, the two encodings hve the sme serch spce size (2 k = k +2 when k = 2), ut the lock-wise opertors retin their modulrity. These networks re constructed y cscding numer of copies of network B in Fig. 1() such tht the source of ech susequent copy of B is replced y n erlier copy s sink. We use fixed-depth inry trees contining 3, 7, 15, nd 31 copies of B (henceforth clled II-3, II-7, II-15, nd II-31, respectively). Prmeters of these networks re given in Tle 1. All these

7 Genetic Representtions for Evolutionry Minimiztion 27 network hve 0 s the minimum numer of coding links, i.e., multicst rte 2 is chievle without coding. We scle up the network size to investigte the pyoff one gets with modulr opertors s the serch spce size increses. We use the NCGA with the decomposition-sed grph trnsformtion nd the mx-flow fesiility testing descried in [9]. For comprison, we lso perform experiments using the two existing pproches y Frgouli et l. [12] ( Miniml 1 ) 3, nd Lngerg et l. [5] ( Miniml 2 ), in oth of which link removl is done in rndom order. For Miniml 1, the sugrph is selected lso y miniml pproch, which strting from the originl grph sequentilly removes the links whose removl does not destroy the chievility. Tle 1. Detils of the networks used in the experiments Network Genotype Numer of Avg. length Serch spce size (log 10 ) length locks of locks (BLS/BTS) I /53.93 I / II /9.63 II /24.08 II /52.98 II / Algorithm Prmeters We set the totl udget of fitness evlutions to 150,000 ( very smll frction of the serch spce size of the networks considered. Preliminry experiments suggested different tournment sizes nd muttion rtes for the two encodings: 10 nd for BLS encoding, nd 100 nd for BTS encoding, respectively. All other prmeters re mtched for the two encodings. We perform 30 runs for ech network with oth encodings. Tle 2 summrizes the prmeters used. Tle 2. GA prmeters used in the experiments Popultion size 150 Tournment size 10(BLS)/100(BTS) Mximum genertions 1000 Mixing rtio/crossover proility 0.8/0.8 Muttion rte 0.006(BLS)/0.012(BTS) When rndomly initilizing the popultion, we insert n ll-one vector, which represents the solution where coding is done everywhere nd thus is fesile y the ssumption tht the given rte R is chievle. The role of the ll-one vector 3 Though minimizing network coding resources is not its min concern, [12] presents n lgorithm to otin sugrph with miniml numer of coding links.

8 28 M. Kim et l. s fesile strting point is crucil to the performnce of the lgorithm s discussed in [8]. 4.3 Experimentl Results Results for the oth experiments re summrized in Tle 3. The tle shows the optiml fitness chieved y ech lgorithm verged over 30 runs. The sttisticl significnce of the difference etween BTS nd BLS encodings is mesured y conducting pired t-tests nd the p-vluesre reported in the lst rowof the tle. Tle 3. Performnce of the lgorithms for ech network. Ech vlue in rckets is stndrd devition. I-50 I-75 II-3 II-7 II-15 II-31 NCGA/BLS 3.33(1.03) 6.43(1.30) 0.93(0.69) 2.20(1.27) 5.57(1.55) 12.43(2.37) (w/o greedy sweep) 3.33(1.03) 39.93(2.74) 0.93(0.69) 2.20(1.27) 5.57(1.55) 12.43(2.37) NCGA/BTS 2.40(0.62) 3.63(0.61) 0.00(0.00) 0.00(0.00) 0.17(0.38) 1.03(0.81) (w/o greedy sweep) 2.40(0.62) 3.63(0.61) 0.00(0.00) 0.00(0.00) 0.17(0.38) 1.07(0.83) Miniml (1.37) 9.50(2.16) 3.00(0.00) 7.00(0.00) 15.50(0.00) 31.00(0.00) Miniml (1.37) 7.90(1.71) 2.13(0.86) 4.37(1.25) 9.90(1.65) 19.97(2.66) p-vlue 8.21e e e e e e 32 Experiment I: For oth networks I-50 nd I-75, the NCGA with greedy sweep, with either of the two encodings, outperforms the two existing miniml pproches. Between the two encodings, BTS encoding gives rise to sustntil performnce gin over BLS encoding with the sttisticl significnce confirmed y the tulted p-vlues. Experiment II: Agin the NCGA with BTS encoding outperforms tht with BLS encoding on verge for ll networks, while either of the two performs significntly etter thn the miniml lgorithms. For networks II-3 nd II-7, the NCGA with BTS encoding finds the optiml (0 coding links) in ll of the 30 runs. For networks II-15 nd II-31, it succeeds to find the optiml solution 25 nd 8 times, respectively. On the other hnd, BLS encoding does not find the optiml numer of coding links in ny of the 30 runs for networks II-7, II-15, nd II-31. The verge performnce of the NCGA with oth encodings is plotted ginst the logrithm of the serch spce size in Fig. 3. The plot suggests liner scling of lgorithms s the serch spce size grows exponentilly. More dt points would lend more confidence to this hypothesis. The curve for BTS encoding hs much smller intercept nd slope thn BLS encoding, implying tht the pyoff of the lock-wise opertors increses s the serch spce size increses. 4.4 Discussion of Results Experiment I clerly indictes tht BTS encoding is etter thn BLS encoding for the networks considered. We cn thus conclude tht the enefits of the smller

9 Genetic Representtions for Evolutionry Minimiztion BTS BLS 10 Averge Performnce Serch spce size (log) Fig. 3. Averge performnce of NCGAs with log of serch spce size serch spce trump the chllenge of the more difficult fitness lndscpe. For network I-50, BTS encoding improves over BLS encoding on verge y single coding link. Though smll, this difference is sttisticlly significnt. For network I-75, without greedy sweep, the verge difference in performnce etween the two lgorithms is much higher, i.e., 34 coding links. This lrge difference in performnce cn e ttriuted to two specific fctors: the much lrger serch spce size (see Tle 1) nd lrger verge lock size. The difference lso indictes tht the informtion on the intermedite solutions tht BLS encoding provides my not e prticulrly useful without gurnteeing tht those intermedite steps ultimtely led to n uncoded trnsmission stte. Experiment II demonstrtes the superiority, y remrkly lrge mrgin, of the lock-wise opertors over the it-wise opertors. It lso indictes tht oth NCGAs scle linerly with n exponentilly growing serch spce size (see Fig. 3), which is remrkle. This prompts due nlysis of the difference etween the two opertors. When pplied to the pir of locks 00 nd 11, the lock-wise crossover cnnot result in either lock 01 or 10. However, for the it-wise crossover, the pir of locks 00 nd 11 my result in 00, 01, 10, or 11. It cn e shown tht with proility 1 4 the two crossovers ehve differently, if the popultion hs equl frequency of ll lock types. Let us recll tht the lock-wise muttion leds to smller numer of chnged its on verge thn the it-wise muttion (Section 3). Nevertheless, the lock-wise muttion exhiits higher explortory power thn the it-wise muttion in the sense tht it is more likely to led to chnges in multiple its. For the lock-wise muttion, given ny lock, the remining three locks re eqully likely to occur on muttion. Thus, if muttion rte is α, the proilities of 0, 1, 2-it chnge re 1 α, 2 3 α, 1 3α, respectively, wheres those proilities in the it-wise cse re (1 α) 2,2α(1 α), α 2, respectively. Provided tht α< 1 3, the proility of 2-it chnge is lrger for the lock-wise muttion. A similr nlysis cn e done for the whole genotype s well. One my speculte tht the etter performnce of the lock-wise opertors is due to the higher explortory power of the lock-wise muttion rther thn the modulrity of the opertors. To confirm the contrry, we consider new set of

10 30 M. Kim et l. Tle 4. Performnce of NCGA with MHD opertors. Refer to Tle 3 for comprison with NCGAs with it-wise or lock-wise opertors. II-3 II-7 II-15 II-31 NCGA/MHD 0.77(0.68) 2.47(1.33) 5.83(1.68) 12.63(3.23) opertors, clled the Mtched Hmming Distnce (MHD) opertors, where the MHD muttion leds to the sttisticlly sme Hmming distnce chnges s the lock-wise muttion, ut exhiits no positionl is s to where the muttion is pplied, nd the MHD crossover is the sme s the it-wise crossover which neither imposes modulrity. From Tle 4 compred with Tle 3, we oserve tht the MHD opertors perform similrly s the it-wise opertors, ut fr worse thn the lock-wise opertors. We cn thus confidently clim tht the respect for modulrity enforced y the lock-wise opertors is the min cuse of the superior performnce of the lock-wise opertors. 5 Conclusions nd Future Work For our suite of network coding prolems, we hve found tht the enefits of the smller serch spce nd modulr opertors trump the chllenge of the more difficult fitness lndscpe. In the future, we will study the effect of exploiting further modulrity with BTS opertors tht cross over t merging node oundries nd perform intr-lock muttions. We could incorporte hierrchicl modulrity using domin knowledge of sprsely connected regions, regions with similr structure or simply neighoring nodes. A hierrchicl structure of crossover oundries could e formed nd pplied with different proilities. These results will inform the GA community nd help push the stte-of-rt in lgorithms for minimizing network coding resources. References 1. Ahlswede, R., Ci, N., Li, S.Y.R., Yeung, R.W.: Network informtion flow. IEEE Trns. Inform. Theory 46(4) (2000) Li, S.Y.R., Yeung, R.W., Ci, N.: Liner network coding. IEEE Trns. Inform. Theory 49(2) (2003) Frgouli, C., Le Boudec, J.Y., Widmer, J.: Network coding: An instnt primer. SIGCOMM Comput. Commun. Rev. 36(1) (2006) Richey, M.B., Prker, R.G.: On multiple Steiner sugrph prolems. Networks 16(4) (1986) Lngerg, M., Sprintson, A., Bruck, J.: The encoding complexity of network coding. In: Proc. IEEE ISIT. (2005) 6. Jggi, S., Snders, P., Chou, P.A., Effros, M., Egner, S., Jin, K., Tolhuizen, L.: Polynomil time lgorithms for multicst network code construction. IEEE Trns. Inform. Theory 51(6) (2005) Ho, T., Koetter, R., Médrd, M., Krger, D.R., Effros, M.: The enefits of coding over routing in rndomized setting. In: Proc. IEEE ISIT. (2003)

11 Genetic Representtions for Evolutionry Minimiztion 31 8.Kim,M.,Ahn,C.W.,Médrd, M., Effros, M.: On minimizing network coding resources: An evolutionry pproch. In: Proc. NetCod. (2006) 9. Kim, M., Médrd, M., Aggrwl, V., O Reilly, U.M., Kim, W., Ahn, C.W., Effros, M.: Evolutionry pproches to minimizing network coding resources. In: Proc. IEEE Infocom (to pper). (2007) 10. Koetter, R., Médrd, M.: An lgeric pproch to network coding. IEEE/ACM Trns. Networking 11(5) (2003) Melnçon, G., Philippe, F.: Generting connected cyclic digrphs uniformly t rndom. Inf. Process. Lett. 90(4) (2004) Frgouli, C., Soljnin, E.: Informtion flow decomposition for network coding. IEEE Trns. Inform. Theory 52(3) (2006)

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