Mnmum Cost Optmzaton of Multcast Wreless Networks wth Network Codng Chengyu Xong and Xaohua L Department of ECE, State Unversty of New York at Bnghamton, Bnghamton, NY 13902 Emal: {cxong1, xl}@bnghamton.edu Abstract Mnmum cost optmzaton for multcast wth network codng has attracted great research nterests. In ths paper, based on an nformaton model that dfferentates ntermedate nodes n a multcast network nto network codng, routng or replcatng nodes, a method s developed to solve the mnmum energy cost multcast problem n wreless networks. Besdes transmsson energy, many other mportant parameters can be convenently optmzed as well, such as the number of packets undergong network codng, the number of network codng nodes, as well as node classfcaton. Some smulatons results are shown to verfy that the proposed method s promsng, especally snce network codng may be expensve or mpossble for certan wreless nodes. Index Terms network optmzaton, wreless network, multcast network, network codng. the ablty to conduct network codng. In practce, some nodes may not be so powerful to conduct network codng, especally n wreless networks where nodes are lmted by battery power, computng power as well as communcaton capablty. Therefore, a more reasonable model should take node dfferentaton nto consderaton. An nterestng node dfferentaton scheme was consdered n [9], where the nodes are classfed nto three dfferent types: routng, replcatng or network codng. Nevertheless, only wre-lne network s consdered n [9]. In ths paper, by explotng the nformaton model wth node classfcaton, we wll set up an optmzaton framework to solve the mnmum energy cost multcast problem n wreless networks. For convenence, Table 1 lsts all the notatons to be used n the rest of ths paper. I. INTRODUCTION II. INFORMATION MODEL WITH NODE DIFFERENTIATION The basc dea of network codng s to make messages sent on a node s output lnk to be some functon of messages that arrved earler on the node s nput lnks. An example of such functons s the XOR functon, whch s the addton operaton (lnear) n fnte feld GF (2 nn ). The capacty of multcast networks wth network codng has been shown n [1] as mn tt TT Mncut(ss, tt), whch s the upper bound of the multcast rate. It has also been shown that lnear network codng s suffcent to acheve the multcast capacty [2-3] and the codng coeffcents necessary to acheve the capacty can be computed n polynomal tme [4-5]. Mnmum cost optmzaton for multcast wth network codng s one of the specal subjects n network codng and has attracted great nterests. When usng network codng, there are two types of mnmum cost optmzaton problems for multcast: ) Fnd the optmal subgraph to code over, and ) determne the code to use over the subgraph [6]. In ths paper, we focus on the frst problem. The optmzaton of sngle multcast and multple multcast connectons were formulated n [7] and the mnmum energy cost problem n wreless network was dscussed. A decentralzed algorthm was proposed n [8] wth whch the ntermedated nodes can determne network codng coeffcents accordng to local nformaton. Nevertheless, the nformaton model used n [7-8] s based on the assumpton that each of the nodes n the network has Consder network (VV, EE), where VV s the set of nodes (vertces), EE s the set of edges (drected lnks). Let cc be the non-negatve capacty of the edge (, jj) EE. In ths paper, we consder only the sngle source multcast problem over a network. Note that multple ndependent source multcast problems can be converted nto sngle source multcast problems [1]. Multcast sesson (ss, TT) stands for a multcast sesson whch has a sender ss VV and a set of recevers TT VV. The number of recevers s TT [1, VV 1], where VV s the number of nodes. If TT s equal to 1, then the problem becomes a uncast problem. On the other hand, f TT s equal to VV 1, t becomes a broadcast problem. To model all the possble recever sets n the multcast problem, let us defne PP as the power set of TT (except the empty set) and QQ as a set contanng all collectons of two or more dsjont sets n PP. We can order the elements n PP and QQ such that PP ll s the ll-th element n PP whle QQ mm s the mm-th element n QQ. If the recever set TT ncludes KK recevers, then there are 2 KK 1 non-emepty sets PP ll, ll = 1,, 2 KK 1. For example, f there are 3 recevers {1, 2, 3} for ths multcast flow, then PP = {{1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3} }, QQ = {{{1}, {2}}, {{1}, {3}}, {{2}, {3}}, {1}, {2,3}, {{2}, {1,3}}, {{3}, {1,2}}, {{1}, {2}, {3}}}. Correspondng to all the sets PP ll, for the flows n each
TABLE I NOTATIONS USED IN THE PAPER (VV, EE) A network, VV s the set of nodes, EE s the set of edges (drected lnks) c j Non-negatve capacty of edge (, jj) EE a j Non-negatve cost per unt flow pass through the edge (, j) E dd j Dstance between wreless nodes and jj (s, T) A sngle source multcast sesson whch has a sender ss VV and a set of recevers TT VV KK Number of recevers n TT PP Power set of T (except the empty set), PP ll s ts ll-th QQ ordered element A set contanng all collectons of two or more dsjont sets n PP, QQ mm s ts mm-th ordered element X j A 2 K 1 dmensonal nformaton flow vector assocated wth the edge (, j). x j (P l ) s ts ll-th element whch represents the nformaton flow common to and only common to the recevers n the set P l RR Transmsson rate r m A routng/replcaton varable assocated wth the set Q m at node, whch s the flow for each recever n the set Q Qm Q beng replcated wth each copy meant for a set n Q m n m A network codng varable assocated wth the set Q m at node, whch s the flow for each set of recevers Q Q m that merges to form one flow that reaches all the recevers z j n the set Q Qm Q Actual nformaton flow n each edge (, j) E edge (, jj), we defne XX as a 2 KK 1 dmentonal nformaton flow vector, where the ll-th element xx (PP ll ) represents the nformaton flow common to and only common to the recevers n the set PP ll that passes through the edge (, jj). Note that XX s the varable that we want to optmze. Note also that xx (PP ll ) s the overall nformaton flow that passes through the edge (, jj) and wll be receved by the recever tt. There are two basc constrants for the multcast network optmzaton problem: A. Edge constrant It means that the amount of nformaton common to all the sets PP ll PP that passes through the edge (, jj) can not be hgher than the capacty of the edge. We can set up the edge constran as PPll PP xx (PP ll ) cc, (, jj) EE. B. Node constrant For the sender ss, the n-lnk flow s 0, whle the out-lnk flow s RR. For the recever tt, the n-lnk flow s RR, whle the out-lnk flow s 0. For any ntermedate node, the n-lnk flow must be equal to the out-lnk flow. The rate RR s the transmsson rate (or the value of ss TT flow where ss s the sender and TT s the set of recevers). In many other research of the max-flow problem, RR has been served as the objectve functon for maxmzaton. In our case, RR s a constrant,.e., the transmsson rate that we want to meet at mnmum energy cost. Whle the rate constrants for the sender and the recevers are straght-forward, the rate constrants for ntermedate nodes are not so clear yet consderng the dfferentaton of the ntermedate node. To match the dfferent functons of these nodes, let us defne two extra varables rr mm and nn mm frst. The former wll be used to descrbe routng/replcatng nodes, whle the latter wll be used to descrbe network codng nodes. The varable rr mm s a routng/replcaton varable assocated wth the set QQ mm at node. It stands for the flow to each recever n the set QQ QQmm QQ that wll be replcated nto multple copes, wth each copy meant for a set n QQ mm. If the node s a routng node, then rr mm = 0. On the other hand, f the node s a replcatng node, then rr mm s equal to the rate of the nput flow. Let QQ mm be the number of sets n QQ mm, then we need to replcate each flow by QQ mm 1 tmes to get QQ mm flows. The varable nn mm s a network codng varable assocated wth the set QQ mm at node. It stands for the flows, each for a set of recevers QQ QQ mm, whch merge nto a sngle flow that wll reach all the recevers n the set QQ QQmm QQ. If the node s a network codng node, the nn mm s equal to the rate of the nput flow and QQ mm flows wll be network coded to a sngle flow. Now let us defne the rate constrants for the three types of ntermedate nodes. For routng nodes, nothng happens to the nformaton flow. For replcatng nodes, t replcates the packets and each copy of the packet n the out-lnk has to reach nodes n a set PP ll QQ mm. For example, for node, f there s one n-lnk flow common to a recever set PP 3 ={1,2,3}, and there are two out-lnks to the recever {1} and the recever set {2,3}, respectvely, then Q m = {1}, {2,3} whch s the set of recever sets. In ths case, QQ mm = 2 and we need to copy the packet QQ mm 1 = 1 tme. Now we have two copes of the packet, one for PP 1 = {1}, and the other for PP 2 = {2,3}. Then for PP 1 we have flow constrant (,jj ) EE xx (PP 1 ) = (jj,) EE xx jjjj (PP 1 ) + rr mm. Smlarly for PP 2 we have (,jj ) EE xx (PP 2 ) = (jj,) EE xx jjjj (PP 2 ) rr mm. Combnng these two cases, for the replcatng node, we have node constrant (,jj ) EE xx (PP ll ) = (jj,) EE xx jjjj (PP ll ) + mm :PP rr mm rr ll QQ mm mm QQ mm For network codng nodes, two or more flows wll be combned together by codng. For example, for node, f there are two n-lnks common to the recever {1} and the recever set {2,3}, and there s one out-lnk common to the recever set PP 3 ={1,2,3}, then Q m = {1}, {2,3}. In ths case, QQ mm = 2 and the two flows merge nto a sngle one. The constrant for PP 1 becomes (,jj ) EE xx (PP 1 ) = (jj,) EE xx jjjj (PP 1 ) nn mm, whereas the constrant for PP 2 s (,jj ) EE xx (PP 3 ) = (jj,) EE xx jjjj (PP 3 ) + nn mm. Combnng these two cases, for the network codng node, we have node constrant (,jj ) EE xx (PP ll ) = (jj,) EE xx jjjj (PP ll ) nn mm + nn mm mm :PP ll QQ mm mm QQ
Fg. 1 Lst of three types of ntermedate nodes and the assocated node constrants. Now, we can develop the node constrants n general. For the sender ss, we have (ss,jj ) EE xx ssss (PP ll ) = RR 1, tt TT, whch means that all the nformaton common to the recever tt passng through all out-lnks of the sender ss s RR 1. It s the rate of the nformaton flow that we want to support. Obvously, RR 1 can not be larger than the value of max flow rate RR. For the recever tt, we have (jj,tt) EE xx jjjj (PP ll ) = RR 1, tt TT, whch means that all the nformaton common to the recever tt passng through all n-lnks of the recever tt s RR 1. For each ntermedate node, we have (,jj ) EE xx (PP ll ) = (jj,) EE xx jjjj (PP ll ) + mm QQ, PP ll PP. The above results are lsted n Fg. 1. A node can be routng/replcatng/network codng node at the same tme. For example, consder an ntermedate node has four n-lnks: one for recever {1}, one for recever {2}, one for recever {3}, one for recever set {4,5}; and four out-lnks: one for recever {1}, one for recever set {2,3}, one for recever {4}, one for recever {5}. Then the ntermedate node s a routng node for the recever {1}, a network codng node for the recever set {2,3} and a replcatng node for the recever set {4,5}. III. MINIMUM COST MULTICAST PROBLEM We assume that the total cost of usng an edge s proportonal to the flow on t and aa s the non-negatve cost per unt flow pass through the edge (, jj) EE. Consderng the edge constrants and node constrants, t s straghtforward to optmze the flow allocatons for mnmzng the cost under the followng optmzaton framework: mmmmmmmmmmmmmmmm aa zz (,jj ) EE subject to, (, jj) EE, tt TT = RR, tt TT = RR, tt TT cc zz xx (PP ll ) ll:tt PP ll xx (PP ll ) 0, PP ll PP, (, jj) EE, rr mm 0, nn mm 0, mm, VV Edge Constrants: PP ll PP xx (PP ll ) cc, (, jj) EE (1) Node Constrants: xx ssss (PP ll ) ll:tt PP ll (ss,jj ) EE 1 xx jjjj (PP ll ) ll:tt PP ll (jj,tt) EE 1 xx (PP ll ) = xx jjjj (PP ll ) + (,jj ) EE (jj,) EE mm QQ PP ll PP, VV {ss, TT} Note that we have ntroduced varables zz correspondng to the actual rate of nformaton flow on each edge (, j) E. It s lmted by the maxmum flow rate to any recever n the edge (, jj) EE. We must also make sure that t s no more than the edge capacty c j. The objectve functon s the sum of zz weghted by the unt cost of transmsson the flow. As a result, the summaton result s the cost to transmt data to all recevers at rate RR 1. The edge constrants P l P x j (P l ) cc and cc zz xx (PP ll ) can be omtted f the cost varables aa are set to extremely large values. In ths case, the optmzaton problem may be easer to solve. We have not specfy the cost. In fact, many natural cost crtera can be used, such as the transmsson power, transmsson energy, number of network codng node (.e., nn(), where nn() = 1 f n m > 0 for some m, and nn() = 0, otherwse), or the number of network codng operatons (.e., nn mm ) [9]. IV. MINIMUM ENERGY MULTICAST OPTIMIZATION IN WIRELESS NETWORKS Wreless networks are a specal type of networks that has multcast advantage, whch means that f data s transmtted from node to node j, then all nodes whose dstance from s smaller than j can receve ths data for free. Ths s due to the broadcastng nature of wreless transmssons, and the transmtted sgnal strength attenuates rapdly along wth transmsson dstance. Ths broadcastng property of wreless transmssons makes the set of out-lnks from a node s a set that nclude all the wreless nodes wthn a certan fxed radus of the node. As shown n [7], the networkng optmzaton model n prevous sectons can be convenently modfed to takng the broadcastng nature of the wreless transmssons nto consderaton. Recall that the varable zz s defned as the actual nformaton flow rate of the edge (, jj). If another node k s farther away from the node than the node j,.e., dd dd,
then the unt cost parameters aa aa. Due to the broadcastng nature of wreless transmssons, the transmsson on the edge (, kk) can also be receved by the node j. Therefore, the flow from the node to the node j just need to satsfy the constrant zz + zz xx (PP ll ) kk (,kk) EE,aa aa {jj } ll:tt PP ll xx (PP ll ), tt TT whch s equvalent to kk (,kk) EE,aa aa zz xx (PP ll ) 0, tt TT. Consderng that wreless nodes are usually prmarly lmted n energy supply, we choose the energy usage as the optmzaton objectve. In ths case, the energy cost per unt flow of the edge (, jj) s aa, whch s proportonal to power of dd αα, where αα s path loss exponental. As a matter of fact, there are at least two ways to defne the energy-related parameters and formulate the energy-related optmzaton framework. The frst approach s the make edge capacty cc constant, whch means the receved sgnal power s constant, but the transmsson power varous accordng to the transmsson dstance. In ths case, the parameters aa are determned accordng to dstances dd. In contrast, an alternatve approach s to make the transmsson power to be constant, whle makng the edge capacty varyng accordng to the transmsson dstances. In our smulatons, we wll smulate both approaches. Based on the general optmzaton framework (1), we can formulate mnmum energy cost optmzaton for wreless networks as follows. mmmmmmmmmmmmmmmm (,jj ) EE aa zz ssssssssssssss tttt kk (,kk) EE,aa aa zz xx (PP ll ) 0, (, jj) EE, tt TT xx (PP ll ) 0, PP ll PP, (, jj) EE rr mm 0, nn mm 0, mm, VV Edge Constrants: PP ll PP xx (PP ll ) cc, (, jj) EE (2) Node Constrants: xx ssss (PP ll ) = RR 1, tt TT ll:tt PP ll (ss,jj ) EE xx jjjj (PP ll ) = RR 1, tt TT ll:tt PP ll (jj,tt) EE (,jj ) EE xx (PP ll ) = (jj,) EE xx jjjj (PP ll ) + mm QQ PP ll PP, VV {ss, TT} Note that EE s a subset of EE, where the major dfference s that f aa = aa then EE just ncludes one of two edges (, kk) and (, jj). For applyng the optmzaton framework (2), we can easly add other constrants as well, such as those assocated wth the node types, e.g., specfyng that the node s not a network codng node by nn mm = 0, mm, or specfyng that the node s a routng node by rr mm = 0, nn mm = 0, mm. We may also add constrants wth respect to other specal wreless transmsson propertes besdes the broadcastng nature addressed n ths paper. However, such study wll be reported elsewhere. V. SIMULATION In ths secton, we use smulaton to study the performance of the mnmum energy cost optmzaton framework. We consder the classcal butterfly network wth wreless transmssons nstead of wrelne transmssons. The network s a sngle-source multcast network wth one sender SS and two recevers YY and ZZ. The wreless transmsson ranges of the network nodes are shown n Fg. 2, where each dashed lne ndcates the broadcast crcle of the node n the center. All the other nodes (except the one n the center pont) nsde a broadcast crcle can receve the data sent by the center node. Consderng the two multcast recevers YY and ZZ, we have PP = {{YY}, {ZZ}, {YY, ZZ}} whch has three elements PP 1 = {YY}, PP 2 = {ZZ}, PP 3 = {YY, ZZ}. We have the set QQ = {{{YY}, {ZZ}}} whch conssts of only one element QQ 1 = {{1}, {2}}. In the optmzaton problem, the optmzaton varables are the nformaton flow rate vector X j assocated wth each edge (, j). Each vector X j has 3 elements correspondng to the recever sets n P, respectvely. Note that the wreless network s treated as a specal wrelne network wth broadcastng, so we can stll use the edge concept for smplfcaton. Fg. 2. Wreless network wth the classcal butterfly network topology. In the frst seres of experments n our smulatons, we set the energy cost coeffcents aa to 1. We also set the edge capacty cc to be unt. Wthout node type constrant, the nformaton flow rate vectors X j after the optmzaton procedure are annotated n Fg. 3. It clearly shows that the upper bound of the multcast rate, mmmmmm tt TT MMMMMMMMMMMM(ss, tt), whch s 2 unt flow per unt tme, can be acheved. Specfcally, the two date packets bb 1 and bb 2 can be transmtted to the two recevers at unt tme. To explan n detals, for the edge SSSS, [0,0,1] means the nformaton flow commons to the recevers set PP 3 = {YY, ZZ} s 1 because only the thrd tem s 1. Node WW s a network codng node, because the nformaton flows on the edge TTTT
(whch s common to PP 2 ) and on the edge UUUU (whch s common to PP 1 ) merge nto a sngle nformaton flow on the edge WWWW (whch s common to PP 3 ). Smlarly, we can fnd that the nodes TT, UU and XX are replcatng nodes. Fg. 3. Informaton flow vectors of the network optmzed wthout node type constrant. The node WW becomes network codng node. In contrast, wth node type constrant (more specfcally, the node WW does not have the ablty of conductng network codng), the nformaton flow vectors after optmzaton are shown n Fg. 4. In ths case, t s well known that the upper bound of the multcast rate cannot be acheved. Instead, the maxmum data rate achevable s 1.5. In other words, three data packets bb 1, bb 2, bb 3 can be transmtted to the destnatons YY and ZZ by usng two unts of tme. Fg. 4. Informaton flow vectors of the network optmzed wth node type constrant. Specfcally, the node WW can not conduct network codng. For the above two cases, the actual rates zz for all edges (, jj) are shown n the frst two rows n Table II. Specfcally, the case 1 s correspondng to Fg. 3, whereas the case 2 s correspondng to Fg. 4. The multcast advantage created by the broadcastng property of the wreless transmssons are clearly seen. For example, the edge TTTT has rate 0.5 nstead of 1 because the node WW can also receved the transmsson from TT to YY. By usng such a lower rate, more energy can be saved. Next, we smulated the cases wth non-unt aa. We just made some nodes movable so as to change the dstances between some nodes. Due to the mportance of the node WW, we moved WW closer to TT, whch made the values of the energy cost coeffcents aa TTTT, aa UUUU, aa WWWW changng to 0.5, 1.5, 1.8, respectvely. The case wth WW as network codng node s lsted as case 3 n Table II, whereas the case wth WW beng no network codng node s lsted as case 4. Both cases gves the same overall rate as case 1 and 2, but wth qute dfferent edge rates. In addton, the uneven rates due to the broadcastng property become even more phenomenon. Specfcally, some edges (such as TTTT) have rate 0, whch means some transmssons can be avoded. Note that n ths case the recevng node (such as WW) can stll receve nformaton because they can hear others transmssons. Ths type of ceasng transmssons surely can save more energy. The above smulatons were conducted wth the frst approach (.e., fxng edge capacty to be unt) dscussed n Secton IV. As the second approach, we also conducted smulaton by lettng edge capactes cc TTTT, cc UUUU, cc WWWW be 1.5, 0.5, 1.8, respectvely. We also let aa = 1/cc for all edges. Some smulaton results are lsted n Table II as the case 5, whch had node constrants. TABLE II. SIMULATION RESULTS OF FIVE DIFFERENT CASES Cases z j No. R Cost ST SU TW UW TY WX UZ XY XZ 1 2 6 1 1 0.5 0.5 0.5 1 0.5 0.5 0.5 2 1.5 4.5 0.75 0.75 0.5 0.5 0.5 0.5 0.5 0.25 0.25 3 2 7.3 1 1 0 1 1 1 0 0.5 0.5 4 1.5 5.15 0.75 0.75 0 0.5 1 0.5 0.5 0.25 0.25 5 1.5 4.78 0.75 0.75 0 0.5 1 0.5 0.5 0.25 0.25 VI. CONCLUSIONS In ths paper, by classfyng the ntermedate nodes n a multcast network nto network codng/replcatng/routng nodes, a method s presented to formulate and solve the mnmum energy cost multcast problem n wreless communcaton networks wth network codng. Both the node dfferentaton and the wreless broadcastng property are addressed. The resulted optmzaton framework may be convenent to nclude more wreless transmsson propertes dfferent from the conventonal wre-lne transmssons. Ths wll be our on-gong work. 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