Energy Benefit of Network Coding for Multiple Unicast in Wireless Networks

Energy Benefit of Net w ork Co ding for Multipl e Unicast in Wir el ess Net w orks Jasp er Goseling Jos H. W eb er IR CTR/CWPC, WMC Group IR CTR/CWPC, WMC Group Delft Univ ersity of T ec hnology Delft Univ ersity of T ec hnology The Netherlands The Netherlands j.goseling@ tudelft.nl j.h.weber@t udelft.nl Abstract W e show that th e maximum p ossible energy b enefit of net w ork co ding for m ul- tiple unicast on wireless net works is at least 3. This imp ro v es the previously kno wn lo w er b ound o f 2 . 4 from [1]. 1 In tro du ction T raditional routing solutions for comm unication net works k eep indep enden t streams of data separate. The idea of net w ork co ding is to allow no des in the net w or k to com bine indep enden t data streams. Some of the b enefits of net work co ding that ha v e b een demonstrated are increased throughput, reduced resource consumption and increased securit y , see e.g. [2] and the references therein. Our in terest is in the reduction in energy consumption in wireless net w orks offered b y net w ork co ding. The p oten tial of net work co ding to reduce energy consum ption is demonstrated using the example giv en in Figures 1 a nd 2 in whic h no des A and C need to exc hange bits x and y . Figure 1 shows a routing solution using 4 transmissions, whic h is the minim um p ossible n um b er if only routing is allo w ed. One can observ e that in this case transmissions 1  and 2  are useful only to no des C a nd A , resp ectiv ely . The netw ork co ding solution from Figure 2 use s 3 transmissions. Net w ork co ding allo ws transmission 3  to b e useful for b oth A and C , increasing effic iency . Without netw ork co ding 4 transmissions ar e required, whereas the net work co ding solution uses 3 transmissions. W e sa y that for this example the energy b enefit of netw ork co ding is 4 3 . The energy benefit of netw ork co ding dep ends on the net w ork top ology and the traffic pat tern. O ne can e.g. sho w that the net w ork obtained by extending the net w or k A B C x x 1  y y 2  Figure 1: Routing solution exc hanging bits x a nd y b etw een no des A and C . T ransmissions 1  and 2  are only useful to no des C a nd A r esp ectiv ely . A B C 3  x y x + y x + y Figure 2: Net w o rk co ding solution. T ransmission 3  is useful for both A and C . The b enefit of net w ork co ding for this configuration is 4 3 . from the prev ious example to a net w ork of man y no des on a lin e, allow s net work co ding to reduce energy consumption by a fa ctor 2. This example w a s first presen ted in [3], where the net w ork co ding b enefits w.r.t. throughput where discussed, t he energy b enefit, ho w ev er, fo llows easily . It w as sho wn in [1] that there exis t net w orks for which this factor is 2 . 4. Our aim is to find the maximum p ossible energy b enefit that net w ork co ding can offer for m ultiple unicast traffic in wireless netw orks. The con tribution in this w ork is a new low er b ound of 3 to this b enefit. In Se ction 2 w e define the net w ork a nd traffic mo del that w e use and state our problem more precisely . An o v erview of know n r esults in the literature is giv en in Section 3 a fter whic h w e presen t our result in Section 4 . Section 5 is used to prov e this result. Sec tion 6 prov ides a discussion on the obtained r esults and p ossible future w ork. 2 Mo d e l and Problem Statemen t Time is slotted. T o simplify notation in Section 5 we a llow no des to tr a nsmit more than once in each time slot. Alternativ ely we could ha v e rescaled time suc h that only one transmission from eac h no de o ccurs in a time slot. All tra nsmissions in the net w or k are broadcasts, i.e. transmissions are receiv ed b y all neigh b ours. The neighbours of a no de are a ll o ther no des in the net w o rk that are within a tra nsmission range that is equal and fixed for a ll no des in the netw ork. T ransmission is noiseless, no errors o ccur and there is no in terference at the re- ceiv ers. Although in terference do es o ccur in realistic net w orks, w e do not tak e it in to accoun t here. If in terference would b e part of the mo del, not all no des could transmit in the same time slot, a t the exp ense of the throug hput, but the num b er of t r a nsmissions that is required would b e the same. Since w e a r e not in terested in throughput, but only in energy consumption in the net w ork, w e do not tak e in terference in to accoun t. The traffic pattern that w e consider is m ultiple unicast. All sym b ols a re fro m the field F 2 , i.e. they are bits a nd additio n corresp onds to the xor op eration. The source of each unicast connection has a sequence of source sym b ols that need to b e deliv ered to the corresp onding receiv er. F or a source x , e.g., w e ha v e x =  . . . x ( t − 1) x ( t ) x ( t + 1) . . .  , with x ( t ) = 0, for t ≤ 0. W e call a net w ork together with a set of unicast connections a configuration. W e are inte rested in the energy consumption in the net w ork, wh ic h we define as the a v erage o v er time of the num b er of transmissions used to deliv er one sym b ol from eac h unicast connection. The energy b enefit of net w ork co ding for a configuration is defined as the ratio of the minim um energy consumption of an y routing solution and the minim um energy consumption of an y net w ork co ding solution, i.e. energy b enefit of net w or k co ding fo r a wireless multiple unicast configuration = minim um energy consumption of an y routing solution minim um energy consumption of an y net w ork co ding solution . In this pa p er we will refer to this ratio as the energy b enefit of netw ork coding , o r simply as the b enefit o f netw ork co ding. K Figure 3: Net w o rk with no des p osi- tioned at hexagonal lattice. Eac h edge of the net w ork consists of K no des. S ( x 1 ) S ( x 2 ) S ( x 3 ) S ( x 4 ) R ( x 1 ) R ( x 2 ) R ( x 3 ) R ( x 4 ) Figure 4: Sources S ( · ) and receiv ers R ( · ) f or first set of unicast connections defined on the net w o rk from Figure 3 . 3 Previous W ork The b est kno wn lo w er b ound on the maxim um energy b enefit of net work co ding ov er all p ossible configura t io ns is 2 . 4 [1]. The net w or k co de that was constructed to sho w this low er b ound, satisfies the prop erty that data sym b ols transmitted b y a no de are linear com binations o nly of source sym b ols that ha v e b een successfully deco ded b y tha t no de. The ratio na le b ehind this is that in this w a y informatio n in the net w o r k can b e constrained to the neigh b ourho o d of the pa t h b et we en the source and destination of the corresp onding unicast ses sion, a net w o r k co de design heuristic that w as in tro duced in [4]. In [5 ] it w as sho wn tha t for random net works the energy b enefit of net work co des satisfying the ab o ve prop erty is upper b o unded by 3. It is not kno wn if 3 is also an upp er b ound on the energy benefit of arbitrary netw ork codes o n arbitra r y configura tions, that is allowing co des tha t do not satisfy the a b o v e prop erty on netw orks with arbitrary top ology and traffic requiremen ts. In this w ork w e presen t a new lo w er b ound o n the energy b enefit of netw ork co ding. In the netw ork co de that w e construct, no des transmit linear c om binations of data sym b ols, for which the corresp onding source sym b ols hav e no t necessarily b een deco ded b y these no des. Our co de therefore do es not satisfy the ab ov e prop ert y . 4 Result W e presen t a result that is based on the configuration that consists of the net w ork in whic h the no des are lo cated on the hexagona l lattice with connectivit y as depicted in Figure 3 , to gether with the unicast sessions that are depicted in Figures 4 , 5 and 6 . The figures depict a netw ork with 4 no des on each edge and 4 unicast sessions of eac h t yp e, i.e. x i , y i and z i , i = 1 , . . . , 4. In general, we will consider net w orks with K no des on the netw ork edges and K sessions of eac h t yp e. The netw ork t o p ology that w e consider is equal t o the o ne used in [1]. Our traffic pattern, ho w eve r, is slightly differen t. W e discuss this in mo r e detail in Section 6 . Lemma 1. The m inimum ener gy c onsumption of any r outing solution for the c onfig- ur ation fr om Figur es 3 – 6 is 3  K 2  . Pr o of. In the minim um cost routing solution, sym b ols from eac h unicast connection follo ws the shortest route from source to r eceiv er, as depicted in Fig ures 4 – 6 . R ( y 1 ) R ( y 2 ) R ( y 3 ) R ( y 4 ) S ( y 1 ) S ( y 2 ) S ( y 3 ) S ( y 4 ) Figure 5: Second set of unicast con- nections defined on the net w ork from Figure 3 . R ( z 1 ) R ( z 2 ) R ( z 3 ) R ( z 4 ) S ( z 1 ) S ( z 2 ) S ( z 3 ) S ( z 4 ) Figure 6: Third set of unicast connec- tions defined on the netw ork from Fig- ure 3 . Lemma 2. F or the c onfig ur ation fr om F igur es 3 – 6 ther e exis ts a network c o ding solu- tion that has ener gy c onsumption 3  K +1 2  − 2  K − 2 2  . W e will prov e Lemma 2 in Section 5 b y constructing a net w ork co de tha t ac hiev es this b ound. The next theorem states our main result. Theorem 3. Ther e exist multiple unic ast wir eless networks for wh ich network c o di n g offers an ener gy b enefit of 3 . Pr o of. The result follows from Lemmas 1 and 2 by taking the limit of K to infinit y , i.e. lim K →∞ 3  K 2  3  K +1 2  − 2  K − 2 2  = 3 . 5 Net w o r k Co de C onstruc t ion In this section w e construct a netw ork co de for whic h the energy consumption is ac- cording to Lemma 2 . W e first in tro duce some notat io n. Le t ˜ x i ( t ) = i − 1 X τ = 0 x i − τ ( t − τ ) , i ∈ { 1 , . . . , K } , with ˜ y j ( t ) and ˜ z k ( t ) defined similarly . Also, let A [ P ] , B [ P ] , . . . , F [ P ] b e the neigh b ours of a no de P as depicted in Figure 7 . The code is defined b y the follo wing prop erties: 1. The data sym b o ls transmitted b y no des are linear com binations of inf o rmation sym b ols of the form ˜ x i ( t − δ x ) + ˜ y j ( t − δ y ) + ˜ z k ( t − δ z ) , where t ∈ N + is the time slot a nd δ x , δ y , δ z ∈ N and i, j, k ∈ { 1 , . . . , K } are p er no de constan ts, i.e. they ma y b e differen t for eac h no de, but are the same for all sym b ols transmitted b y a sp ecific no de. A [ P ] B [ P ] F [ P ] P C [ P ] E [ P ] D [ P ] Figure 7 : The neighbours of a no de P . i =1 , δ x =0 j = 1 , δ y =0 k =1 , δ z =0 Figure 8: V alues for i , j , k , δ x , δ y and δ z for some no des in the net w ork. Re- maining v alues follow from ( 1 ). 2. Let P ( t ) = ˜ x i ( t − δ x ) + ˜ y j ( t − δ y ) + ˜ z k ( t − δ z ) b e the sym b ol sent by no de P in time slot t . The sym b ols transmitted by its neigh b ours in that same time slot are A [ P ]( t ) = ˜ x i +1 ( t − δ x ) + ˜ y j − 1 ( t − δ y + 1) + ˜ z k ( t − δ z − 1) , B [ P ]( t ) = ˜ x i +1 ( t − δ x − 1) + ˜ y j ( t − δ y + 1) + ˜ z k − 1 ( t − δ z ) , C [ P ]( t ) = ˜ x i ( t − δ x − 1) + ˜ y j + 1 ( t − δ y ) + ˜ z k − 1 ( t − δ z + 1) , D [ P ]( t ) = ˜ x i − 1 ( t − δ x ) + ˜ y j + 1 ( t − δ y − 1) + ˜ z k ( t − δ z + 1) , E [ P ]( t ) = ˜ x i − 1 ( t − δ x + 1) + ˜ y j ( t − δ y − 1) + ˜ z k +1 ( t − δ z ) , F [ P ]( t ) = ˜ x i ( t − δ x + 1) + ˜ y j − 1 ( t − δ y ) + ˜ z k +1 ( t − δ z − 1) . (1) 3. The exception to the ab o v e tw o rules comes from all no des that are at an edge or corner of the net w ork. These no des transmit three data sym b ols in eac h time slot. If ( 1 ) dictat es that a no de should tr ansmit P ( t ) = ˜ x i ( t − δ x ) + ˜ y j ( t − δ y ) + ˜ z k ( t − δ z ), and the no de is at an edge or corner, it t r ansmits three differen t sym b ols: P x ( t ) = ˜ x i ( t − δ x ), P y ( t ) = ˜ y j ( t − δ y ) and P z ( t ) = ˜ z k ( t − δ z ). F or no tational conv enience later on let P ( t ) = P x ( t ) + P y ( t ) + P z ( t ) . (2) Note that P ( t ) is not actually transmitted by no des at edges or corners of the net w o rk, but only a notational shortcut. 4. Let R b e the receiv er of source z k , i.e. R is a no de on the left edge of the net w ork. Supp ose t hat in time slot t , R tra nsmits R z ( t ) = ˜ z k ( t − δ z ). In that same time slot no de R deco des source sym b ol z k ( t − δ z ). This implies that, after time slot δ z , one source sym b ol fr om z k is deco ded eac h time slot. Similar deco ding pro cedures are used at all other receiv ers. 5. The only thing tha t remains to b e specified is the v alue of i , j , k , δ x , δ y and δ z for all no des. W e o nly sp ecify some v alues at the corners of the net w ork. These are giv en in F igure 8 . The remaining v alues follow from ( 1 ). W e need to show t ha t the sc heme is v alid, i.e. that a ll no des are able to pro duce the required linear combinations and that all receiv ers are able t o deco de. W e need to analyze differen t cases, depending on the lo cat io n of the no de. W e distinguish b et w een no des tha t are a t corners of the net w o rk, at edges of the net w or k and the remaining no des, whic h w e r efer to as in ternal no des. Since o ur construction is symmetric and homogeneous w e will consider only the no de in the top corner, an arbitra ry no de at the left edge, a nd a n arbitrary internal no de. Claim 1. L et P b e any internal no de. The symb o l P ( t + 1) tr ansmitte d by P in time slot t + 1 satisfies P ( t + 1) = A [ P ]( t − 1) + B [ P ]( t )+ C [ P ]( t − 1) + D [ P ]( t ) + E [ P ]( t − 1) + F [ P ]( t ) + P ( t − 2) . (3) Pr o of. Assum e that P ( t ) = ˜ x i ( t − δ x ) + ˜ y j ( t − δ y ) + ˜ z k ( t − δ z ). W e hav e A [ P ]( t − 1) + B [ P ]( t ) = y j ( t − δ y + 1) + ˜ z k ( t − δ z − 2) + ˜ z k − 1 ( t − δ z ) . C [ P ]( t − 1) + D [ P ]( t ) = ˜ x i ( t − δ x − 2) + ˜ x i − 1 ( t − δ x ) + z k ( t − δ z + 1) . E [ P ]( t − 1) + F [ P ]( t ) = x i ( t − δ x + 1) + ˜ y j ( t − δ y − 2) + ˜ y j − 1 ( t − δ y ) . The result follo ws from ˜ x i ( t − δ x + 1) = x i ( t − δ x + 1) + ˜ x i − 1 ( t − δ x ) and equiv alent relations for ˜ y j ( t − δ y + 1) and ˜ z k ( t − δ z + 1). Note that if some of P’s neigh b ours are on the b order of the net w o rk w e require ( 2 ). Claim 2. L et Q b e any no de on the le ft e dge of the network. Assume Q x ( t ) = ˜ x i ( t − δ x ) . The symb ols tr ansmitte d by Q in time slot t + 1 satisfy Q x ( t + 1 ) = x i ( t + 1 − δ x ) + E [ Q ] x ( t − 1) , (4) Q y ( t + 1 ) = B [ Q ] y ( t ) (5) and Q z ( t + 1 ) = B [ Q ] z ( t ) + C [ Q ]( t − 1) + D [ Q ]( t )+ E [ Q ] x ( t − 1) + Q x ( t − 2) . (6) Pr o of. Assum e that Q y ( t ) = ˜ y j ( t − δ y ) and Q z ( t ) = ˜ z k ( t − δ z ). Since Q is o n the left edge of the net w o r k it has only neighbours B [ Q ], C [ Q ], D [ Q ] and E [ Q ]. Noting that x i ( t + 1 − δ x ) is av ailable as a source sym b ol, the relations fo r Q x ( t + 1) and Q y ( t + 1) are readily v erified. F or Q z ( t + 1 ) we consider C [ Q ]( t − 1) + D [ Q ]( t ) = ˜ x i ( t − δ x − 2) + ˜ x i − 1 ( t − δ x ) + z k ( t − δ z + 1) and note that B [ Q ] z ( t ) = ˜ z k − 1 ( t − δ z ) and E [ Q ] x ( t − 1) = ˜ x i − 1 ( t − δ x ). Claim 3. L et R b e the no de in the top c orner of the network. The symb ols tr ansmitte d by R in time slot t + 1 satisfy R x ( t + 1 ) = x K ( t + 2 − K ) + E [ Q ] x ( t − 1) , (7) R y ( t + 1 ) = y 1 ( t + 1) (8) and R z ( t + 1 ) = D z [ R ]( t ) . (9) Pr o of. Note that from ( 1 ) and Figure 8 it follo ws that for R : i = K , j = 1, δ x = K − 1 and δ y = 0. The pro o f of ( 7 ) is equiv alen t to the pro of o f ( 4 ). Relations ( 8 ) and ( 9 ) can b e easily v erified. Pr o of of L emma 2 . First we need to prov e that t he sc heme is v alid. F rom Claims 1 – 3 it fo llo ws that a ll no des can pro duce the required linear com binations of sym b ols to transmit. W e also need to sho w that source sym b ols can b e deco ded. Consider no de Q on the left edge of the netw ork that is the receiv er of z k . Supp ose that in time slot t it transmits Q z ( t ) = ˜ z k ( t − δ z ). It can reco v er z k ( t − δ z ) as z k ( t − δ z ) = Q z ( t ) + B [ Q ] z ( t − 1) . No de R in the top corner do es not ha v e a neigh b our B [ R ], but it needs to de co de z 1 ( t − δ z ) for whic h ˜ z 1 ( t − δ z ) = R z ( t ) = z 1 ( t − δ z ). The con tributions t o the energy consumption are 1 for eac h of the  K − 2 2  in ternal no des and 3 for eac h o f the  K +1 2  −  K − 2 2  no des at the b o rder of the netw ork. 6 Discuss ion W e ha v e show n that the energy b enefit of net w ork co ding for m ultiple unicast in wireless net w o rks is at least 3. The net work to p ology that w e use in our constructiv e pro of is the same as used in [1] in whic h a lo w er b ound of 2 . 4 w as obtained. Th e difference is in the traffic pattern used. In [1] the num b er of unicast sessions is smaller than considered in this pap er. It is men tioned in [1] that for the n um b er of unicast sessions used in this pap er there do es not seem to b e a v alid net w o rk co de for whic h: 1) in ternal no des in the net w ork t r a nsmit only once p er deco ded s ource sym b ol and 2 ) data sym b ols transmitted b y a no de are linear com binations only of source sym b ols tha t hav e b een success fully deco ded b y that no de, i.e. satisfying the pro p ert y discussed in Section 3 . W e hav e obtained a v alid co de satisfying 1), but not 2). It is not kno wn if a co de satisfying b oth prop erties exists. The only know n upp er b ound to the energy b enefit of netw ork co ding for m ultiple unicast in wireless net works comes f r om [5], in whic h only a restricted class of net work co des is considered. It is an op en problem to find general upp er b ounds. References [1] M. Effros, T. Ho, and S. Kim, “A tiling approach to net w o rk co de design for wireless net w o rks,” in IEEE Information The ory Worksh op , 20 0 6, pp. 62 – 66. [2] C. F rago uli, J. Widmer, and J.-Y. Le Boudec, “Net w ork co ding: an instant primer,” A CM S IGCOMM Com p uter Co mmunic ation R eview , v ol. 36, no . 1, pp. 63–68, 2006 . [3] Y. W u, P . A. Chou, and S.-Y. Kung, “Info r mation exc hange in wireless netw orks with net w ork co ding and phys ical-lay er broadcast,” Microsoft Researc h, T ec h. Rep. MSR-TR-2004- 78, 2004. [4] S. Katti, H. Ra h ul, W. Hu, D . K a tabi, M. M´ edard, and J. Crow croft, “X OR s in the air: practical wireless net work co ding,” in Pr o c. of A CM SIGCOMM , 2006, pp. 243–254. [5] J. Liu, D. G o ec kel, and D. T o wsley , “Bounds on the g ain of net w ork co ding and broadcasting in wireless net w orks,” in Pr o c. of IEEE INFOCOM , 2007, pp. 6– 12.