Some bounds on the capacity of communicating the sum of sources

Some bound s on the capacity of communicating the sum of sources Brijesh Kumar Rai, Bikash K umar Dey and Sagar Shen vi Departmen t of Electrical Engineering Indian Institute of T echnolo gy Bombay Mumbai, India, 40 0 076 { bkrai, bikash,sagar s } @ee.iitb .ac.in Abstract — W e consider directed acyclic networks with multi- ple sourc es and multiple terminals where eac h sour ce generates one i.i.d. random process ov er an abelian gro up and all the terminals want to r ecover the sum of these random processes. The different source processes ar e assumed to be independent. The solvability of such networks has been considered in some prev ious works. In this paper we inv esti gate on the capacity of such n etworks, referre d as sum-networks , and present some bounds in terms of min-cut, and the numbers of sources and terminals. I . I N T RO D U C T I O N The seminal w ork by Ahlswede et al. [1] started a ne w regime of comm unication in a network where inter mediate nodes are allo wed to c ombine incoming information to construct o utgoing symb ols/packets. This has b een popu larly known as network coding. It was sh own th at the capacity of a multicast network un der network c oding is the min imum of the min- cuts o f the in dividual termina ls fro m the sourc e. The mu lticast capacity un der routing may b e strictly less than th at with co ding. The area has subsequen tly seen rapid developments. Lin ear coding was pr oved to be sufficient to ach iev e capacity of a m ulticast ne twork in [ 2]. Koetter and M ´ edard [3] proposed a dif ferent framew ork of rand om and determ inistic linea r network codin g, and Jag gi et. al [4] pr oposed a poly nomial time algo rithm for de signing a linear n etwork cod e for a multicast n etwork. Th e capacity of networks with routin g and network codin g was in vestigated in [5], [6], [7]. In this paper , we consider a directed ac yclic n etwork with multiple sources and terminals where the sources generate one rando m process each and the termin als r equire the sum of those pro cesses. W e call such a network as a sum-n etwork . The alphabet of the sourc e processes is assumed to be a finite abelian gr oup G , and the sum is defined as the oper ation in G . W e allo w frac tional vector network cod ing where the number k of sums communic ated to the terminals may be different fr om the vector d imension l . The ca pacity is then defined natur ally as the suppremu m of all rates k /l wh ich are achiev able. Wh en the alphabet is a field or more generally a module ov er a commu tativ e r ing with iden tity , the capacity achieved by using only linear cod es over tha t rin g is referred as the linear cod ing ca pacity . The pr oblem of distributed fun ction comp utation h as b een considered previously in the li terature in different fl av o rs (see [8], [9], [10], [ 11], [12], [ 13] for e xample). In th e context of network coding, and along the same line a s our present work, commun icating the sum of the sour ces has been co nsidered in several past works. Ramamoor thy ([14]) showed that if the number of sources or the n umber of terminals is no t more than two, then the sum of th e sources can be co mmunic ated if and only if each source-term inal pa ir is connected. On the other ha nd, ther e are networks ([1 5], [1 6]) with more than two sou rces and terminals whe re the su m can n ot be commun icated at rate on e even though every source-term inal pair is co nnected . In [15], [17], [ 18], the au thors sho wed th e richness of th is pr oblem as a c lass by showing existence of networks which are linearly solvable on ly over finite fields of characteristics belon ging to a giv en finite o r Co- finite set of primes, existence o f network which is solvably equiv alent to any (non-func tion) general network coding, and thus equi valent to any given system of polynomial eq uations [19]. It was also shown th at by using a co de construction originally giv en in [2 0], any fractio nal co ding solution of a sum-network also naturally provides a fraction al cod ing solution o f the same r ate f or the r ev erse network. The case of on e terminal and mor e ge neral fu nctions have been considered in [21], [22]. In this pape r , we con sider the prob lem of comm unicating the sum o f the sou rces over a network to a set of terminals and in vestigate the capacity of such networks. The exact characterizatio n of th e capac ity seems to be diffi cult and we present some bo unds, and find the capacity exactly for some interesting networks with three source s and th ree terminals. The pa per is organ ized as follows. In Section II, we formally introduce the system model a nd some p reliminary definition. The results of the paper , that is, the bounds on the capacity of sum-networks are presented in Section III. W e end with a discu ssion in Sectio n IV. I I . S Y S T E M M O D E L A N D D E FI N I T I O N S W e consider a directed acyclic multigraph G = ( V , E ) , where V is a finite set of nodes and E ⊆ V × V is the set o f edg es in the network. For any edge e = ( i, j ) ∈ E , the n ode j is called the head of the edge and the no de i is called the tail of the ed ge; and are deno ted as head ( e ) and tail ( e ) respecti vely . For each no de v , I n ( v ) = { e ∈ E : head ( e ) = v } is the set of incoming edges at the node v . Similar ly , Ou t ( v ) = { e ∈ E : tail ( e ) = v } is the set of outgoin g edges from the no de v . Each edge in the network is c apable of carrying a symbol from the alphabet in each use. Each edge is used once per unit time and is assumed to b e zero-erro r and zero -delay commun ication chan nel. A n etwork code is an assignm ent of an edg e function to each ed ge and a decodin g func tion to e ach term inal. In a ( k , l ) fractio nal network co de, k symbols generated at each source are blocked and encode d into l -length vectors on the ou tgoing ed ges. All the internal edges also carry l -leng th vectors. Thus for a ( k , l ) fra ctional network code over G , an edge fun ction for an edge e , with tail ( e ) = v , is defined as f e : G k → G l , if v ∈ S (1) and f e : G l | I n ( v ) | → G l , if v / ∈ S. (2) A decod ing functio n fo r a terminal v is defined as g v : G l | I n ( v ) | → G k . (3) The g oal in a sum-n etwork is th at the terminals should be able to recover the sum of the k -len gth vectors gener ated at the sources. A ( k , l ) fractional network code over G is called a k -length or a k - dimension v ector n etwork code over G if k = l and called a scalar network code over G if k = l = 1 . A network code is called a linear n etwork co de when the alph abet is a field, o r more generally a m odule over a co mmutative r ing with iden tity , and all th e edge functio ns and the deco ding fun ctions are linear functions over th e alphabet field or the r ing. No te that ev en if the alp habet is an abelian gro up, o ne can talk ab out a lin ear solution by considerin g the abelian group as a module ov er th e integer ring and then a co de can be linea r over the integer r ing. A network has a solution over G using a ( k, l ) fractional network code over G if the demand of each term inal node is fu lfilled u sing some ( k, l ) fractional network cod e over G . The ratio k/l is the rate o f th e ( k , l ) fractional network code. A rate k /l is said to be achievable if ther e is a ( k, l ) fractional solution for th e network. The suppremum of all achiev able rates is defined to be the ca pacity . The linear network codin g capacity o f a network is the supp remum of all rates th at are ach iev ab le using fractiona l linear network codes. Clearly , the n etwork cod ing ca pacity of a network is greater th an or equal to the linear network coding cap acity . A sum-network is said to be solvable (resp. lin early solvable) if it has a (1 , 1) codin g (re sp. linear coding) solution . I I I . C A PAC I T Y O F S U M - N E T W O R K S First, we men tion the following simple lower bound on the capacity of any sum-network. Theor em 1 : T he capacity of a sum- network is boun ded by the min imum of the min-cu ts of all so urce-term inal pairs. That is, C apaci ty ≤ min i,j ( min-cut ( s i − t j )) . Pr oo f: For any sour ce s i of the network , let u s fix the source p rocesses of the o ther sou rces to the all-zer o (”z ero” being the identity element of the alphab et group) sequen ce. Then the problem red uces to the multicast problem f rom the source s i to all the terminals, and the cap acity of this problem is the minimum of the min-cuts from S i to all th e terminals. The overall cap acity of the sum-network must be less than or equal to each o f these mu lticast capacities f or different i . In [17], [ 18], the re verse of a sum-network was considered where the direction o f the edges a re re versed and the role of sources and ter minals is interch anged. It was shown, u sing a c ode-con struction originally described in a basic f orm in [20] as the dual cod e in the language of codes on graphs, that if a sum-network has a ( k , n ) fr actional linear solution, then from such a network code, one can also construct a ( k , n ) f ractional linear solution of the r ev erse sum-n etwork. This means that the linear coding capacity of the reverse sum-network is the same as the linear codin g cap acity o f th e original sum-network. Our lo wer bou nds on the capacity of sum-networks have varying d egree of tigh tness depe nding on the number of sources and the n umber of terminals o f the network. So we present these bo unds in different subsections d ealing with various numbers o f sou rces and terminals. For the rest of the paper, m an d n will d enote the number of so urces and the num ber of termin als resp ectiv ely . A. Th e case of min { m, n } = 1 If the sum-network has on ly one so urce, then the netw ork is a mu lticast network. T he capacity of a multicast network is k nown to be equal to the minimu m of the min-cu ts of the source-ter minal pairs, and thus the cap acity achieves the min-cut upper bou nd. Moreover , th is capacity is achieved by linear codes if alphab et is a finite field. Now , over a finite field, for the case of n = 1 , let us consider the reverse network of a sum-network obtained by reversing th e direction of th e edges an d intercha nging the role o f the sources and the term inals. Th e reverse network is a multicast ne twork and th us has linear coding capacity equ al to the m inimum of the min-cu ts of the source-termin al pairs. So the linear coding capacity o f the original one-ter minal sum-network is the minimum of the min-cuts of the sou rce-term inal pairs. Since the c oding capacity is also upp er bo unded by the m in- cut, the coding cap acity of a one-term inal sum-network is the minimum of the min-cuts of the sou rce-term inal pairs. So we have Theor em 2 : T he capacity (an d the linear coding capacity) of a one-source or one-terminal sum-network is the m inimum of the min- cuts of all sour ce-termin al pairs. B. Th e case of min { m, n } = 2 It was proved in [14] that for a network with min { m, n } = 2 where every sour ce-terminal pair is con nected, i t is possible to co mmunica te th e sum of the sources to the terminals (at rate 1 ) . Which means that for min { m , n } = 2 , C apacity ≥ 1 if th e minimu m of the min-cu ts of th e sou rce-term inal pairs is at least 1 . So, the m in-cut bound is tight in this c ase if the min-cu t is 1 . Howe ver , if the min-cut is greater than 1 , then it is not known if this upp er bound is achie vable. Howe ver, we can always achieve the half of th e m in-cut upper bound by time-sharing. For example, f or m = 2 , each source can communicate its symb ols in one slot at the rate of min-cu t and then after two time-slots, th e terminals ca n add the symb ols r eceiv ed f rom the two sources. So, we ha ve Theor em 3 : For min { m, n } = 2 , the capacity of a sum - network is bounded as C apaci ty ≥ max { min { 1 , mi n i,j ( min-cut ( s i − t j )) } , 0 . 5 × min i,j ( min-cut ( s i − t j )) } . C. The ca se of m = n = 3 The case of m = n = 3 is intriguin g. On one han d, these are the smallest values of m, n for which there is a network (c alled S 3 in [ 15] and shown in Fig. 1 ) where every source-term inal pair is conn ected, i.e., which h as min- cut ≥ 1, b ut still do es n ot h av e a linear [15] o r non-linear [16] solution o f rate 1 . So, these are the smallest parameters for which the min -cut upper bo und is kn own to be not achiev able. (Th ough it is still not clea r at this po int if th e min-cut up per boun d may still be achiev ab le in the limit as the su ppremu m of achie vable rates.) On the other hand, from elaborate inv estigation of possible networks with these parameters, there seems to be very limited typ es of ne tworks. The S 3 and its extensions (essentially th e network shown in Fig. 2) seem to be the on ly ”no n-solvable” sum-networks fo r m = n = 3 . The network (let us call it X 3 ) shown in Fig. 3 was pr esented in [1 7] an d w as sho wn to be solvable by scalar linear cod e over a ll fields except the binary field F 2 . 3 1 s s 3 s 2 u 2 u 1 v 2 v 1 1 t t t 2 Fig. 1. The network S 3 First, we give a gene ric lower bou nd on th e capacity of any sum -network with min { m, n } = 3 with min-cut ≥ 1 . Theor em 4 : T he linear cod ing cap acity of any sum- network with min { m, n } = 3 with min-cu t ≥ 1 is at least 2 / 3 . 1 3 s 1 s 3 s u u 1 u 2 v v t 1 t t 2 2 3 2 Fig. 2. The network S ′ 3 u u s s s 1 2 3 3 2 u 1 v 1 v 2 v 3 3 t 2 t 1 t Fig. 3. The network X 3 Pr oo f: W itho ut loss of generality , let u s assume that the numb er of terminals is 3 (otherwise consider the r ev erse network). Let us consider tw o sym bols at each source: X i 1 , X i 2 at s i for i = 1 , 2 , . . . , m . Let the two sums be denoted as S um 1 = P m i =1 X i 1 . and S um 2 = P m i =1 X i 2 . If we take two ter minals at a time, the resulting network has a capacity ≥ 1 as discussed in Section III-B using scalar linear network coding as propo sed in [1 4]. Now , the two sums S u m 1 and S um 2 can be communicated to all th e terminals in three time slots. In the first time slot, S um 1 is commu nicated to t 1 and t 2 . In the secon d time slot, S um 2 is comm unicated to t 2 and t 3 . In th e third tim e slot, S um 1 + S um 2 = P m i =1 ( X i 1 + X i 2 ) is communicated to t 1 and t 3 . Ha ving received S um 1 (respectively S um 2 ) an d S um 1 + S um 2 , the terminal t 1 (respectively t 3 ) can recover S um 2 (respectively S um 1 ) as well. So all the termin als recover the two sum s in thr ee time slots, thus achieving a rate 2 / 3 using linear cod ing. It was proved in [ 16] tha t if a sum- network with m = n = 3 has two edg e-disjoint path s between any source-ter minal pairs, then the network is lin early so lvable, that is, rate 1 is achiev able by scalar line ar coding. This gives the fo llowing bound . Pr op osition 5 : T he linear coding capacity of any sum - network with m = n = 3 with min-cut ≥ 2 is at least 1 . Now we show that the network S 3 and its extension S ′ 3 shown in Fig. 2 b oth have capacity exactly 2 / 3 whereas th eir min-cut upp er bound is 1 . So, there is a gap between the capacity and the min- cut up per boun d. Theor em 6 : T he capacity an d linear codin g capacity of S 3 and S ′ 3 is 2 / 3 . Pr oo f: Clearly , the network S ′ 3 is obtained from S 3 by add ing one direct edge from s 1 to t 1 , and sub dividing the edg e ( s 2 , t 1 ) and adding o ne ed ge into it fro m s 1 . So , the capacity and th e lin ear co ding cap acity of the n etwork S ′ 3 is at least that of S 3 . By the previous theor em, the rate 2 / 3 is achievable b y linear network coding in S 3 . No w we will show that the capacity of S ′ 3 is boun ded from above b y 2 / 3 . This will prove that both the n etworks ha ve the same capacity and linear coding capacity and that th ese are both 2 / 3 . Consider any ( k , l ) fractional network co ding solu tion of S ′ 3 . Let X 1 , X 2 , X 3 ∈ G k be th e me ssage blo cks gene rated at the thr ee sou rces. Let the edg es ( u 1 , v 1 ) and ( u 2 , v 2 ) carry the functions f ( X 1 , X 3 ) and g ( X 2 , X 3 ) resp ectiv ely . For any fixed values of X 1 and X 2 , th e set of m essages received by the terminal t 1 should be a one-one function of X 3 since the term inal can recover the sum X 1 + X 2 + X 3 which is a one-on e function of X 3 . Since the m essages o n ( s 1 , t 1 ) an d ( u 3 , t 1 ) are fixed by the values o f X 1 and X 2 , the message on ( v 1 , t 1 ) and thus f ( X 1 , X 3 ) must b e a one-one f unction of X 3 for a fixed value of X 1 . Now clear ly , for t 2 to be able to recover the sum, the function g should be such that one c an recover X 2 + X 3 from g ( X 2 , X 3 ) . Since t 2 recovers the sum X 1 + X 2 + X 3 , and it can re cover X 2 + X 3 from the message in ( v 2 , t 3 ) , it ca n also recover X 1 by subtracting. Now , t 3 receives f ( X 1 , X 3 ) on ( v 1 , t 3 ) (WLOG) and this is a o ne-one function of X 3 for any g iv en X 1 . So, having re covered X 1 , t 3 can r ecover X 3 from f ( X 1 , X 3 ) . Then by using g ( X 2 , X 3 ) receiv ed on ( v 2 , t 3 ) and the value o f X 3 , t 3 can also re cover X 2 . So, t 3 can recover all the origin al messages X 1 , X 2 , X 3 . Now ( X 1 , X 2 , X 3 ) takes a total of | G | 3 k possible values as a triple. On the oth er hand { ( u 1 , v 1 ) , ( u 2 , v 2 ) } is a cut between the sources and t 3 , and this cut can carry at most | G | 2 l possible different m essage-pairs. So, we ha ve | G | 2 l ≥ | G | 3 k ⇒ k /l ≤ 2 / 3 . The fo llowing o bservations lead us to b eliev e that th e network S ′ 3 is essentially the on ly ma ximal extension o f S 3 which has the same cap acity . 1) Furth er subdividing ( u 3 , t 1 ) and ad ding an edge from it to t 2 makes the network X 3 a subg raph of the resulting network, and thus the capacity of the network increases to 1 . 2) Also su bdividing ( s 1 , t 2 ) and adding an edge f rom it into t 1 does not change its capacity since there is already an edge ( s 1 , t 1 ) and the ne w edge can not carry any extra inform ation to t 1 . 3) Also sub dividing ( s 1 , t 2 ) and adding an e dge to it f rom s 2 giv es a strictly richer n etwork th an X 3 , an d thus the capacity of the network increa ses to 1 . 4) Instead of the edge ( s 1 , t 1 ) , if an edge ( s 2 , t 2 ) is added, then the resulting network (shown in Fig. 4) is strictly richer th an X 3 because the edges ( s 1 , t 2 ) and ( s 2 , t 2 ) can jointly carry mo re informatio n than an ed ge from u 3 to t 2 . So the resultin g network has capacity 1 even though it does not ha ve a binar y scalar solution (like X 3 ). These observations also lead u s to believe that Conjectur e 7 : The cap acity of a sum -network with m = n = 3 is either 0 , 2 / 3 or at lea st 1 . s s 3 s u 1 u 3 u 2 2 1 v 1 2 v t 1 t 3 t 2 Fig. 4. The network X ′ 3 D. Th e case of m, n > 3 This is the most ill-und erstood class of sum- networks. W e only ha ve what we suspect to b e a very loo se lo wer boun d on the capac ity of this class of networks. This lower b ound is o btained by similar co ding by time-sharing scheme as in the pro of of Th eorem 4 . Theor em 8 : T he linear cod ing capacity of a sum- network with min { m, n } ≥ 2 and min -cut ≥ 1 is at least 2 / min { m, n } . Pr oo f: The case of min { m, n } = 2 follows fr om Theorem 4. For min { m, n } > 2 , with out loss of generality , let us assume that n ≤ m . For even n , we c an group the terminals in to n/ 2 pa irs an d in each time slot com municate the sum of the source sym bols to one pair o f terminals. So, in n/ 2 time slots we can c ommun icate one sum of the sou rce symbols to all the ter minals thus ach ieving a rate 2 /n . For odd n , we can grou p the term inals into ( n − 3) / 2 pairs and one trip le. W e can communicate on e sum to ea ch pair in one time slot. So , we c an communicate two sums to a ll the pairs in ( n − 3) slots. Then using the same sche me as in the proof of Theor em 4, we can comm unicate two sum to the grou p of three te rminals in three time slots. So, in overall n time Solv abil ity Capaci ty min-cut = 1 min-cut > 1 min-cut = 1 min-cut > 1 min { m, n } = 1 Solv able Solv able = 1 = min-cut min { m, n } = 2 Solv able Solv able = 1 ≥ min-cut / 2 (l oose/ti ght?) m = n = 3 Networ k de pendent Solv able ≥ 2 / 3 (tight) ≥ max { 1 , min-cut / 3 } (loose!) min { m, n } = 3 Networ k de pendent ? ≥ 2 / 3 (ti ght) ≥ m in-cut / 3 (loose!) min { m, n } > 3 Networ k de pendent ? ≥ 2 / min { m, n } (loose!) ≥ min-cut / min { m, n } (loose!) T ABLE I S O LV A B I L I T Y A N D B O U N D S O N TH E CA PA C I T Y slots, we can comm unicate two sums to all the term inals in the network. This gives us a rate 2 / n = 2 / min { m, n } . W e belie ve that th is bou nd is very lo ose an d th ere is scope for improvement. Even th ough we failed to come up with an y achiev able scheme of h igher rate, we also failed to constru ct a network satisfying the bou nd with eq uality . I V . D I S C U S S I O N Some upper and lower bou nds on the capacity of commu- nicating the sum of sources to a set of terminals are presented in this paper . A decreasing degree of tigh tness is observed or suspec ted in these bound s as the number s of sources an d terminals increase. W e summarize the b ound s in T ab le I to bring out this o bservation. The p arenthe tic comments in th e table entries indicate the tig htness of the b ound as known o r conjecture d (indicated with an exclam ation (!) mark .) The interrog ation (?) mark as an entry indicates that nothin g is known abou t th e c ase. V . A C K N O W L E D G M E N T The work of B. K. Rai was sup ported in part b y T ata T eleservices I IT Bombay Center of Excellence in T elecomm (TICET). The work of B. K. Dey and S. Shen vi was supported in part by T ata T eleservices IIT Bombay Center of Excellence in T ele comm (TICET) and Bharti Centre for Communica tion. 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