Some results on communicating the sum of sources over a network

Some results on communicating the sum of s ources o v er a network Brijesh Kumar R ai, Bikash Kumar De y , and A bhay Karandikar Department of Electrical Eng ineering Indian Institute o f T echn ology Bombay Mumbai, India, 4 00 076 { bkrai,bikash,k arandi } @ee.iitb .a c.in Abstract —W e consider the pr oblem of communicat ing the sum of m sources to n terminals in a directed acyclic network. Recently , it was shown that f or a network of unit capacity links with either m = 2 or n = 2 , the sum of t he sources c an be communicated to the te rminals if a nd only if every source-terminal pair is connect ed in the network. W e show in this paper that f or any finite set of primes, there ex ists a network where the sum of the sources can be co mmunicated t o t he te rminals only over finite fields of charact eristic belonging to t hat set. As a corollary , this gives net works where the sum c an not be communicat ed over any finite field e ven t hough every source is connected to every ter minal. I . I N T R O D U C T I O N The semina l w ork by Ah lswede et al. [1] showed that in a multicast network, higher rates are achiev able if the intermediate nodes in the network perform s ome pro- cessing of the incoming information before forwarding. This has been pop ularly known a s network coding since then. Even thoug h the multicast capa city of a network under rou ting is difficult to c ompute or charac terize in general, the authors showed in [1] that under network coding, the multicas t capacity is given by the minimum of the min-cut capacities of the indi vidual terminals. Li et a l. [2] showed that the mu lticast ca pacity is also achiev able by linear network c oding, i.e., with each intermediate no de comp uting linear c ombinations of the incoming me ssages for trans mitting o n the outgoing links. An a lgebraic formulation was prese nted in [3]. It was also sh o wn that linear network cod es can be designed to be robust ag ainst link failure. J aggi et al. [4] gave a polynomial time algorithm for designing multicas t network codes . Recently , the prob lem of communicating the sum of sources to s ome termi nals was c onsidered by Ra mamoor- thy [5]. It was shown tha t if there are tw o sou rces o r two terminals in the network, then the sum of the so urces can b e communicated to the terminals if a nd o nly if ev ery source is conne cted to ev ery t erminal. Whereas t his condition is also neces sary for any number of source s and terminals, it may not be sufficient. Further , no neces sary an d s uf ficient condition is known for a rbitrary number of sources an d terminals. The p roblem of distrib uted function computation in general has be en c onsidered in d if ferent c ontexts in the past. Distributed event dete ction and d ata a ggregation techniques in se nsor n etwork ha s b een of significa nt interest [6], [7] since the av ailability of cheap efficient sensors . Distri buted compu tation of the su m/parity of the binary source s in a lar ge network was first conside red by Ga llager [8]. The re h as bee n significa nt interest in computation of such functions over random ge ometric graphs motiv ated by wireless sen sor network, e. g., [9], [10]. T here is also signific ant interest in the computation of su ch functions over arbitrary n etwork graphs , e .g., [11]. In this paper , we cons ider a d irected acyclic network. W e show that for every finite set o f prime nu mbers, there exists a directed acyclic network o f unit-capacity links with some sources an d terminals s o that the s um of the sources can be co mmunicated to a ll the terminals if and only if the cha racteristic of the a lphabet fi eld belongs to the giv en set. As a corollary , we find a ne twork of 3 sources and 3 terminals where the sum of the sources can no t be c ommunicated to the terminals over any fin ite field e ven though every s ource is conn ected to every terminal in the network. This example network w as also independ ently fou nd by Ra mamoorthy and Langberg [12]. In Se ction II, we introduc e the s ystem model. The results of this pap er are presented in Section III. W e conclude the paper with a dis cussion in Sec tion IV. I I . S Y S T E M M O D E L The network is repres ented by a directed acyc lic gra ph G = ( V , E ) whe re V is a finite set denoting the vertices of the network, E ⊆ V × V is the se t of edges. Among the vertices, there a re m source s s 1 , s 2 , · · · , s m ∈ V , and n terminals t 1 , t 2 , · · · , t n ∈ V in the network. In general, each terminal node may have a requirement of recovering some part of the sou rce me ssages or their functions. W e conside r a network where ea ch termi nal node requ ires to recover the s um of the source messages . For any edge e = ( i, j ) ∈ E , the node j will be called the head of the edge and the no de i will be called the tail of the edge ; a nd they will be denoted as head ( e ) and tail ( e ) res pectiv ely . Through out the pa per , p , po ssibly with subscripts, will denote a pos iti ve prime integer , and q will de note a power of a prime. Le t F q denote the alphabet field. Each link in the network is assumed to be capable of c arrying a symbol from F q in each use. Each symbol interval uses the ch annel once and this time is taken a s the unit time. For any e dge e ∈ E , let Y e ∈ F q denote the messag e trans mitted through e . In sc alar linear network coding, ea ch node comp utes a linear c ombination of the incoming s ymbols for transmission on an ou tgoing link. That is, Y e = X e ′ : head ( e ′ )= tail ( e ) β e ′ ,e Y e ′ (1) when tail ( e ) is not a source nod e. Here β e ′ ,e ∈ F q are called the local coding co ef ficients. A source node computes a linear combination of some data sy mbols generated at that source for transmission on an outgoing link, that is, Y e = X j : X j generated at tail ( e ) α j,e X j (2) for some α j,e ∈ F q if tail ( e ) is a source nod e. W e assume that each s ource ge nerates one sy mbol from F q per unit time. So, there is only one term in t he summation in (2) an d α j,e can be taken to be 1 without loss of generality . T he de coding ope ration at a termi nal in volves taking a linear combination of the incoming me ssages to recover the required da ta. In vector linear network coding , the data stream gen - erated at ea ch source node is blocked in vectors of length N . The c oding operations are similar to Eq. (1) and Eq. (2) with the diff erence that, now Y e , Y e ′ , X j are vectors from F N q , and β e ′ ,e , α j,e are matrices from F N × N q . It is known that sc alar linear ne twork c oding may gi ve be tter throughput in so me networks tha n that is achiev able by routing. V ector linear network c oding may gi ve further improvement over scalar linear network coding in some networks [13], [14], [15]. I I I . R E S U L T S A comp lete bipartite g raph K m,n has m nodes in one partition and n node s in the other . W e will assu me that the m no des in one partition are source s and the n nodes in the o ther partition are the terminals in the correspond ing network. All the e dges are a ssumed to be directed from the source partition to the terminal partition. Clearly , in the network K m,n , each source node can broad cast its messa ge to all the terminals an d each terminal node can thu s recov er a ny func tion of t he source messag es. In particular , e ach terminal no de c an rec over the sum of the source mess ages. W e now de fine a s pecial class o f networks. W e defi ne a network S m △ = ( V ( S m ) , E ( S m )) which has four laye rs of vertices V ( S m ) = S ∪ U ∪ V ∪ T . The first layer of no des are the m s ource nodes S △ = { s 1 , s 2 , . . . , s m } . The se cond and third layers have m − 1 nodes each, and they a re denoted as U △ = { u 1 , u 2 , . . . , u m − 1 } an d V △ = { v 1 , v 2 , . . . , v m − 1 } respec - ti vely . The last layer consists of the m terminal nod es T △ = { t 1 , t 2 , . . . , t m } . For every i = 1 , 2 , . . . , m − 1 , there is an e dge from s i to u i , from u i to v i , and from v i to t i . T hat is, ( s i , u i ) , ( u i , v i ) , ( v i , t i ) ∈ E ( S m ) for e ach i = 1 , 2 , . . . , m − 1 . For every i, j = 1 , 2 , . . . , m − 1 , i 6 = j , there is a n edge from s i to t j . Finally , for every i = 1 , 2 , . . . , m − 1 , there is an edge from s m to u i and from v i to t m . So, the set of ed ges is given by E ( S m ) = ∪ m − 1 i =1 { ( s i , u i ) , ( u i , v i ) , ( v i , t i ) } ∪ { ( s i , t j ) : i, j = 1 , 2 , . . . , m − 1 , i 6 = j } ∪ { ( s m , u i ) : i = 1 , 2 , . . . , m − 1 } ∪ { ( v i , t m ) : i = 1 , 2 , . . . , m − 1 } The ne twork is shown in Fig. 1. Now we define a method of combining two networks to obtain a larger network. W e call this method cr iss- cr ossing . Le t N 1 be a directed a cyclic network with s ome source nodes S 1 ⊆ V ( N 1 ) and some terminal nodes T 1 ⊆ V ( N 1 ) . Similarly let N 2 be a directed acyc lic network with some source n odes S 2 ⊆ V ( N 2 ) and s ome terminal n odes T 2 ⊆ V ( N 2 ) . W e ass ume that the nod es of N 1 and N 2 are labe led such that V ( N 1 ) ∩ V ( N 2 ) = φ . t t t t s s s s 1 2 m−1 m 1 2 m−1 m u u u v v v 1 1 2 2 m−1 m−1 Fig. 1. The network S m The c risscrossed n etwork N 1 ⊲ ⊳ N 2 has the node set V ( N 1 ⊲ ⊳ N 2 ) = V ( N 1 ) ∪ V ( N 2 ) , and the edge set E ( N 1 ⊲ ⊳ N 2 ) = E ( N 1 ) ∪ E ( N 2 ) ∪ ( S 1 × T 2 ) ∪ ( S 2 × T 1 ) . That is, other than the edges of N 1 and N 2 , their crisscross has edge s from the s ources of N 1 to the terminals of N 2 , and from the source s of N 2 to the terminals of N 2 . The crisscross of the ne tworks S 4 and K 2 , 3 is sh own in Fig. 2 for example. Now we presen t our main res ult. Theorem 1: For a ny finite set P = { p 1 , p 2 , . . . , p l } of positiv e prime numbers, there exists a directed acyclic network of unit-capa city edg es where it is p ossible to c ommunicate the sum of the source message s at rate one with linear network coding if and only if the characteristic of the alpha bet field belon gs to P . Proof: Define m = p 1 p 2 . . . p l + 2 . W e prov e that the network S m satisfies the con dition in the the orem. First, it may be n oted that every source-terminal pair in the network S m is c onnected . T his is clearly a nec essary condition for b eing able to communicate the sum of the source mess ages to eac h terminal nod e over any field. First we pro ve that if it is pos sible to c ommunicate the sum of the sou rce mess ages by vec tor linear n etwork coding over F q to all the terminals in S m , then the characteristic of F q must be from P . As in Eq. (1) and Eq. (2), the messa ge carried b y a n edg e e is d enoted by Y e . For i = 1 , 2 , . . . , m , the me ssage vector generated by the source s i is de noted by X i ∈ F N q . Ea ch terminal t i computes a linear combination R i of the received vectors. Local cod ing coe f ficients use d at dif ferent layers in the network are denoted by d if ferent symbols for clarity . The mess age vectors carried by different edges and the correspond ing local coding coe f ficients are as below . Y ( s i ,t j ) = α i,j X i for 1 ≤ i, j ≤ m − 1 , i 6 = j (3a) Y ( s i ,u i ) = α i,i X i for 1 ≤ i ≤ m − 1 (3b) Y ( s m ,u i ) = α m,i X m for 1 ≤ i ≤ m − 1 (3c) Y ( u i ,v i ) = β i, 1 Y ( s i ,u i ) + β i, 2 Y ( s m ,u i ) for 1 ≤ i ≤ m − 1 (3d) R i = m − 1 X j = 1 j 6 = i γ j,i Y ( s j ,t i ) + γ i,i Y ( v i ,t i ) for 1 ≤ i ≤ m − 1 (4a) R m = m − 1 X j = 1 γ j,m Y ( v j ,t m ) . (4b) Here all the coding c oefficients α i,j , β i,j , γ i,j are N × N matrices ov er F q , and the me ssage vectors X i and the messag es ca rried by the links Y ( .,. ) are length- N vectors over F q . W ithout loss of ge nerality (w .l.o.g.), we assume that Y ( v i ,t i ) = Y ( v i ,t m ) = Y ( u i ,v i ) . By a ssumption, each terminal dec odes the sum of all the source messag es. That is, R i = m X j = 1 X j for i = 1 , 2 , . . . , m (5) for all values o f X 1 , X 2 , . . . , X m ∈ F N q . From eq uations (3) and (4), we have R i = m − 1 X j = 1 j 6 = i γ j,i α j,i X j + γ i,i β i, 1 α i,i X i + γ i,i β i, 2 α m,i X m (6) for i = 1 , 2 , . . . , m − 1 , and R m = m − 1 X j = 1 γ j,m β j, 1 α j,j X j + m − 1 X j = 1 γ j,m β j, 2 α m,j X m . (7) Since (5) is true for all v alues of X 1 , X 2 , . . . , X m ∈ F N q , equations (6) a nd (7) imply γ j,i α j,i = I for 1 ≤ i, j ≤ m − 1 , i 6 = j (8) γ i,i β i, 1 α i,i = I for 1 ≤ i ≤ m − 1 (9) γ i,i β i, 2 α m,i = I for 1 ≤ i ≤ m − 1 (10) γ i,m β i, 1 α i,i = I for 1 ≤ i ≤ m − 1 (11) m − 1 X i =1 γ i,m β i, 2 α m,i = I (12) t t s s 1 2 1 2 u u v v 1 1 2 2 s t v u t s 3 3 4 4 s s t t t 5 5 6 6 7 K 2,3 S 4 3 3 Fig. 2. The network S 4 ⊲ ⊳ K 2 , 3 where I denotes the N × N ide ntity ma trix over F q . All the cod ing ma trices in equations (8),(9),(10), (11) are in verti ble sinc e the right hand side of the eq uations are the identity matrix. Equations (9) and (10) imply β i, 1 α i,i = β i, 2 α m,i for 1 ≤ i ≤ m − 1 . So, (12) gives m − 1 X i =1 γ i,m β i, 1 α i,i = I (13) Now , us ing (11), we get m − 1 X i =1 I = I (14) ⇒ ( m − 1) I = I (15) ⇒ ( m − 2) I = 0 . (16) This is true if and only if the cha racteristic o f F q divi des m − 2 . S o, the s um o f the source can be communicated in S m by vec tor linear n etwork coding at rate 1 only if the c haracteristic of F q belongs to P . Now , if the ch aracteristic o f F q belongs to P , then for any b lock length N , in particular for s calar n etwork coding for N = 1 , ev ery co ding matrix in (3a)-(3d) ca n be chos en to be the iden tity matrix. T he terminals then can recover the s um of the source messages by taking the sum of the incoming mess ages, i.e., b y tak ing γ j,i = I for 1 ≤ j ≤ m − 1 and 1 ≤ i ≤ m in (4a) and (4b). When the set P is a sing leton, The orem 1 giv es, as a special c ase, the following corollary . Cor ollary 2: For a ny prime numb er p , the s um of the source messag es can be commun icated to all the terminals by vector linear networ k c oding in the network S p +2 only over fie lds of c haracteristic p . Moreov er , over fields of charac teristic p , the s um c an be communicated in this network by sc alar network cod ing. When the set P is e mpty , The orem 1 g i ves, as a special ca se, a network wh ere the the sum of the source messag es can not be communicated using l inear n etwork coding over any fie ld. Cor ollary 3: In the network S 3 , the s um of the sources can not be c ommunicated to the terminals over any fi nite field. It was p roved in [5] that if m < 3 or n < 3 , then a ny network whe re every s ource-terminal p air is conne cted allows the sum o f the source s to be commu nicated to the terminals by linear network coding . The network S 3 , shown in Fig. 3, is an example of a network with m, n ≥ 3 where the su m of the source s can not be com- municated to the terminals by linear netw ork coding even though e very source-terminal pair is conne cted. This example was also found indepen dently by Rammoorthy and La ngberg [12]. Let P ( N ) deno te the se t of characte ristics of fields over which the sum of the s ources ca n be communicated to the terminals by linear n etwork c oding in the network N . W e have the following results. Theorem 4: For any two networks N 1 and N 2 , P ( N 1 ⊲ ⊳ N 2 ) = P ( N 1 ) ∩ P ( N 2 ) . Proof: For any two ne tworks N 1 and N 2 , a nd for a ny field F , it is po ssible to communica te the sum of the sources to the terminals in the c risscrossed network N 1 ⊲ ⊳ N 2 by linear ne twork cod ing over F if and only if it is pos sible to communica te the su m o f the source s to the terminals in the indi vidual networks N 1 and N 2 by linear network coding over F . So the result follo ws.  Cor ollary 5: For any network N , a nd a ny positive integers m, n , P ( N ⊲ ⊳ K m,n ) = P ( N ) . For any p ositi ve integer m , let Π( m ) d enote the set of prime f actors of m . Then T heorem 1 states that P ( S m ) = Π( m − 2) . The orem 4 then gives Cor ollary 6: For any two p ositi ve integers m and n , P ( S m ⊲ ⊳ S n ) = Π(gcd ( m − 2 , n − 2)) Theorem 4 toge ther with the two corollaries allow construction of a large clas s of ne tworks having arbitrary finite P ( N ) and arbitrary size. For example, for any positiv e m, n > 3 , one can construct a network by crisscrossing S 3 with an appropriate complete bipartite network to get a network with m sources and n terminals where the sum of the sources can not be communicated to the terminals by linea r network c oding ov er a ny fie ld. Such networks c an also be con structed by crisscross ing networks with disjoint Π . For instance, for any two prime numbers p 1 and p 2 , the sum of the sources can not be communicated to the terminals in the network S p 1 +2 ⊲ ⊳ S p 2 +2 . t s 2 2 u v 2 2 t s 1 1 u v 1 1 s t 3 3 Fig. 3. The network S 3 I V . D I S C U S S I O N W e h av e co nstructed networks wh ere the sum of the sources ca n b e communica ted by linear network cod ing only over fields o f a s pecified finite set of characteristics. A nec essary and s uf ficient con dition for being ab le to communicate t he sum of the sou rces over a field to a ll the terminals is still not kn o wn. Suc h a cond ition for fields of both zero and n on-zero chara cteristic a re of interest. Communication of the sum over fields like R and C of zero characteristic fields are of interest, for example, in sensor n etwork when the rea l measureme nt values are transmitted as analog values. Many o ther func tions of the source messag es ma y also be of prac tical interest. V . 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