Feasible alphabets for communicating the sum of sources over a network

0.12 MB·English

by  Brijesh Kumar Rai

Feasible alphabets for communicating the sum of
sources over a network
Brijesh Kumar Rai and Bikash Kumar Dey
Department of Electrical Engineering
Indian Institute of Technology Bombay
Mumbai, India, 400 076
{bkrai,bikash}@ee.iitb.ac.in
9
Abstract—We consider directed acyclic sum-networks with m Itis worth mentioningthatthe problemof distributedfunc-
0
sourcesandnterminalswherethesourcesgeneratesymbolsfrom tion computation in general has been considered in different
0
an arbitraryalphabetfieldF,and theterminalsneedtorecover
2 contexts in the past. The work in [10], [11], [12], [13], [14]
the sum of the sources over F. We show that for any co-finite
is only to mention a few.
n set of primes, there is a sum-network which is solvable only
a over fields of characteristics belonging to that set. We further Given a multiple unicast network, its reverse network is
J construct a sum-network where a scalar solution exists over all obtained by reversing the direction of all the links and inter-
5 fields other than the binary field F2. We also show that a sum- changing the role of source and destination for each source-
1 networkissolvableoverafieldifandonlyifitsreversenetwork destinationpair.Itisknown([15],[16])thatamultipleunicast
is solvable over the same field.
networkislinearlysolvableifandonlyifitsreversenetworkis
]
T linearlysolvable.However,therearemultiple-unicastnetworks
I. INTRODUCTION
I which are solvable by nonlinear network coding but whose
.
s After its introduction by the seminal work by Ahlswede et reverse networks are not solvable ([15], [16]).
c al. [1], the field of network coding has seen an explosion of In this paper, we consider a directed acyclic network with
[
interestanddevelopment.See,forinstance,[2],[3],[4],[5]for unit-capacity links. We prove the following results.

1 some early development in the area. The work by Dougherty • For every co-finite set of prime numbers, there exists a
v et al. in [6], [7] are specially relevant in the context of this directedacyclicnetworkofunit-capacitylinkswithsome
8
paper for the nature of results and the proof techniques. They sources and terminals so that the sum of the sources
9
1 definedanetworkwithspecificdemandsoftheterminalstobe can be communicated to all the terminals using vector
2 scalarlinearsolvable(resp.vectorlinearsolvable)overa field or scaler network coding if and only if the characteristic
1. Fq if there exists a scalar linear network code (resp. vector of the alphabet field belongs to the given set. This result
0 linear network code) over Fq which satisfies the demands of complements the result in [9].
9 all the terminals. A prime p is said to be a characteristic of • Weconstructanetworkwherethesumofthesourcescan
0 a network if the network is solvable over some finite field of be communicated to the terminals over all fields except
:
v characteristic p. They showed that for any finite or co-finite the binary field F . This shows that whether the sum of
2
i setofprimes,thereexistsanetworkwherethegivensetisthe the sources can be communicated to the terminals in a
X
set of characteristics of the network. network using scalar linear network coding over a field
r
a In most of the past work, the terminal nodes have been doesnotdepend onlyon the characteristic of the field. It
consideredtorequiretherecoveryofallorpartofthesources’ may also depend further on the order of the field.
data. A more general setup is where the terminals require • The sum of the sources can be communicated to the
to recover some functions of the sources’ data. Recently, terminals in a network over some alphabet field using
the problem of communicating the sum of sources to some linear network coding if and only if the same is true for
terminals was considered in [8], [9]. We call such a network the reverse network.
as a sum-network. It was shown in [8] that if there are two
Proof techniques of this paper are similar to that in [9].
sources or two terminals in the network, then the sum of the In Section II, we introduce the system model. The results
sources can be communicated to the terminals if and only of this paper are presented in Section III and Section IV. We
if every source is connected to every terminal. While this conclude the paper with a short discussion in Section V.
condition is also necessary for any number of sources and
terminals, it may not be sufficient. In [9], the authors showed II. SYSTEMMODEL
thatforanyfinitesetofprimenumbers,thereexistsanetwork A sum-network is represented by a directed acyclic graph
where the sum of the sources can be communicated to the G = (V,E) where V is a finite set denoting the vertices
terminals using scalar or vector linear network coding if and of the network, E ⊆ V × V is the set of edges. Among
only if the characteristic of the alphabet field belongs to the the vertices, there are m sources s ,s ,··· ,s ∈ V, and
1 2 m
given set. n terminals t ,t ,··· ,t ∈ V in the network. For any edge
1 2 n
e=(i,j)∈E, the node j will be called the head of the edge a network S∗ =△ (V(S∗),E(S∗)) which has four layers of
m m m
andthenodeiwillbecalledthetailoftheedge;andtheywill verticesV(S∗)=S∪U∪V ∪T.The first layer of nodesare
m
be denoted as head(e) and tail(e) respectively. Throughout the m−1 source nodes S =△ {s ,s ,...,s }. The second
1 2 m−1
the paper, p, possibly with subscripts, will denote a positive and third layers have m−1 nodeseach, and they are denoted
prime integer, and q will denote a power of a prime. Let Fq as U =△ {u ,u ,...,u } and V =△ {v ,v ,...,v }
1 2 m−1 1 2 m−1
denotethealphabetfield.Eachlinkinthenetworkisassumed
respectively. The last layer consists of the m terminal nodes
to be capableof carryinga symbolfromF in eachuse. Each
q △
T = {t ,t ,...,t }. For every i = 1,2,...,m−1, there is
linkisusedonceineverysymbolintervalandthistimeistaken 1 2 m
astheunittime.EachsourcegeneratesonesymbolfromF in an edge from si to ti, ui to vi, vi to ti, and from vi to tm.
q That is, (s ,t ),(u ,v ),(v ,t ),(v ,t ) ∈ E(S∗) for each
every symbol interval, and each terminal requires to recover i i i i i i i m m
the sum of the source symbols (over F ). i = 1,2,...,m−1. Also for every i,j = 1,2,...,m−1,
q
For any edge e ∈ E, let Y ∈ F denote the message i6= j, there is an edge from si to uj. So, the set of edges is
e q
given by
transmitted through e. In scalar linear network coding, each
node computesa linear combinationof the incomingsymbols
E(S∗)= ∪m−1 {(s ,t ),(u ,v ),(v ,t ),(v ,t )}
for transmission on an outgoing link. That is, m i=1 i i i i i i i m
∪ {(s ,u ):i,j =1,2,...,m−1,i6=j}
i j
Ye = X αe′,eYe′ (1)
e′:head(e′)=tail(e) The network is shown in Fig. 1.
when tail(e) is not a source node. Here αe′,e ∈Fq are called s s s
thelocalcodingcoefficients.Asourcenodecomputesalinear 1 2 m−1
combination of some data symbols generated at that source
for transmission on an outgoing link, that is,
Ye = X βj,eXj (2) u1 u2 um−1
j:Xj generated at tail(e)
for some β ∈ F if tail(e) is a source node. Since each
j,e q
source generates one symbol from F per unit time, there is
q
only one term in the summation in (2) and β can be taken v v v
j,e 1 2 m−1
tobe1withoutlossofgenerality.Thedecodingoperationata
terminalinvolvestakinga linear combinationof the incoming
messages to recover the required data.
In vector linear network coding, the data stream generated t t t t
1 2 m−1 m
at each source node is blocked in vectors of length N. The
codingoperationsaresimilarto(1)and(2)withthedifference
that, now Ye,Ye′,Xj are vectors from FNq , and αe′,e,βj,e are Fig.1. Thenetwork Sm∗
matrices from FN×N. It is known that scalar linear network
q Now we present a lemma which will be used to prove one
codingmaygivebetterthroughputinsomenetworksthanthat
of the main results of this paper.
is achievable by routing. Vector linear network coding may
Lemma 1: For any positive integer N, the network S∗ is
givefurtherimprovementoverscalarlinearnetworkcodingin m
N-lengthvectorlinearsolvableifandonlyifthecharacteristic
some networks [17], [5], [18].
of the alphabet field does not divide m−2.
A sequence of nodes (v ,v ,...,v ) is called a path,
1 2 l
Proof: Firstwenotethateverysource-terminalpairinthe
denoted as v → v → ··· → v, if (v ,v ) ∈ E for
1 2 l i i+1 networkS∗ isconnected.Thisisclearlyanecessarycondition
i = 1,2,...,l − 1. Given a network code on the network, m
l−2α is called the path gain of the path forbeingabletocommunicatethesumofthesourcemessages
Qi=1 (vi,vi+1),(vi+1,vi+2) to each terminal node over any field.
v →v →···→v .
1 2 l
We nowprovethatif itispossibletocommunicatethe sum
Asum-networkissaidtobeN-lengthvectorlinearsolvable
of the source messages using vector linear network coding
over F if there is a N-length vector linear network code so
q over F to all the terminals in S∗, then the characteristic of
that each terminal recovers the sum of the N-length vectors q m
over F generated at all the sources. Scalar linear solvability Fq must not divide m−2. As in (1) and (2), the message
q
carried by an edge e is denoted by Y . For i = 1,2,...,m,
of a sum-network is defined similarly. e
the message vector generated by the source s is denoted by
i
III. RESULTS X ∈FN. Each terminalt computesa linear combinationR
i q i i
In [9], a special network S was defined where the sum of the received vectors.
m
of the sources can be communicated to the terminals using Local coding coefficients/matrices used at different layers
scalar or vector linear network coding only over fields of in the network are denoted by different symbols for clarity.
characteristics dividing m −2. For m ≥ 3, we now define The message vectorscarried by differentedges and the corre-
sponding local coding coefficients are as below. Equations (8) imply β = β for 1 ≤ i,j,k ≤ m − 1,
j,i k,i
j 6=i6=k.So,letusdenotealltheequalco-efficientsβ ;1≤
Y = α X for 1≤i≤m−1, (3a) j,i
(si,ti) i,i i j ≤m−1,j 6=i by βi. Then (10) can be rewritten as
Y = α X
(si,uj) i,j i m−1
for 1≤i,j ≤m−1,i6=j, (3b) γ β =I for 1≤j ≤m−1. (11)
X i,m i
m−1
i=1
Y(ui,vi) = X βj,iY(sj,ui) i6=j
j=1 Equation (11) implies
j6=i
for 1≤i≤m−1, (3c) γ β =γ β for 1≤i,j ≤m−1,i6=j.
i,m i j,m j
Then (11) gives
R = γ Y +γ Y
i i,1 (si,ti) i,2 (vi,ti)
for 1≤i≤m−1, (4a) (m−2)γ β =I
1,m 1
m−1 ⇒γ β =(m−2)−1I. (12)
1,m 1
R = γ Y . (4b)
m X j,m (vj,tm)
j=1 Equation (12) implies that the matrix γ1,mβ1 is a diagonal
matrix and all the diagonal elements are equal to (m−2)−1.
Here all the coding coefficientsα ,β ,γ are N×N ma-
i,j i,j i,j
trices over F , and the message vectors X and the messages Buttheinverseof(m−2)existsoverthealphabetfieldifand
q i
carried by the links Y are length-N vectors over F . only if the characteristic of the field does not divide (m−2).
(.,.) q
So, the sum of the source messages can be communicated in
Without loss of generality (w.l.o.g.), we assume that
S∗ by N-lengthvectorlinear networkcodingoverF onlyif
Y = Y = Y and α = α = I for m q
(vi,ti) (vi,tm) (ui,vi) i,i i,j the characteristic of F does not divide (m−2).
1≤i,j ≤m−1,i6=j, where I denotes the N ×N identity q
Now,ifthecharacteristicofF doesnotdivide(m−2),then
matrix. q
foranyblocklengthN,inparticularforscalarnetworkcoding
By assumption, each terminal decodes the sum of all the
for N = 1, every coding matrix in (3a)-(3c) can be chosen
source messages. That is,
to be the identity matrix. The terminals t ,t ,··· ,t then
1 2 m−1
m−1 canrecoverthesumofthesourcemessagesbytakingthesum
R = X for i=1,2,...,m (5)
i X j of the incoming messages, i.e., by taking γi,1 = γi,2 = I for
j=1 1 ≤ i ≤ m − 1 in (4a). Terminal t recovers the sum of
m
for all values of X ,X ,...,X ∈FN. the source messages by taking γ ,1 ≤ i ≤ m−1 in (4b)
1 2 m−1 q i,m
From equations (3) and (4), we have as diagonal matrices having diagonal elements as inverse of
(m−2). The inverse of (m−2) exists over F because the
m−1 q
Ri = X γi,2βj,iXj +γi,1Xi (6) characteristic of Fq does not divide (m−2).
Lemma 1 gives the following theorem.
j=1
j6=i Theorem 2: For any finite set P = {p1,p2,...,pl} of
positive prime numbers, there exists a directed acyclic sum-
for i=1,2,...,m−1, and
network of unit-capacity edges where for any positive integer
N, the network is N-length vector linear solvable if and only
m−1 m−1 
R = γ β X if the characteristic of the alphabet field does not belong to
m Xi=1 i,mXj=1 j,i j P.
j6=i  Proof: ConsiderthenetworkSm∗ form=p1p2...pl+2.
ThisnetworksatisfiestheconditioninthetheorembyLemma
m−1m−1 
1.
= γ β X . (7)
X X i,m j,i j We note that the alphabet field in Theorem 2 may also
j=1 i=1 
i6=j  be an infinite field of non-zero characteristic. In particular,
the theorem also applies to the field of rationals F (X) over
Since (5) is true for all values of X ,X ,...,X ∈ FN, q
1 2 m q F . So, the sum-network in Theorem 2 is also solvable using
q
equations (6) and (7) imply
linear convolutional network code over F if and only if the
q
γi,2βj,i =I for 1≤i,j ≤m−1,i6=j, (8) characteristic of Fq is not in P.
Now we define another sum-network G with the set
γ =I for 1≤i≤m−1, (9) 1
i,1 of vertices V(G ) =△ ∪3 {s ,u ,v ,t }, edges E(G ) =△
m−1 1 i=1 i i i i 1
γ β =I for 1≤j ≤m−1. (10) {(ui,vi)|i = 1,2,3}∪{(si,uj),(vi,tj)|i,j = 1,2,3,i 6= j}.
X i,m j,i The network is shown in Fig. 2. The nodes s ,s ,s are
i=1 1 2 3
i6=j the sources and the nodes t ,t ,t are the terminals in the
1 2 3
Allthecodingmatricesinequations(8),(9)areinvertiblesince network. The symbols generated at the sources are denoted
the right hand side of the equations are the identity matrix. by X,Z, and W respectively.
X Z W
Note that equation (17) requires that the coding coefficients
s s s
1 2 3 α, β and γ be non-zero. This requirement can also be seen
as natural since if any of these coefficients is zero, then a
particular source-terminal pair will be disconnected.
Since all the terminals must recover the sum of the source
u1 u2 u3 messages, i.e., R =R =R =X +Z+W, we have
1 α β 1 1 γ 1 2 3
β+γ−1 = 1, (18a)
v1 v2 v3 γ+α−1 = 1, (18b)
α+β−1 = 1. (18c)
Now, over the binary field the values of α, β and γ must all
t t t be 1. Putting α = β = γ = 1 in equation (18), we have
1 1 γ−1 α−1 2 1 1 3 β−1 1 = 0. This gives a contradiction. So, it is not possible to
R1 R2 R3 communicate the sum of the sources to the terminals in this
network over the binary field F using scalar linear network
2
Fig.2. Thenetwork G1 coding.
Now,weconsideranyotherfinitefieldF (q 6=2).Weshow
q
The following lemma gives our second main result.
that over F , the conditions in equation (18) are satisfied for
q
Lemma 3: The sum-network G is scalar linear solvable some choice of α,β, and γ.
1
over all fields other than F2. Since q > 2, let α ∈ Fq be any element other than 0 and
Proof: The message vectors carried by different edges 1. Also, take γ =1−α−1 and β =(1−α)−1. Clearly, they
and the corresponding local coding coefficients are as below. satisfy (18a)-(18c). Hence, it is possible to communicate the
Without loss of generality, we assume sum of the source messages to the terminals over Fq.
It is worth noting that though the sum can not be commu-
Y = Y =X, (13a)
(s1,u2) (s1,u3) nicated in this network by scalar network coding over F2, it
Y(s2,u1) = Y(s2,u3) =Z, (13b) is possible to do so by vector network coding over F2 using
Y = Y =W, (13c) any block length N > 1. This follows because it is possible
(s3,u1) (s3,u2)
to communicate the sum over the extension field F using
and 2N
scalar network coding.
Y = Y +αY , (14a)
(u1,v1) (s2,u1) (s3,u1) IV. REVERSIBILITY OF NETWORKS
Y = Y +βY , (14b)
(u2,v2) (s3,u2) (s1,u2) Given a sum-network (recall the definition from Sec. I)
Y = Y +γY , (14c)
(u3,v3) (s1,u3) (s2,u3) N, its reverse network N′ is defined to be the network with
where α,β,γ ∈Fq. the same set of vertices, the edges reversed, and the role of
Also, w.l.o.g, we assume that sources and terminals interchanged. It should be noted that
since N may have unequal number of sources and terminals,
Y = Y = Y , (15a)
(u1,v1) (v1,t2) (v1,t3) the number of sources (resp. terminals) in N and that in N′
Y(u2,v2) = Y(v2,t1) = Y(v2,t3), (15b) maybedifferent.Forexample,thereversenetworkS∗′ ofS∗
m m
Y(u3,v3) = Y(v3,t1) = Y(v3,t2). (15c) has m sources and m−1 terminals and so the problem in
S∗′ is to communicate the sum of the source messages (say,
Since there is onlyonepath s2 →u3 →v3 →t1 fromsource m
Y ,...,Y ) to the m−1 terminals. In this section, we show
s to terminal t and also one path s → u → v → t 1 m
2 1 3 2 2 1
that for any sum-network N and any alphabet field F , the
from source s to t with path gainsγ and 1 respectively,the q
3 1
sum-network N is N-length vector linear solvable over F if
recovered symbol R at t must be q
1 1
and only if its reverse network N′ is N-length vector linear
R1 = Y(v2,t1)+γ−1Y(v3,t1). (16a) solvable over Fq.
Similarly, the recovered symbols R and R should be Consider a generic sum-network N depicted in Fig. 3.
1 2
ConsiderthecutsC andC showninthefigure.Wecallthese
R = Y +α−1Y , (16b) 1 2
2 (v3,t2) (v1,t2) cuts, the source-cut and the terminal-cut of the sum-network
R3 = Y(v1,t3)+β−1Y(v2,t3). (16c) respectively. The transfer function from C1 to C2 is defined
to be the m×n matrix T over F which relates the vectors
The coding coefficients are depicted in Fig. 2 for clarity. q
X=(X ,X ,...,X ) and R=(R ,R ,...,R ) as
From equations (13), (14), (15) and (16) it follows that 1 2 m 1 2 n
R = (β+γ−1)X +Z+W, (17a) R=XT.
1
R2 = X+(γ+α−1)Z +W, (17b) In case of N-length vector linear coding,
R = X+Z+(α+β−1)W. (17c) X ,X ,...,X ,R ,R ,...,R ∈ FN, X ∈ FmN,
3 1 2 m 1 2 n q q
and R ∈ FnN. The transfer matrix is a mN ×nN matrix V. DISCUSSION
q
which is easier viewed as an m×n matrix of N×N blocks. We havepresentedsomeresultsoncommunicatingthe sum
The (i,j)-th element (‘block’ for vector linear coding) of the of source messages to a set of terminals. It was shown in
transfer matrix is the sum of the path gains of all paths from
[7] that there is a 1 − 1 correspondence between systems
X to R . The following lemma follows directly.
i j of polynomial equations and networks. This is a key result
Lemma 4: A sum-network N is N-length vector linear which implies existence of networks with arbitrary finite or
solvable if and only if there is an N-length vector linear
co-finite characteristic set. Though sum-networks have very
networkcodesothateachelement/blockofthetransfermatrix
specific demands by the terminal nodes and thus are more
from the source-cut to the terminal-cut is the N ×N identity restrictedasaclass,acompletecharacterizationofthesystems
matrix.
of polynomialequationswhich have equivalentsum-networks
Now consider the reverse network N′ of N. Let us denote is notyetknown.Investigationin this directionis in progress.
the sourcesymbolsin N′ asY ,Y ,...,Y and therecovered
1 2 n
symbolsattheterminalsasR′,R′,...,R′ .Letusdenotethe VI. ACKNOWLEDGMENTS
1 2 m
network coding coefficients of N′ by βe,e′ for any two edges This work was supported in part by Tata Tele-services IIT
e,e′ ∈ E(N′) so that head(e) = tail(e′). Let us denote the Bombay Center of Excellence in Telecomm (TICET) and
edge in N′ obtained by reversing the edge e ∈ E(N) by e˜. Bharti Centre for Communication.
Clearly, there is a 1−1 correspondence between the paths in
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