A class of index coding problems with rate 1/3

0.51 MB·

by  Prasad Krishnan

1
A class of index coding problems with rate
3
Prasad Krishnan and V. Lalitha,
Signal Processing and Communications Research Centre,
International Institute of Information Technology, Hyderabad.
Email: {prasad.krishnan,lalitha.v}@iiit.ac.in
Abstract—An index coding problem with n messages has by modelling the unavailable side-information as interference.
symmetric rate R if all n messages can be conveyed at rate Theideaistoassignprecodingmatricestothemessagevectors
6 R. In a recent work, a class of index coding problems for which suchthatallreceiverscandecodeeveninthepresenceofinter-
1 symmetric rate 1 is achievable was characterised using special
3 ference(therebyrequiringsomedegreeoflinearindependence
0 properties of the side-information available at the receivers. In
2 this paper, we show a larger class of index coding problems between the precoding matrices, and hence reducing the rate).
(which includes the previous class of problems) for which However, at the same time the interference at the receivers
n symmetric rate 1 is achievable. In the process, we also obtain mustbeas‘aligned’aspossible(inordertoreducetheamount
Ja a stricter necessa3ry condition for rate 13 feasibility than what is oflinearindependencerequired,i.e.toincreasetherate).This
known in literature.
technique was further explored in [3], [12], [13] and several
5
2 classes of index coding instances with certain feasible rates
I. INTRODUCTION wereidentifiedbasedonthepropertiesoftheinterferenceseen
] by the receivers.
T Index Coding, introduced in [1], considers the problem of
Thisworkbuildsprimarilyupontheresultsin[12].In[12],
I efficiently broadcasting a number of messages available at a
. a necessary and sufficient condition for the feasibility of rate
s source,toreceiversthatalreadypossesssomepriorknowledge
c half2 in a groupcast index coding problem was established
[ of the messages. In the index coding framework, the source based on the properties satisfied by two graphs obtained from
1 is allowed to encode the messages (and thereby use the the interference structure of the problem, called the conflict
channelefficiently)whilesatisfyingthereceiverdemands.The
v graph and the alignment graph. Also, a sufficient (but not
general class of groupcast index coding problems consists of
9 necessary)conditiononthestructureofthesegraphswasgiven
8 n messages generated at a source, where each message is for rate 1 feasibility. Prior work [2] also gives a necessary
6 demanded by at least one receiver. Index coding problems 3
(but not sufficient) condition for rate 1 feasibility based on
6 where each receiver demands a unique message are called 3
0 the interference structure of the problem. The relevant details
single unicast index coding problems and are the most widely
. of theprior work arediscussed elaborately inthe forthcoming
1 studied class.
sections of this paper.
0
Although index coding continues to be open in general,
6
several researchers have made inroads into characterising the
1
rate of index coding1 and presenting achievable schemes. The A. Contributions
:
v
landmarkpaper[2]famouslyconnectedthescalarlinearindex • Firstly,werevisethedefinitionoftheconflictgraphgiven
i
X codingproblemtofindingaquantitycalledminrankassociated in [12]. The conflict graph definition in [12] does not
r with the side-information graph related to the given single capture the interference structure of the index coding
a unicastindexcodingproblem.Upperandlowerboundsonthe problem sufficiently. In order to rectify this, we define
rate for single unicast index coding have been presented via the conflict hypergraph, which is shown to capture the
graph theoretic ideas like clique cover, chromatic number [1], interference structure sufficiently. (Subsection III-B)
[2],localchromaticnumber[3],fractionalcliquecoveringand • Ourmainresultshowsthatrate 1 isachievableinagiven
3
hyperclique covering [4]–[6], and recently, the ‘generalized index coding problem under certain conditions on the
interlinked cycle cover’ [7]. Many of these papers naturally topology of the alignment graph and conflict hypergraph
leadtoconstructionsof(scalarandvector)linearindexcodes. of the index coding problem. The sufficient condition
Linear codes however are not always found to be optimal which we present for rate 1 feasibility is looser than the
3
[8]. Random coding approaches to index coding were studied sufficient condition shown in [12]. Therefore, this class
in [9]. Bounds on the rate of groupcast index coding were of index coding problems is bigger than the previously
presented in [10]. known class of rate 1 feasible problems. The achievabil-
3
Interference alignment, well known as a powerful tool to ity of rate 1 in such problems is shown by presenting a
3
study degrees of freedom in wireless interference networks, construction of an index code by random generation of
was employed to the linear index coding problem in [11], precoding vectors over a large field. (Subsection IV-F)
1roughly,theratioofinformationconveyedbyeverymessagetothenumber 2for every two F (some finite field) symbols transmitted through the
oftimesthebroadcastchannelisused channel,oneFsymbolofeachmessageisconveyed.
• Towards obtaining our main result, we also obtain a If we have a linear index code of rate R, then we can
necessaryconditionforrate 1 feasibility,whichisstricter represent the encoding function as follows.
3
than the prior condition from [2] (Subsection IV-D). Our
n
(cid:88)
feasibility conditions can thus be seen as being midway E(W ,W ,...,W )= V W ,
1 2 n i i
between those of [12] and [2].
i=1
Notations: Throughout the paper, we use the following nota- whereeachV isaL×LRmatrixwithelementsfromF.Inthe
i
tions. Let [1 : m] denote {1,2,...,m}. For a set of vectors case of scalar linear index coding, we have LR=1. Finding
A, sp(A) denotes their span. For a vector space V, dim(V) ascalarlinearindexcodeoflengthL(i.e.,withafeasiblerate
denotesitsdimension.AnarbitraryfinitefieldisdenotedbyF. 1/L) is equivalent to finding an assignment of these L-length
A vector from the m-dimensional vector space Fm is said to vectors V s to the n messages such that the receivers can all
i
be picked at random if it is selected according to the uniform
decode their demanded messages, i.e.,
distribution on Fm.
(cid:32) n (cid:33)
(cid:88)
D V W ,S(j) =D(j), ∀ j ∈[1:T].
II. REVIEWOFINDEXCODING j i i
i=1
Formally, the index coding problem (over some field F)
Inthecaseofascalarlinearindexcode,theencodingfunction
consistsofabroadcastchannelwhichcancarrysymbolsfrom
E fixes the vectors assigned to the messages. However not
F, along with the following.
every vector assignment is a valid index coding encoding
• A set of T receivers function,asthedecodingfunctionsmaynotexist.Inanumber
• A source which has messages W = {Wi,i ∈ [1 : n]}, of proofs in this paper, we start with some vector assignment
each of which is modelled as a vector over F.
andshowthatitleadstoanencodingfunctionofavalidindex
• For each receiver j, a set D(j)⊆W denoting the set of code. Therefore, in such proofs we refer to the initial vector
messages demanded by the receiver j. assignment as a valid encoding function E by abusing the
• For each receiver j, a set S(j) ⊆ W\D(j) denoting notation.
the set of side-information messages available at the jth
receiver. Remark 2. We restrict our attention to scalar linear index
This general class of index coding problems is known as codes for the rest of this paper. However we believe that our
groupcast index coding problems. results can be extended to vector linear index codes as well.
Definition1(IndexcodeofsymmetricrateR). Anindexcode
A. Modelling unavailable side-information as interference
ofsymmetricrateRforagivenindexcodingproblemconsists
of an encoding function Definition 4 (Interfering sets and messages, conflicts). For
some receiver j and for some message W ∈ D(j), let
E:FLR×FLR×...×FLR →FL, k
Interf (j) (cid:44) W\(W ∪S(j)) denote the set of mes-
(cid:124) (cid:123)(cid:122) (cid:125) k k
n times sages (except W ) not available at the receiver j. The sets
k
for some L ≥ 1, mapping the n LR-length message vectors Interf (j),∀k are called the interfering sets at receiver j.
k
(Wi ∈ FLR) to some L-length codeword which is broadcast If receiver j does not demand message Wk, then we define
through the channel, as well as decoding functions Interf (j)(cid:44)φ.IfamessageW isnotavailableatareceiver
k i
D :FL×FLR×FLR×...×FLR →FLR×FLR×...×FLR j demanding at least one message Wk (cid:54)=Wi, then Wi is said
j
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) to interfere at receiver j, and Wi and Wk are said to be in
|S(j)| times |D(j)| times conflict.
at the receivers j = [1 : T], mapping the received codeword
For a set of vertices A ⊆ W, let VE(A) denote the vector
and the side-information messages to the demanded messages
space spanned by the vectors assigned to the messages in A,
D(j), i.e.,
under the specified encoding function E. If A=φ, we define
Dj(E(W1,...,Wn),S(j))=D(j), ∀ j ∈[1:T]. VE(A) as the zero vector.
Remark 1. We could in general have different rates for Definition 5 (Resolved conflicts). For a given assignment of
different messages, but in this paper we restrict our attention vectors to the messages (or equivalently, for a given encoding
tosymmetricrates.Thereforeanyratereferredtointhispaper function E), we say that conflicts within a subset W(cid:48) ⊆ W
is the symmetric rate. are resolved, if
Definition 2 (Achievable rates and rate R feasibility). For a Vk ∈/VE(Interfk(j)∩W(cid:48))
given index coding problem, a rate R is said to be achievable ∀W ∈W(cid:48),∀ receivers j ∈[1:T], (1)
k
if there exists an index code of rate R, and the index coding
problem is said to be rate R feasible. where V is the vector assigned to W under the encoding
k k
function E. If (1) holds for W(cid:48) =W, then all the conflicts in
Definition 3 (Scalar index codes and linear index codes). If
the given index coding problem are said to be resolved.
a rate R=1/L is achievable, the associated index code is a
scalar index code of length L. If the encoding and decoding We now state a simple lemma, rephrased from [11], which
functions are linear, then we have a linear index code. is easily proved.
Lemma 1. For any encoding function E, successful decoding B. Capturing interference in conflict hypergraphs
at the receivers is possible if and only if all the conflicts are
The following example illustrates that the conflict graph
resolved.
definition does not capture the directionality of the conflicts.
Proof: Let V be the vector assigned to W under the
k k Example 1. Consider two single unicast index coding prob-
encoding function E.
lems with four messages. In the first problem, the interfer-
If part: At all receivers j, consider that we have V ∈/
k ing sets are as follows Interf (1) = W ,Interf (2) =
1 3 2
VE(Interfk(j)), ∀k ∈ [1 : n]. What is received at a W ,Interf (3) = W ,Interf (4) = {W ,W ,W }.
receiverj isthecodeword(cid:80)n V W ,fromwhichitwantsto 1 3 2 4 1 2 3
i=1 i i In the second problem, Interf1(1) = φ,Interf2(2) =
obtain W . As receiver j can always subtract the contribution
k W ,Interf (3) = {W ,W },Interf (4) = {W ,W ,W }.
from the side-information messages from (cid:80)n V W , it only 1 3 1 2 4 1 2 3
i=1 i i All other interfering sets corresponding to the receivers are
remains to be shown that W can be decoded from
j empty. These two problems have the same alignment and
(cid:88) (cid:88) conflictgraphs(seeFig.1(a)).Inparticular,theconflictgraphs
V W =V W + V W , (2)
i i k k i i are the same because the definition of the conflict graph does
i:Wi∈/S(j) i:Wi∈Interfk(j) not model the directionality of the absent side-information
messages. Note that they both have different solutions. We
where the equality is because
leave it to the reader to check that the first problem is rate 1
2
{W ∈/ S(j)}={W ∈Interf (j)}∪{W }. feasible, while the second problem is not rate 1 feasible, but
i i k k 2
has a rate 1 solution.
3
BecauseoftheassumptionthatVk ∈/ VE(Interfk(j)),receiver To overcome this issue, we define the conflict hypergraph
j can get W from (2).
k as follows.
Only if part: Consider now that there is some receiver j
and some message k such that Vk ∈VE(Interfk(j)). Clearly, Definition 8 (Conflict hypergraph). The conflict hypergraph
solving for W from (2) does not lead to an unique solution. is an undirected hypergraph with vertex set W (the set of
k
Hence decoding fails. This concludes the proof. messages), and its hyperedge set defined as follows.
By Lemma 1, it should also be clear that if there is an • For any receiver j demanding any message Wk, Wk
assignment of L-length vectors Vis to the messages Wis such and Interfk(j) are connected by a hyperedge, which is
that the condition in Lemma 1 is satisfied, then these vectors denoted by {W ,Interf (j)}.
k k
naturally define an index code of length L for the given index
For example, the two problems presented in Example 1 are
coding problem.
now represented using different conflict hypergraphs in Fig.
1(b) and Fig. 1(c) (the alignment graphs remains the same).
III. ARELOOKATFEASIBILITYOFRATE 1 Note that even though this definition for the conflict hyper-
2
graph does not explicitly contain direction, the directionality
A. Alignment and conflict graphs of [12]
of interference seen by any receiver is modelled correctly
In [12], the authors defined the notions of alignment graph whenever the number of interfering messages is more than
and conflict graph whose properties were used to characterise one. This is sufficiently general as we see in the following
index coding problems for which rate 1 is feasible. Both of Lemma.
2
these graphs have the same vertex set, which is the set of Lemma2. Supposetwoindexcodingproblems,denotedbyI
1
messages W. and I , are modelled by the same conflict hypergraph. Then
2
any index coding solution for I is an index coding solution
Definition 6 (Alignment graph and alignment sets - [12]). In 1
for I .
the alignment graph, the vertices W and W are connected 2
i j
byanedge(calledanalignmentedge,showninourfiguresby Proof: Let E be the encoding function of the given
asolidedge)whenthemessagesWi andWj arenotavailable index code for I . Let V be the vector assigned to W . By
1 k k
at a receiver demanding a message other than W and W .
i j Lemma 1, we must have that for Vk ∈/ VE(Interfk(j)), ∀k ∈
A connected component of the alignment graph is called an [1 : n], ∀j ∈ [1 : T] in I . Now assume that the same
1
alignment set. index code is used for I . For any receiver j and message
2
W with |Interf (j)| ≥ 2 in I , the conflict hyperedge
It is easy to see that the alignment sets define a partition of k k 2
{W ,Interf (j)} present in the conflict hypergraph of I is
the alignment graph. Also, the messages in Interf (j), for k k 2
k present also in that of I as both I and I have the same
all messages k at all receivers j are fully connected in the 1 1 2
conflict hypergraph. In other words, message Interf (j) is
alignment graph. k
an interfering set of a receiver j in I also and hence we
1
Definition 7 (Conflict Graph - [12]). In the conflict graph, must have Vk ∈/ VE(Interfk(j)), which means that Wk is
Wi and Wj are connected by an edge (called an conflict recoverable at j in I2 also.
edge, shown by a dotted edge) if Wi is not available at a The only case left to check is when |Interfk(j)| = 1
receiver demanding Wj, or Wj is not available at a receiver in I2 for some receiver j and message Wk. Let us assume
demanding Wi. that Interfk(j) = Wi. Because I1 and I2 share the same
1 1 1
2 2 2
4 3 4 3 4 3
(a) (b) (c)
Fig. 1. (a) Alignment and conflict graphs for the two index coding problems discussed in Example 1. Edges in alignment graph are shown in solid
lines and those in conflict graph are shown in dotted lines. (b) Conflict hypergraph for the first index coding problem in Example 1. The hyperedges are
{1,3},{2,1},{3,2},{4,{1,2,3}}.(b)Conflicthypergraphforthesecondproblem.Thehyperedgesare{2,1},{3,{1,2}},{4,{1,2,3}}.
conflicthypergraphs,thisconflict(insomedirection)ispresent be assigned to any message as this means that the message
in I also. By assignment E, we must have that V and V cannot be decoded by any receiver).
2 i k
are linearly independent, which ensures that this conflict is If part: Suppose that there are no internal conflicts. We
resolved in I also, irrespective of its directionality. assume a large field F. For each alignment set, we indepen-
2
dently generate a random 2 × 1 vector over F and assign
it to the vertices of the alignment set. Because of random
C. A new lemma and its application: Rate half feasibility
generation,wecanassumethatanyassignedvectorisnon-zero
condition from [12]
and any two assigned vectors are linearly independent with
Towards showing a necessary and sufficient condition for high probability (whp). Let E denote the associated encoding
rate 21 feasibility,thefollowingdefinitionforinternalconflicts functionandVk denotethevectorassignedtovertexWk.Since
was given in [12]. therearenointernalconflicts,weonlyhavetocheckconflicts
betweenalignmentsets.ForanyvertexW ,thesetInterf (j)
Definition 9 (Internal conflict [12]). A conflict between two k k
foranyreceiverjwhichdemandsW belongstoa(i)different
messageswithinanalignmentsetiscalledaninternalconflict. k
and (ii) unique alignment set than Align(k) ((i) is because
Thefollowingtheoremwasprovedin[12]onrate 1 feasible therearenointernalconflicts,(ii)isbecauseallthemessages
2
index coding problems. in Interf (j) must be in the same alignment set). Since any
k
Theorem1. Anindexcodingproblemisrate 1 feasibleifand two alignment sets get independent vectors (whp), we have
only if there are no internal conflicts. 2 that Vk ∈/ VE(Interfk(j)), and the same argument is true for
all receivers j and all messages W . Hence this assignment
k
The following lemma plays a crucial role in our proof of of vectors ensures successful decoding by Lemma 1.
Theorem 1, which is included for the sake of completeness as Only if part: Suppose that there is some internal conflict
it is not available in its complete form in prior literature. We (represented in the conflict graph as an edge between node
will also see in Subsection IV-F that it is central to proving k(cid:48) and node k) in an alignment set Align(k). Because k(cid:48),k
our main theorem which deals with the feasibility of rate 13. are part of the same alignment set (connected component)
Align(k), there lies a path from k(cid:48) to k, given by an ordered
Lemma 3. Let U ,U ,...,U be N sets of vectors, such that
1 2 N
set {k(cid:48),i ,i ,...,i ,k}, such that every adjacent pair of
dim(sp(U ∩U )) = K,∀j ∈ [1 : N −1]. Then the space 1 2 N−1
j j+1
spanned by ∪N U has dimension K if and only if each U elements ({k(cid:48),i1}, etc.) belong to the interfering set of some
j=1 j j
receiver.
spans a vector space of dimension K.
In some assignment corresponding to a rate 1 solution,
2
dimenPsriooonf:KIf,apnadrt:diSmup(sppo(sUeje∩acUhj+U1j))sp=anKs,a∀vje∈cto[1r :spNac−e1o]f. lteot tVhek(cid:48),vVeir1t,ic..e.s,V{iNk−(cid:48),1i,1V,ik2,b..e.,itNhe−1n,okn}-.zeWroevdeecfitonres tahsesigsneetsd,
tChlueOsanrdllyyi,miaf(lsplpat(hr∪te:Njv=Seu1cpUtpojor))sse=pdaciKmes.(sspp((∪UNj)sUare))e=xacKtlyanedqual, and UU1N(cid:44)(cid:44){{VVikN(cid:48),−V1,i1V}k,}.Ul (cid:44) {Vil−1,Vil},∀l ∈ [2 : N −1] and
j=1 j
dim(sp(U ∩U ))=K,∀j ∈[1:N −1]. (3)
j j+1
U2 i2 . . . iN-2 UN-1
By (3), dim(sp(U )) ≥ K,∀j. If some dim(sp(U )) > K,
j j i i
then dim(sp(∪N U )) > K, and we have a contradiction. 1 N-1
j=1 j U U
Thus, we must have that dim(sp(U ))=K,∀j. 1 N
j
We now use Lemma 3 and Lemma 1 to prove Theorem 1. k' k
Proof of Theorem 1: Corresponding to any vertex W
k
in the alignment graph, let Align(k) denote the alignment set
it belongs to (this is unique as the alignment sets partition Fig. 2. Since k(cid:48) and k are in conflict, they must be assigned linearly
independent vectors. This requires that at least one of the sets U1,...,UN
the alignment graph). We first note that in any (scalar linear)
hastobetwodimensional.
index coding scheme for the given problem, all the vertices
must be assigned some non-zero vectors (zero vector cannot Supposesomedim(sp(U ))=2forsomel.Thenareceiver
l
j (atwhichthemessagescorrespondingtoU areunavailable)
l
1 2
‘sees’ an interfering space of dimension 2. Thus we need to
assign a vector linearly independent from U (which itself
l 5
has dimension 2) to the corresponding demanded message of 1
2
receiver j. This can be possible only if the assigned vectors 3
4
are of length at least 3, i.e. the rate can be at most 1. 5 6
3
Therefore, for a rate 1 index coding assignment, all 3
2 4 6
U should spanning a space of dimension 1. Thus we
l
have dim(sp(U )) = 1,∀l and dim(sp(U ∩ U )) = (a) (b)
l l l+1
Fig. 4. (a) Alignment graph and conflict hypergraph for the problem in
1,∀l ∈ [1 : N − 1]. By Lemma 3, we should thus have
Example 2, which is rate 1 infeasible. (b) Alignment graph and conflict
dimHo(swpe(v∪eNlr=1Uk(cid:48)l))a=nd1. k are in conflict, which means hypergraphfortheproblem3inExample3,whichisrate 31 feasible.
that they should be assigned linearly independent vec-
tors, i.e., dim(sp({V ,V })) = 2 which means that {W ,W ,W },Interf (6) = {W ,W ,W }. The alignment
k(cid:48) k 1 3 4 6 1 2 4
dim(sp(∪N U ))>1. Thus there is a contradiction and thus graph and the conflict hypergraph corresponding to the
l=1 l
any internal conflicts forces the rate to be less than 1. This problem are given in Fig. 4(a). It can be seen that the
2
concludes the proof. conflict hypergraph of this problem does not have an
acyclic subset of messages of size 4. If we assume that the
IV. FEASIBILITYOFRATE 13 problem is rate 13 feasible, then it has to be necessarily
true that both sp({V ,V ,V }) and sp({V ,V ,V }) are
From Section III, the following is clear. 1 3 4 1 2 4
two dimensional. We leave it to the reader to check that
• If there are no conflicts (not even conflicts between sp({V ,V ,V }) must also be two dimensional. However, the
two alignment sets) in the alignment graph, rate 1 is 1 2 3
conflict between the messages W ,W and W are resolved
1 2 3
achievable(thisisthecasewhenanyreceiverdemanding
only if sp({V ,V ,V }) is three dimensional. Thus, the
1 2 3
amessagehasalltheothermessagesasside-information).
problem is rate 1 infeasible.
• For rate 1 infeasible index coding problems, rate 1 is 3
2
feasible if and only if there are no internal conflicts.
B. A known sufficient condition for rate 1 feasibility
Towardsobtainingourmainresult,whichcharacterisesaclass 3
of index coding problems which are rate 1 feasible, we first In [12], the following theorem about the achievability of
3 rate 1 was proved (this is a special case of Corollary 9 of
give a prior known necessary condition for feasibility of rate 3
1. [12]).
3
Theorem 3. Consider a rate 1 infeasible index coding prob-
2
A. A known necessary condition for rate 1 feasibility lem with no acyclic subset of size 4. If none of its alignment
3
sets have both forks (a fork is a vertex connected by three or
Suppose there are 4 messages {W ,k =1,..,4} such that
ik more edges) and cycles, then the index coding problem is rate
message W is demanded by some receiver (say receiver j )
ik k 1 feasible.
and {Wik(cid:48) :k(cid:48) <k}⊆Interfik(jk). Following [12], we call 3
suchasetofmessages{Wik,k =1,..,4}asanacyclicsubset As mentioned in [12], the condition that there are no
of messages of size 4. The following theorem can be obtained alignment sets with both forks and cycles of length 3 means
from the results in [2]. that there is no interfering set Interf (j) at any receiver of
k
Theorem2. Anindexcodingproblemwhichisrate 1 feasible size ≥ 4, as such a set would mean that there is both a
3 cycle and fork within an alignment set (since the messages
cannot have an acyclic subset of messages of size 4.
in Interf (j) are fully connected in the alignment set).
k
Therefore Theorem 3 characterises a rather limited class of
indexcodingproblemswhichare 1 feasible.Example3shows
3
1 2 3 4 anindexcodingproblemwhichdoesnotsatisfytheconditions
of Theorem 3 but is rate 1 feasible.
3
Example 3 (Example to illustrate that the condition in Theo-
Fig. 3. From the structure of the conflict hypergraph, it can be seen rem 3 is not necessary). Consider a single unicast index cod-
that vectors {V1,...,V4} assigned to the four messages must be linearly ingproblemwithsixmessages.Theinterferingsetsoftheprob-
independent.
lem are as follows: Interf (1) = {W ,W },Interf (2) =
1 4 6 2
{W ,W },Interf (3) = φ,Interf (4) = φ,Interf (5) =
The following example however shows that Theorem 2 is 1 4 3 4 5
{W ,W },Interf (6) = {W ,W ,W }. The alignment
not a sufficient condition for rate 1 feasibility. 2 3 6 3 4 5
3 graph and the conflict hypergraph corresponding to the prob-
Example 2. Consider a single unicast index coding lem are given in Fig. 4(b). It can be seen that the alignment
problem with six messages. The interfering sets graph of this problem has both forks and a cycle. Consider
are as follows: Interf (1) = W ,Interf (2) = 3 linearly independent 3×1 vectors V ,V ,V . We note that
1 4 2 1 2 3
{W ,W },Interf (3)=W ,Interf (4)=φ,Interf (5)= the following assignment of vectors resolves all conflicts: (i)
1 3 3 1 4 5
vectorV tomessagesW andW ,(ii)vectorV tomessages we cannot add another triangular interfering set to a type-2
1 1 3 2
W and W , (iii) vector V to messages W and W . Thus, alignment set and still maintain connectivity (as in Definition
4 5 3 2 6
the problem is rate 1 feasible. 12).
3
By definition, every type-2 alignment set must be a subset
C. Triangular interfering sets and Type-2 alignment sets of a (regular) alignment set, and there could be many type-
2 alignment sets within any alignment set. Given an index
Inthissubsection,wedevelopanewframeworkforstudying
the rate 1 feasibility of groupcast index coding problems. coding problem, we can identify type-2 alignment sets as
3 follows. Within any alignment set, we identify a triangular
Towards this, we define the notions of a triangular interfering
interfering set of messages (if there is no such set then there
set and a type-2 alignment set.
is no type-2 alignment set inside that alignment set). Then we
Definition 10 (Triangular Interfering Sets). A subset W(cid:48)(cid:48) repeatedly add triangular interfering sets which are adjacent
of size three of the set of messages W is said to be a to the existing connected triangular interfering sets. When we
triangular interfering set if all the messages in W(cid:48)(cid:48) interfere can no longer add such adjacent triangular interfering sets,
simultaneously at some receiver, and at least two of the then we have our type-2 alignment set.
messages in W(cid:48)(cid:48) are in conflict.
Definition 11 (Adjacent Triangular Interfering Sets). Two D. A new stricter necessary condition for rate 1 feasibility
3
distinct triangular interfering sets W and W are said to be
1 2 We now prove a necessary condition for rate 1 feasibility
adjacent if they ‘meet’ at a conflicting edge, i.e., W1∩W2 = 3
based on the vectors assigned to type-2 alignment sets. This
{W ,W } such that W and W are in conflict.
i j i j
theorem is another application of the key lemma, Lemma 3.
Definition 12 (Connected triangular interfering sets, Type-2
Theorem 4. In any rate 1 solution to a given index coding
alignment sets). Two triangular interfering sets W1 and W2 3
problem,allthemessagesinanytype-2alignmentsetmustbe
aresaidtobeconnectedifthereexistsapath(i.e.,asequence)
assigned vectors from a vector space of dimension two.
of adjacent triangular interfering sets starting from W and
1
ending at W2. A type-2 alignment set is a maximal set of Proof:LetEbetheencodingfunctionofarate 1 solution.
3
triangular interfering sets which are connected to each other. Consideratype-2alignmentsetW(cid:48) withtriangularinterfering
sets W ,i∈[1:S].
i
Suppose for any triangular interfering set W of W(cid:48), we
i
1
have dim(VE(Wi)) = 3. Since all the vertices in Wi inter-
2 2 fere at some receiver (say, a receiver which requests some
1
other message W ), the message W must be assigned a
j j
4 vector which is linearly independent from those assigned to
the messages in W , and thus we need at least 4 linearly
i
3
4 3 independentvectors,andhencetheratehastobe≤1/4.Thus
notriangularinterferingsetWi ofW(cid:48) hasdim(VE(Wi))=3.
(a) (b) However, any triangular interfering set W of W(cid:48) must have
i
dim(VE(Wi))=2, as Wi has a conflict.
8
Suppose dim(VE(W(cid:48))) > 2. Consider three messages
7 Wj1,Wj2, and Wj3 in W(cid:48), that have been assigned three
linearlyindependentvectors,belongingtosomethreetriangu-
2
1
AligTnympeen-2t Set lar interfering sets (not necessarily different), Wi1,Wi2, and
4 W respectively.Wehavealreadyshownthatwecannothave
i3
W = W = W . So at least two of the three triangular
3 6 i1 i2 i3
interfering sets are different.
5 Suppose all three sets Wi1,Wi2, and Wi3 are different.
Because the three triangular interfering sets are within the
sametype-2alignmentset,itmustbethecasethatthereexists
9
apathconsistingofadjacenttriangularinterferingsetsstarting
(c) fromWi1,throughWi2 anduptoWi3.LetN bethenumberof
Fig. 5. (a) Messages 1,2,3 interfere at a receiver which demands 4 and triangularinterferingsetsonthispath(countedaswegoalong
messages2,3areinconflict.Thus,{1,2,3}formatriangularinterferingset. the path; repetitions are counted separately). For i ∈ [1 : N],
(b) Triangular interfering sets {1,2,3},{2,3,4} are adjacent as they both
have common conflict edge {2,3}. The receivers at which these triangular letUi denotethesetof3vectorsassignedtotheith triangular
interfering sets interfere are not explicitly shown in the figure. (c) Type-2 interfering set in this path. Fig. 6 illustrates this scenario for
alignmentsetisthesubsetofmessagesindicatedintheellipse,thetriangu- the type-2 alignment set example shown in Fig. 5(c).
lar interfering sets include {1,2,3},{2,3,4},{3,4,5},{4,5,6},{2,4,7}.
These5triangularinterferingsetsareconnected. By the previous arguments, we have that dim(sp(Ui)) =
2,∀i. Also, dim(sp(U ∩U )) = 2,i ∈ [1 : N −1], as the
i i+1
Examples to illustrate the above definitions are shown in ith and the (i+1)th triangular interfering sets are adjacent by
Fig. 5. Note that the maximality in Definition 12 means that constructionofthepath.Therefore,byLemma3,itmustbethe
Definition 14 (‘Restricted’ versions of alignment graphs,
7 alignment sets, and internal conflicts). The alignment graph
and the alignment sets of the restricted index coding problem
1 2 U3 Type-2 IW(cid:48) are called the W(cid:48)-restricted alignment graph and W(cid:48)-
U U Alignment Set restrictedalignmentsetsrespectively.AW(cid:48)-restrictedinternal
1 2 4 conflict is a conflict between any two messages within a
U
4 restricted alignment set of W(cid:48).
3 6
U
U 6
5
5 1 1
2 2
Fig.6. Considermessages1,7,6.Todeterminethedimensionofthespan
ofthesemessages,considerthefollowingpathoftriangularinterferingsets:
{1,2,3},{2,3,4},{2,4,7},{2,3,4},{3,4,5},{4,5,6} and Ui,1 ≤ i ≤ 3 5 3
6arethesetsof3vectorsassignedtoith triangularinterferingset. 4
casethatdim(sp(∪N U ))=2.However,thevectorassigned 6 6
i=1 i
to the message Wjk belongs to ∪Ni=1Ui for k = 1,2,3, and (a) (b)
according to our assumption the vectors assigned to these Fig.7. (a)Alignmentgraphandconflicthypergraphforanexampleindex
three messages are linearly independent vectors. Thus there codingproblem.(b)Alignmentgraphandconflicthypergraphfortherestricted
index coding problem, restricted to messages 1,2,3,6. The conflict edge
is a contradiction, which means that we cannot have three
{1,2}isarestrictedinternalconflict.
messages in three different triangular interfering sets which
have been assigned linearly independent vectors. A similar
The proof of the following theorem is a direct application
claim can be proved if the three messages come from two
of Theorem 1, and hence is skipped.
different triangular interfering sets.
Thus, no three messages in a type-2 alignment set can be Theorem 5. The restricted index coding problem IW(cid:48) is rate
assigned linearly independent vectors. In other words, any 1 feasible if and only if there are no W(cid:48)-restricted internal
2
type-2 alignment set W(cid:48) in a rate 1 solution must have conflicts.
3
dim(VE(W(cid:48)))=2. Corollary 1. For a given index coding problem I, there exists
Theorem4isstricterthanTheorem2,asTheorem2applies
an assignment of 3×1 vectors from a two dimensional vector
onlytoanacyclicsubsetofmessagesofsize4,whichbasically space(overalargeenoughfieldF)tothemessagesinasubset
is equivalent to a triangular interfering set. Theorem 4 on W(cid:48) such that the conflicts within W(cid:48) are resolved, if and only
the other hand considers a ‘connected component’ of such if the restricted index coding problem I is rate 1 feasible.
triangular interfering sets, and is therefore more strict. We W(cid:48) 2
leave it to the reader to verify that the problem in Example Proof: We first recall that conflicts within W(cid:48) are said to
2, while ‘passing’ the condition of Theorem 2, ‘fails’ the be resolved if (1) is satisfied.
condition of Theorem 4. If part: The proof for the if-part follows the achievability
scheme shown in the proof of Theorem 1, with the difference
that we assign to the messages in W(cid:48) randomly generated
E. Restricted index coding problems and rate 1 feasibility
2 3×1 vectors (not 2×1 vectors as in Theorem 1) from a two
Theorem 4 prescribes that type-2 alignment sets must be dimensional space over a suitably large field F. Because I
W(cid:48)
‘two-dimensional’ in a rate 13 code. In this subsection, we is rate 12 feasible, such an assignment resolves the conflicts
give a necessary and sufficient condition for achieving this within W(cid:48).
two-dimensionality.Forthispurpose,werequirethenotionof Onlyifpart:Supposethereisanassignmentof3×1vectors
a restricted index coding problem. from a two dimensional vector space to messages in W(cid:48) such
Definition13(RestrictedIndexCodingproblem). LetIdenote that all conflicts within W(cid:48) are resolved. Then we can always
anindexcodingproblemwithmessagesetW.ForsomeW(cid:48) ⊆ obtaina2×3matrixAsuchthatpremultiplyingallthevectors
W, a W(cid:48)-restricted index coding problem is defined as the assigned to W(cid:48) with A gives us a rate 21 (length 2) index
index coding problem IW(cid:48) consisting of coding solution for IW(cid:48).
The following theorem gives a necessary and sufficient
• The messages W(cid:48).
• The subset TW(cid:48) (of size TW(cid:48)) of the receivers of I which condition for assigning vectors from a two dimensional space
to the type-2 alignment sets (i.e., for satisfying the necessary
demand messages in W(cid:48).
condition of Theorem 4).
• Foreach j ∈TW(cid:48) thedemandsets DW(cid:48)(j) andtheside-
information sets SW(cid:48)(j) are restricted within W(cid:48), i.e., Theorem 6. Let W(cid:48) be a type-2 alignment set of the given
index coding problem I. If I is rate 1 feasible, then I must
D (j)=D(j)∩W(cid:48). 3 W(cid:48)
W(cid:48) be rate 1 feasible which holds if and only if there are no
SW(cid:48)(j)=S(j)∩W(cid:48). W(cid:48)-restri2cted internal conflicts.
Proof: The proof follows by combining the claims of
Theorem 4, Corollary 1 and Theorem 5.
Assign independently Assign a random
F. Anewclassofindexcodingproblemswithrate 1 feasibility generated random (3 x 1) vector to all
3 (3 x 1) vector to each the vertices
vertex
Wenowprovethemainresultofthispaper,whichconnects
1
5
allthepreviouslyprovedresultsandwidenstheclassofindex
codingproblemsforwhichrate 1 isachievable.Becauseofthe 4
3 2 7
frameworkwehavedevelopedintheprevioussubsections,the
6
3
proof of this theorem is simpler than Theorem 3, while also
subsuming that result.
Theorem 7. A rate 1 infeasible index coding problem I is
2
rate 1 feasible if every alignment set of I satisfies either of 9
3 8
the following conditions.
1) It does not have both forks and cycles. Assign independently 11
generated random (3 x 1)
2) It is a type-2 alignment set with no restricted internal vector from a two
dimensional space 10
conflicts. to each vertex
Proof: 12
We first make some observations before giving the achiev-
able index coding scheme.
Observation1:ConsideranalignmentsetAofIwhichdoes
Fig.8. ThreetypesofalignmentsetsdiscussedintheproofofTheorem7
not contain both forks and cycles and is also not a triangular
areshown.Inthetype-2alignmentset,thereceiversatwhichthetriangular
interfering set. We claim that there is no triangular interfering interfering sets interfere are not explicitly shown in the figure. Since there
setwithinA.Thisisbecauseifthereisatriangularinterfering should not be any restricted internal conflict within a type-2 alignment set,
no triangular interfering set interferes at a receiver inside the same type-2
set (say W ) then there should be at least one more message
1 alignmentset.
which is not included within W . However this would mean
1
that there is both a fork and a cycle within A, which is not
allowed. and the technique of assignment, which follows, depends on
Observation2: Nowsupposetherearethreemessagesin A the type of the alignment set.
such that all of them interfere with a particular receiver. Then Alignment set which has no three messages interfering at
there must necessarily be no conflicts in-between the three any receiver: For each message in such an alignment set, we
messages. This is because if there were conflicts in between assign an independently generated random 3×1 vector (over
the three messages, then these three messages will form a a large field F).
triangular interfering set. Thus no three messages from A Alignment set which consists only of three messages inter-
havingatleastoneconflictin-betweenthemselvescaninterfere fering at any receiver without any conflicts in-between: We
at the same receiver. randomly generate a 3 × 1 vector and assign it to all the
Observation 3: By a similar argument as in Observation 1, messages in such an alignment set.
ifAhasthreemessagesinterferingatareceiver,thenitcannot Alignment set which is a type-2 alignment set without
have any other message than these three messages (because if restricted internal conflicts: Let W(cid:48) be the type-2 alignment
it did, then we would have both cycles and forks within A. set under concern. For each W(cid:48)-restricted alignment set, we
Observation4:ConsideranalignmentsetBofIwhichdoes assignanindependentlygeneratedrandom3×1vectorfroma
not contain both forks and cycles, but is also a triangular in- twodimensionalspace.Notethatthisresolvesalltheconflicts
terferingset.Then,bydefinitionBmustbeatype-2alignment betweenthemessageswithinW(cid:48),byTheorem5andCorollary
set. For the sake of this proof, we consider an alignment set 1.
such as B as being under the class of type-2 alignment sets.
Allthemessagesfallunderoneofthesealignmentsets,and
We thus have three kinds of alignment sets in I.
hence all of the messages have been assigned vectors at this
1) Alignment sets which have no three messages interfering point. Let E denote encoding function corresponding to this
at any receiver. assignmentandV denotethevectorassignedtomessageW .
k k
2) Alignmentsetswhichconsistsonlyofthreemessages,all Wenow showthatthis assignmentresolvesall theconflictsin
three interfering at some receiver, without any conflicts I.
in-between. (these three messages may interfere at other Consider a receiver j which requests a message W . We
k
receivers also, but at least one common receiver where proceed on a case by case basis, depending on the size of
they all interfere exists). Interf (j). For each case we check whether the condition,
k
3) Alignment sets which are also type-2 alignment sets Vk ∈/ VE(Interfk(j)) (whp), is met. We call this condition as
without restricted internal conflicts. the no conflict condition for the sake of this proof.
We now give the achievability index coding scheme by Case 1: |Interf (j)| ≤ 2 : There are two cases here,
k
assigning vectors independently to each alignment set of I, either W and Interf (j) are in the same alignment set or
k k
they are in different alignment sets. If W and Interf (j) connection to topological interference management problems
k k
are in different alignment sets, then W and the messages follows from [12]. From [2], it is known that the length
k
Interf (j) have been assigned independently and randomly of scalar linear index codes for the single unicast index
k
generated 3×1 vectors. Thus the no conflict condition is met. coding problem is known to be equal to minrank of the side-
Now, W and Interf (j) are in the same alignment set. information graph. Thus, our results also imply a class of
k k
This can only be an alignment set where no three messages graphswhoseminrankisequalto3.Thisisapromisingresult
interfere at any receiver, or a type-2 alignment set. In the as the general minrank problem is known to be NP hard [14].
former case, the no conflict condition is met as any three
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V. DISCUSSION
Inthiswork,wepresentedaclassofindexcodingproblems
for which rate 1 is feasible. This class of problems is larger
3
than what was previously known. We believe that the frame-
work developed in this work in order to obtain our results
can be leveraged to settle the rate 1 feasibility completely. In
3
particular, we conjecture that the the necessary condition for
rate 1 feasibility of Theorem 6 is also sufficient.
3
Conjecture. A given index coding problem is rate 1 feasible
3
if and only if all the type-2 alignment sets have no restricted
internal conflicts.
Developing conditions for feasibility of rates of the form
1 would also be an interesting area of future study. The
m