A New Index Coding Scheme Exploiting Interlinked Cycles
AA New Index Coding Scheme ExploitingInterlinked Cycles
Chandra Thapa, Lawrence Ong, and Sarah J. Johnson
School of Electrical Engineering and Computer Science, The University of Newcastle, Newcastle, AustraliaEmail: [email protected], [email protected], [email protected]
Abstract —We study the index coding problem in the unicastmessage setting, i.e., where each message is requested by oneunique receiver. This problem can be modeled by a directedgraph. We propose a new scheme called interlinked cycle cover,which exploits interlinked cycles in the directed graph, fordesigning index codes. This new scheme generalizes the existingclique cover and cycle cover schemes. We prove that for a class ofinfinitely many digraphs with messages of any length, interlinkedcycle cover provides an optimal index code. Furthermore, theindex code is linear with linear time encoding complexity.
Index Terms —Index coding, unicast, optimal broadcast rate,linear codes, interlinked cycles.
I. I
NTRODUCTION
We consider a source sending message packets through anoiseless broadcast channel to multiple receivers, each know-ing some packets a priori, which is known as side information .One can exploit the side information to reduce the numberof coded packets to be sent by the source, for all receiversto decode their requested messages. This is known as theindex coding problem and was introduced by Birk and Kolin 1998 [1]. The problem can be modeled by a digraph (i.e.,directed graph). The aim is to find an optimal scheme, whichprovides the minimum number of coded packets. Birk andKol used graph theory to find upper and lower bounds tothe minimum number of coded packets. Subsequently, tighterbounds were found using various approaches including graphtheory [1]–[5], Shannon random coding [6], [7], numericalapproaches, i.e., linear programming [8], and interferencealignment [9], [10]. However, the index coding problem re-mains open to date.Among graph-theoretic approaches, clique cover [1] andcycle cover [2]–[4] are useful as they provide insights on howto code on specific graph structures (as opposed to numericalapproaches) and they are valid for message packets of anylength (as opposed to random coding). However, they code ondisjoint cycles and cliques on the digraph, ignoring useful sideinformation captured in interlinked cycles. In this paper, wepropose a new scheme, called interlinked cycle cover (
ICC ),to exploit interlinked cycles. The
ICC scheme turns out to bea generalization of clique cover and cycle cover.Index codes generated by
ICC are scalar linear codes. Linearcodes simplify encoding and decoding process over non-linear
This work is supported by the Australian Research Council under grantsFT110100195, FT140100219, and DP150100903. codes. Ong [11] [12], found some classes of graphs wherescalar linear codes are optimal. These classes of graphs haveeither five vertices or fewer, or the property that the removal oftwo vertices results in a maximum acyclic induced subgraph(MAIS). In fact, optimal linear codes for a digraph can befound using the minrank function [2], which is, however, NP-hard to compute [13] in general. In this paper, we characterizea class of digraphs for which scalar linear codes generated by
ICC are optimal.
A. Our Contributions
1) We propose a new index coding scheme,
ICC , whichgeneralizes the cycle cover and the clique coverschemes.2) We show that for some digraphs,
ICC can outperformexisting techniques for message packets of finite length.3) We characterize a class of digraphs where
ICC is optimal(over all codes, including non-linear index codes).II. D
EFINITIONS
Suppose we have an index coding problem in which a sourcewants to send n message packets X = { x , x , . . . , x n } to n receivers, where each receiver is requesting a unique messagepacket x i (i.e., unicast), and each receiver has some sideinformation, S i ⊆ X \{ x i } . This problem can be described bya digraph D = ( V, A ) , where V = { v , v , . . . , v n } is the setof vertices representing the n receivers. An arc ( v i → v j ) ∈ A exists from vertex v i to vertex v j if receiver v i has packet x j (requested by receiver v j ) as its side information. If vertex v i has an out-neighborhood N + D ( v i ) , then the side informationof v i is S i = { x j : v j ∈ N + D ( v i ) } . For simplicity, we use theterm “messages” to refer to message packets in the remainderof this paper. Definition 1: (Valid index codes)
Let x i ∈ { , } t for all i , and for some integer t ≥ , i.e., each message contains t binary bits. Given an index coding problem D , a valid indexcode ( F , { G i } ) is defined as follows:1) An encoding function for the source, F : { , } nt →{ , } p , which maps X to a p -bit index for some integer p .2) A decoding function G i for every receiver v i , G i : { , } p × { , } | S i | t → { , } t , that maps the receivedindex F ( X ) and its side information S i to the requestedmessage x i . a r X i v : . [ c s . I T ] A p r ath P Path P Path P k Path P , Path P , Path P ,k Path P k, Path P ,k Path P k, Fig. 1. An
ICC
Digraph.
The broadcast rate of the ( F , { G i } ) index code is thenumber of transmitted bits per received message bits at everyuser, or the number of coded packets (of t bits), denoted by (cid:96) t ( D ) (cid:44) pt . Thus, the optimal broadcast rate for a given indexcoding problem D with t -bit messages is β t ( D ) = min F pt =min F (cid:96) t ( D ) (cid:44) (cid:96) ∗ t ( D ) . For a given index coding problem D ,the minimum optimal broadcast rate over all t is defined as β ( D ) = inf t β t ( D ) . Definition 2: (Path and cycle)
In a digraph, a path com-prises a sequence of distinct (except possibly the first andlast) vertices, say u , u , . . . , u L , and, an arc ( u i → u i +1 ) for each consecutive pair of vertices ( u i , u i +1 ) for all i ∈{ , . . . , L − } . Here, u is called the initial vertex, and u L the terminal vertex of the path. If the initial vertex and terminalvertex of a path are the same, then it is called a cycle .III. C ONSTRUCTION OF INTERLINKED CYCLE COVER ( ICC ) A. Definition of
ICC digraphs
We now construct a class of digraphs, which we call
ICC digraphs. In an
ICC digraph, there are two types of paths,Type-I and Type-II, where the terminal vertex of each Type-Ipath has an out-degree of k − , and the terminal vertex ofeach Type-II path has an out-degree of , for some k ≥ (seeFig. 1). More specifically, an ICC digraph D = ( V, A ) with n vertices consists of • k paths of Type-I, where k is a positive integer, • k ( k − paths of Type-II, and • interconnecting arcs in between paths of Type-I and Type-II, or between paths of Type-I.The k paths of Type-I are denoted by P i for i =1 , , . . . , k . Each P i contains a sequence of n i ≥ vertices { v i , v i , , . . . , v in i } , and arcs { ( v ia → v ia +1 ) : for all a =1 , , . . . , ( n i − } . Similarly, the k ( k − paths of Type-II are denoted by P i,j for each ordered pair ( i, j ) from { , , . . . , k } where i (cid:54) = j . Each P i,j contains a sequence of n ij ≥ vertices { v ij , v ij , . . . , v ijn ij } and arcs { ( v ija → v ija +1 ) : for all a =1 , , . . . , ( n ij − } .We now define interconnecting arcs between different paths:For each ( i, j ) , if n ij ≥ , one interconnecting arc connectsthe terminal vertex of P i to the initial vertex of P i,j , i.e., ( v in i → v ij ) ∈ A , and another arc connects the terminal vertexof P i,j to some vertex of P j , i.e., ( v ijn ij → v jq i ) ∈ A for some v jq i ∈ { v j , v j , . . . , v jn j } . Otherwise, ( n ij = 0 , i.e., P i,j = ∅ ),then one interconnecting arc connects the terminal vertex of P i directly to some vertex of P j , i.e., ( v in i → v jq i ) ∈ A . Werequire that the initial vertex v j of each path P j has at leastone in-degree . Fig. 1 is a graphical representation of ICC digraphs.The sets of vertices of all paths P i and P i,j are mutuallydisjoint. So, the total number of vertices in D is n = (cid:88) i n i + (cid:88) i,j s.t. i (cid:54) = j n ij . (1)Let x ia denote the message requested by receiver v ia , and x ija that requested by v ija . B. Code construction for
ICC digraphs
For any
ICC digraph D , we propose a valid index codethat maps n message packets (of t bit each) to (cid:96) ICC ( D ) codedsymbols (of t bits each), consisting of1) coded symbols obtained by the bitwise XOR ( ⊕ ) of eachmessage pair requested by adjacent vertices of paths P i for all i ∈ { , , . . . , k } , and n i ≥ , w ia = x ia ⊕ x ia +1 , for a = 1 , , . . . , ( n i − , (2)(if n i = 0 or , then no w ia is constructed),2) coded symbols obtained by the bitwise XOR of eachmessage pair requested by adjacent vertices of paths P i,j for all i (cid:54) = j, i, j ∈ { , , . . . , k } , and n ij ≥ , w ija = x ija ⊕ x ija +1 , for a = 1 , , . . . , ( n ij − , (3)(if n ij = 0 or , then no w ija is constructed),3) coded symbols obtained by the bitwise XOR of themessage requested by the terminal vertex of P i,j andthat by v jq i of P j for all i (cid:54) = j , w ijn ij = x ijn ij ⊕ x jq i , (4)(if n ij = 0 , then no w ijn ij is constructed), and4) a coded symbol obtained by the bitwise XOR of mes-sages requested by the terminal vertex of all paths P i , w (cid:48) = k (cid:77) i =1 x in i . (5) Remark 1:
The encoding of the above code requires lessthan or equal to t ( n − bit-wise XOR operations.Now, the index code constructed for the ICC digraph is W = { ( w ia ) ∀ i, ∀ a , ( w ijb ) ∀ ij, ∀ b , ( w ijn ij ) ∀ ij , w (cid:48) } . The total num-ber of coded symbols, each of t -bits, in W is, We can show that our results also apply to the digraphs without thisrestriction.
ICC ( D ) = 1 + (cid:88) i ( n i −
1) + (cid:88) i,j s.t. i (cid:54) = j ( n ij −
1) + (cid:88) i,j s.t. i (cid:54) = j
1= 1 + (cid:88) i n i + (cid:88) i,j s.t. i (cid:54) = j n ij − k = n − k + 1 . (6)Let us show that all vertices in D can decode their respectiverequested messages from W . From (2), in any path P i , allvertices v ia , except the terminal vertex v in i , can decode theirrequested messages. This is because by construction, for each a = 1 , , . . . , ( n i − , vertex v ia has message x ia +1 as sideinformation.From (3), in any path P i,j , all vertices v ija , except theterminal vertex v ijn ij , can decode their respective messages.This is because by construction, for all a = 1 , , . . . , ( n ij − ,each vertex v ija has message x ija +1 as side information.Similarly, v ijn ij knows x jq i . Thus from (4) the terminal vertexof each P i,j can decode its message.For n ij ≥ , and i (cid:54) = j , we evaluate the following: n j − (cid:77) a = q i w ja ⊕ n ij − (cid:77) b =1 w ijb ⊕ w ijn ij = n j − (cid:77) a = q i ( x ja ⊕ x ja +1 ) ⊕ n ij − (cid:77) b =1 ( x ijb ⊕ x ijb +1 ) ⊕ ( x ijn ij ⊕ x jq i )= ( x jq i ⊕ x jn j ) ⊕ ( x ij ⊕ x ijn ij ) ⊕ ( x ijn ij ⊕ x jq i )= ( x ij ⊕ x jn j ) (cid:44) w (cid:48)(cid:48) ij . (7)Similarly, we evaluate the following: (cid:77) h ∈{ ,...,k }\{ i } s.t. n ih =0 (cid:32) n h − (cid:77) a = q i w ha (cid:33) = (cid:77) h ∈{ ,...,k }\{ i } s.t. n ih =0 (cid:32) n h − (cid:77) a = q i ( x ha ⊕ x ha +1 ) (cid:33) = (cid:77) h ∈{ ,...,k }\{ i } s.t. n ih =0 ( x hq i ⊕ x hn h ) = Y i ⊕ Y (cid:48) i , (8)where, Y i (cid:44) (cid:76) h ∈{ ,...,k }\{ i } s.t. n ih =0 x hq i , and Y (cid:48) i (cid:44) (cid:76) h ∈{ ,...,k }\{ i } s.t. n ih =0 x hn h .Again, we evaluate the following: (cid:77) h ∈{ ,...,k }\{ i } s.t. n ih ≥ w (cid:48)(cid:48) ih = (cid:77) h ∈{ ,...,k }\{ i } s.t. n ih ≥ ( x ih ⊕ x hn h ) = Z i ⊕ Z (cid:48) i , (9)where, Z i (cid:44) (cid:76) h ∈{ ,...,k }\{ i } s.t. n ih ≥ x ih , and Z (cid:48) i (cid:44) (cid:76) h ∈{ ,...,k }\{ i } s.t. n ih ≥ x hn h .On the other hand, we can expand w (cid:48) as: w (cid:48) = k (cid:77) i =1 x in i = x in i ⊕ (cid:77) h ∈{ ,...,k }\{ i } s.t. n ih =0 x hn h ⊕ (cid:77) h ∈{ ,...,k }\{ i } s.t. n ih ≥ x hn h = x in i ⊕ Y (cid:48) i ⊕ Z (cid:48) i . (10)Now, using (10), (8), and (9), we evaluate the following: ( x in i ⊕ Y (cid:48) i ⊕ Z (cid:48) i ) ⊕ ( Y i ⊕ Y (cid:48) i ) ⊕ ( Z i ⊕ Z (cid:48) i ) = x in i ⊕ Y i ⊕ Z i . (11)From (11), the terminal vertex of each P i , i.e., v in i , can decodeits requested message x in i because by construction, if n ij ≥ (for the term Z i ), then v in i has x ij as side information, and if n ij = 0 (for the term Y i ), then v in i has x jq i as side information. Therefore, from (2), (3), (4) and (5) all the vertices in D candecode their requested messages. Hence, the index code W isa valid index code . Definition 3: ( Saved packets ) The term saved packets (orsimply savings) is the number of packets saved (i.e., n − (cid:96) t ( D ) )by sending coded packets (coded symbols) rather than sendinguncoded message packets. Remark 2: If k = 1 , then there exists only a single path P in the ICC digraph. Thus a valid index code in this casewill be w a = x a ⊕ x a +1 , for a = 1 , , . . . , ( n i − , and w (cid:48) = x n i . Here, the number of coded symbols equals thenumber of vertices, and so no saved packets is obtained.IV. R ESULTS
A. The
ICC
Scheme
Now, we formally state our proposed
ICC scheme:
Definition 4: ( Interlinked Cycle Cover ( ICC ) scheme ) Forany digraph, the ICC scheme finds a set of disjoint
ICC subgraphs. It then (a) codes each of these
ICC subgraphs usingthe code construction described in Section III.B, and (b) sendsuncoded messages requested by all remaining vertices (i.e.,vertices which are not in any of these disjoint
ICC subgraphs).We denote an
ICC digraph with k number of Type-I pathsas a k - ICC digraph. Using the
ICC scheme on an
ICC digraph,we have the following:
Lemma 1:
For a k - ICC digraph D with t -bit messages, forany k ≥ and any t ≥ , the total number of saved packetsusing the ICC scheme is k − , i.e., n − (cid:96) ICC ( D ) = k − . Proof:
Subtracting (cid:96)
ICC ( D ) of (6) from n we get n − (cid:96) ICC ( D ) = n − ( n − k + 1) = k − . (12)We can generalize this to an arbitrary digraph: Theorem 1:
For any digraph D , a valid index code of length (cid:96) ICC ( D ) = n − (cid:80) ψi =1 ( k i − can be constructed using the ICC scheme, where ( k i − is the saving in each disjoint k i - ICC subgraphs, and ψ is the number of disjoint ICC subgraphs.
Proof:
For any digraph D containing ψ number of disjoint ICC subgraphs, each k i - ICC subgraph gives a saving of k i − (Lemma 1), where i ∈ { , . . . , ψ } . The total savings is the sumof savings in all disjoint ICC subgraphs, i.e., (cid:80) ψi =1 ( k i − .Therefore, (cid:96) ICC ( D ) = n − (cid:80) ψi =1 ( k i − . Remark 3:
The
ICC subgraphs found by the
ICC scheme arenot unique. So, finding the best (cid:96)
ICC ( D ) involves optimizingover all choices of disjoint ICC subgraphs in D . B. ICC includes cycle cover and clique cover as special casesTheorem 2:
The
ICC scheme includes the cycle coverscheme and the clique cover scheme as special cases.
Proof:
Let us consider a cycle having L verticesand ( L − arcs for some integer L ≥ , i.e., { v , ( v → v ) , v , . . . , ( v L − → v L ) , v L , ( v L → v L +1 ) , v L +1 , . . . , ( v L − → v L ) , v L , ( v L → v ) , v } , where ≤ L < L . For this cycle, the cycle cover scheme providesa valid index code of length ( L − [3], [4], i.e., ( x ⊕ x ) , ( x ⊕ x ) , . . . , ( x L − ⊕ x L ) , ( x L ⊕ x L +1 ) , ( x L +1 ⊕ x L +2 ) . . . , ( x L − ⊕ x L − ) , ( x L − ⊕ x L ) . (13) ath P v v v L − v L Path P v L +1 v L +2 v L − v L (a) Path P Path P Path P k v v v L (b)Fig. 2. Special cases of ICC digraphs (a) with k = 2 , n = L , n = L − L , n = n = 0 , v q = v , and v q = v L +1 forming a cycle ,and (b) with any k = L ≥ , n i = 1 ∀ i, and n ij = 0 ∀ i (cid:54) = j forming aclique. Here the saving is always one packet. This cycle can be viewedas a - ICC digraph, which is shown in Fig. 2(a). Using the
ICC scheme we get a valid index code of length (cid:96)
ICC ( D ) = n − k + 1 = L − , i.e., ( x ⊕ x ) , ( x ⊕ x ) , . . . , ( x L − ⊕ x L ) , ( x L +1 ⊕ x L +2 ) , ( x L +2 ⊕ x L +3 ) , . . . , ( x L − ⊕ x L ) , ( x L ⊕ x L ) . (14)Both the valid index codes from cycle cover and from ICC areof the same length. The difference (indicated in red) is that ( x L ⊕ x L +1 ) does not appear in the ICC code, and ( x L ⊕ x L ) does not appear in the cycle-cover code. But one can generate ( x L ⊕ x L +1 ) from the existing code symbols of the ICC codes and vice versa (the proof is straightforward).Furthermore, consider any digraph D with a total of n vertices and | C | disjoint cycles. The saving by cycle coveris one packet for each cycle. The messages of vertices notcovered by these selected cycles are sent uncoded. So, thetotal savings is the sum of savings for all disjoint cycles. Thelength of a valid index code from cycle cover is therefore (cid:96) cyc ( D ) = n − | C | (cid:88) r =1 n − | C | . (15)Similarly, for the same digraph D , considering each cycle asa - ICC subgraph, Theorem 1 gives (cid:96)
ICC ( D ) = n − | C | (cid:88) i =1 ( k i −
1) = n − | C | (cid:88) i =1 (1) = n − | C | . (16)Hence, both schemes return the same index code length forany digraph D , if the ICC scheme assigns one
ICC subgraphto each disjoint cycle. Moreover, the index codes from bothschemes are equivalent (using the above argument). Thisproves that cycle cover is also a special case of
ICC .To prove clique cover as a special case of
ICC , let usconsider a clique of L vertices { v , . . . , v L } where, L ≥ .The valid index code for this clique using the clique coverscheme is of length one, i.e., ( x ⊕ x ⊕ . . . ⊕ x L ) . The cliquecan be viewed as a L - ICC digraph, which is shown in Fig.2(b). The
ICC scheme gives the same valid index code as thatgiven by clique cover.Furthermore, consider any digraph D with n vertices and | χ | disjoint cliques, where each clique r ∈ { , , . . . , | χ |} consists of n r vertices. The saving by clique cover is n r − packetsfor each clique r . The messages corresponding to vertices notcovered by these disjoint cliques are sent uncoded. So, thetotal savings is the sum of savings for each disjoint clique.The length of a valid index code from clique cover is (cid:96) cc ( D ) = n − | χ | (cid:88) r =1 ( n r −
1) = n − | χ | (cid:88) r =1 n r + | χ | . (17)Similarly, for the same digraph D , considering each clique asa n i - ICC subgraph, the length of a valid index code by the
ICC scheme using Theorem 1 is (cid:96)
ICC ( D ) = n − | χ | (cid:88) i =1 ( n i −
1) = n − | χ | (cid:88) i =1 n i + | χ | . (18)Hence, both schemes return the same index code length for anydigraph D , if the ICC scheme assigns one
ICC digraph to eachdisjoint clique. Moreover, the index codes from both schemesare equivalent. This proves that clique cover is a special caseof
ICC . C. ICC is optimal for any
ICC digraph
We first prove the following lemma:
Lemma 2:
In an
ICC digraph, any cycle that contains avertex v ∈ P i must also contain the terminal vertex v in i , andany cycle that contains a vertex v ∈ P i,j must also containthe terminal vertex v jn j . Proof:
For any cycle containing v , there must be a path,say P , from v back to itself.(Case 1) If v ∈ P i (where P i is not a cycle), then thepath P must leave P i . By construction, any arc that leaves P i originates from v in i . Hence, P must contain v in i . So, any cyclethat contains v ∈ P i must also contain v in i .(Case 2) If v ∈ P i,j (where P i,j is again not a cycle), thenthe path P must leave P i,j . There is only one arc leaving P i,j , which is from v ijn ij ∈ P i,j to v jq i ∈ P j . Note that v / ∈ P j . Repeating the argument for Case 1, the path P mustgo through v jn j before going back to v (to form a cycle). Soany cycle that contains v ∈ P i,j must also contain v jn j .With the above lemma, we now show the following: Theorem 3:
For any t ≥ , the linear index code given bythe ICC scheme is optimal for any
ICC digraph, i.e., (cid:96) ∗ t ( D ) = (cid:96) ICC ( D ) . Proof:
It has been shown [2] that for any digraph D andany message length t , (cid:96) ∗ t ( D ) ≥ MAIS ( D ) , where MAIS ( D ) isthe order of a maximum acyclic induced subgraph of D . Toobtain MAIS ( D ) , one has to remove the minimum number ofvertices from D to make it acyclic.Consider a k - ICC digraph D . From Lemma 2, we knowthat any cycle must contain the terminal vertex of a Type-Ipath, say v in i . Note that any outgoing arc from v in i terminatesat a vertex in either (a) P i,j for some j , or (b) P j for some j (cid:54) = i . Using the same argument in the proof of Lemma 2, anycycle that contains v in i must also contain v jn j for some j (cid:54) = i .So, every cycle must contains at least two terminal vertices ofType-I. Therefore, removing ( k − terminal vertices of Type-Ipaths makes D acyclic. This gives, MAIS ( D ) ≥ n − ( k − . v v v v v (a) v v v v v v v v v v = (b)Fig. 3. ICC digraphs: (a) D with n = 6 , k = 3 , n = n = n = 2 andall n ij = 0 , and (b) D with n = 5 , k = 3 , n = 1 , n = n = 2 , andall n ij = 0 . The removal of any k − or fewer vertices from an ICC digraph cannot make the digraph acyclic. This can be provedby the following reasoning. The removal of any vertex, say v (which must belong to some path P i or path P j,i ), to breakcycles containing v is no better than the removal of v in i (whichalso breaks those cycles). This is due to Lemma 2. It followsthat the removal of any k − or fewer vertices cannot be betterthan the removal of k − or fewer terminal vertices. Even if k − terminal vertices are removed, say { v in i : i = 3 , , . . . , k } without loss of generality, P , P , , P , and P , form a cycle,which is not removed. Thus, k − is the least possible removalto make an ICC digraph acyclic, i.e.,
MAIS ( D ) ≤ n − ( k − .Combining the upper and lower bounds, we have MAIS ( D ) = n − k + 1 ≤ (cid:96) ∗ t ( D ) . (19)From Lemma 1 we get, (cid:96) ICC ( D ) = n − k + 1 ≥ (cid:96) ∗ t ( D ) . (20)From (19) and (20), we get (cid:96) ∗ t ( D ) = (cid:96) ICC ( D ) .For any ICC digraph D , β t ( D ) = (cid:96) ICC ( D ) = n − k + 1 ,which is independent of t . This means β ( D ) = inf t β t ( D ) = n − k + 1 = (cid:96) ICC ( D ) , and we have the following: Corollary 1:
For any
ICC digraph, the
ICC scheme achieves β ( D ) . D. ICC can outperform existing techniques
For some digraphs,
ICC can outperform existing tech-niques such as clique cover ( cc ) [1], fractional clique cover( fcc ) [8], partial clique cover ( pcc ) [1], fractional partialclique cover ( fpcc ) [14], cycle cover ( cyc ) [2]–[4], localchromatic number ( lc ) [5], and local time sharing bounds ( b ( R LTS ( D )) and b LTS ( D )) [14]. Here are two examples:For the ICC digraph D in Fig. 3(a), a valid index code fromthe ICC scheme is { x ⊕ x , x ⊕ x , x ⊕ x , x ⊕ x ⊕ x } ,which is of length four i.e. (cid:96) ICC ( D ) = 4 . For this digraph, β ( D ) = (cid:96) ICC ( D ) = 4 < (cid:96) fpcc ( D ) = 4 . < (cid:96) lc ( D ) = (cid:96) pcc ( D ) = (cid:96) cyc ( D ) = 5 < (cid:96) cc ( D ) = (cid:96) fcc ( D ) = 6 .Similarly, for the ICC digraph D in Fig. b ) , a valid indexcode from the ICC scheme is { x ⊕ x , x ⊕ x , x ⊕ x ⊕ x } ,which is of length three i.e. (cid:96) ICC ( D ) = 3 . For this digraph, β ( D ) = (cid:96) ICC ( D ) = 3 < b LTS ( D ) = b ( R LTS ( D )) = 7 / < (cid:96) lc ( D ) = 4 .Furthermore, some of the existing techniques (e.g., pcc , lc )use maximum distance separable (MDS) codes, which requires t to be sufficiently large.We now describe a class of digraphs where the ICC schemeoutperforms the local chromatic number in the order of theorder of the digraph (i.e., the number of vertices). Consider adigraph D with even number of vertices, n = 2 k , where k isany positive integer. Furthermore, the vertices can be groupedinto two sets, without loss of generality, say V = { v , . . . , v k } and V = { v k +1 , . . . , v n } , such that for each i ∈ { , . . . , k } , v k + i knows a message requested by v i , and v i knows messagesrequested by all V \ { v k + i } . We can show that the gap (cid:96) lc ( D ) − (cid:96) ICC ( D ) for this type of digraphs grows linear with n . Note that D in Fig. 3(a) belongs to this class of digraphswith k = 3 . V. C ONCLUSION
For unicast index coding problems, we designed a newcoding scheme, called interlinked cycle cover (
ICC ), whichexploits interlinked cycles in the digraph. Our proposed
ICC scheme includes clique cover and cycle cover as special cases.We proved that this scheme gives an optimal index code for aclass of digraphs, namely,
ICC digraphs, and it can outperformexisting schemes. R
EFERENCES[1] Y. Birk and T. Kol, “Informed-source coding-on-demand (ISCOD) overbroadcast channels,” in
Proc. IEEE INFOCOM , vol. 3, San Francisco,CA, Mar. 1998, pp. 1257–1264.[2] Z. Bar-Yossef, Y. Birk, T. S. Jayram, and T. Kol, “Index coding withside information,”
IEEE Transactions on Information Theory , vol. 57,no. 3, pp. 1479–1494, Mar 2011.[3] M. J. Neely, A. S. Tehrani, and Z. Zhang, “Dynamic index coding forwireless broadcast networks,”
IEEE Transactions on Information Theory ,vol. 59, no. 11, pp. 7525–7540, Nov 2013.[4] M. A. R. Chaudhry, Z. Asad, A. Sprintson, and M. Langberg, “On thecomplementary index coding problem,” in
Proc. IEEE Int. Symp. Inf.Theory (ISIT) , July 2011, pp. 224–248.[5] K. Shanmugam, A. G. Dimakis, and M. Langberg, “Local graphcoloring and index coding,” in
Proc. IEEE International Symposiumon Information Theory (ISIT) , July 2013, pp. 1152–1156.[6] F. Arbabjolfaei, B. Bandemer, Y.-H. Kim, E. Sasoglu, and L. Wang,“On the capacity region for index coding,” in
Proc. IEEE InternationalSymposium on Information Theory (ISIT) , July 2013, pp. 962–966.[7] S. Unal and A. B. Wagner, “General index coding with side informa-tion: Three decoder case,” in
Proc. IEEE International Symposium onInformation Theory (ISIT) , July 2013, pp. 1137–1141.[8] A. Blasiak, R. D. Kleinberg, and E. Lubetzky, “Broadcasting with sideinformation: Bounding and approximating the broadcast rate,”
IEEETransactions on Information Theory , vol. 59, no. 9, pp. 5811–5823,Sept 2013.[9] H. Maleki, V. R. Cadambe, and S. A. Jafar, “Index coding – Aninterference alignment perspective,” May 2012. [Online]. Available:http://arxiv.org/pdf/1205.1483v1.pdf[10] S. A. Jafar, “Topological interference management through indexcoding,” Sep 2013. [Online]. Available: http://arxiv.org/pdf/1301.3106v2.pdf[11] L. Ong, “Linear codes are optimal for index-coding instances with five orfewer receivers,” in
Proc. IEEE International Symposium on InformationTheory (ISIT) , June 2014, pp. 491–495.[12] ——, “A new class of index coding instances where linear coding isoptimal,” in
Proc. IEEE International Symposium on Network Coding(NetCod) , June 2014, pp. 1–6.[13] R. Peeters, “Orthogonal representaions over finite fields and the chro-matic number of graphs,”
Combinatorica , vol. 16, no. 3, pp. 417–431,Sept 1996.[14] F. Arbabjolfaei and Y.-H. Kim, “Local time sharing for index coding,”in