Generalized Alignment Chain: Improved Converse Results for Index Coding

Transcription

1 Generalized Alignent Chain: Iproved Converse Results for Index Coding Yucheng Liu and Parastoo Sadeghi Research School of Electrical, Energy and Materials Engineering Australian National University, Canberra, ACT, 260, Australia Eails: {yucheng.liu, arxiv: v [cs.it] 26 Jan 209 Abstract In this work, we study the inforation theoretic converse for the index coding proble. We generalize the definition for the alignent chain, introduced by Maleki et al., to capture ore flexible relations aong interfering essages at each receiver. Based on this, we derive iproved converse results for the single-server centralized index coding proble. The new bounds uniforly outperfor the axiu acyclic induced subgraph bound, and can be useful for large probles, for which the generally tighter polyatroidal bound becoes coputationally ipractical. We then extend these new bounds to the ulti-server distributed index coding proble. We also present a separate, but related result where we identify a saller centralized index coding instance copared to those identified in the literature, for which non-shannon-type inequalities are necessary to give a tighter converse. I. INTRODUCTION Index coding, introduced by Birk and Kol in [], investigates the noiseless broadcast rate of n essages fro a server to ultiple receivers. Each receiver wants to decode a unique essage and has prior knowledge of soe other essages as its side inforation. During the past two decades, there has been substantial progress in theory and applications of index coding. See [2] and the references therein. However, in general, the index coding proble reains open. In contrast to the classic single-server centralized index coding (CIC) proble, in the ore general distributed index coding (DIC) proble different subsets of the n essages are stored at ultiple servers. Such counication odel has clear applications for practical circustances where the inforation is geographically distributed over ultiple locations. See [3] [5] and references therein. In this paper we focus on deriving inforation theoretic converse results for both CIC and DIC probles. For the CIC proble, the axiu acyclic induced subgraph (MAIS) bound was proposed in [6], and the polyatroidal (PM) bound based on the polyatroidal axios of the entropy function was presented in [7], [8]. Both bounds have been extended to the DIC proble [4], [5]. The PM bound is generally tighter than the MAIS bound. However, the coputational coplexity of the PM bound is uch higher than that of the MAIS bound, which can be prohibitively high for probles with a large nuber of essages. Therefore, it is iperative to find a iddle ground between the easily coputable yet looser MAIS bound and the highcoplexity yet tighter PM bound. The internal conflict bound for the CIC proble, introduced in [9], [0] based on the alignent chain odel, can soeties be useful. However, it does not subsue the MAIS bound and these two bounds can outperfor each other for different instances of the proble. In this paper we first generalize the internal conflict bound for the CIC proble by extending the alignent chain odel. The new converse results (Th., Cor. in Section IV-A), which are of closed-for given the generalized alignent chains, uniforly outperfor the internal conflict bound and the MAIS bound. We show by exaples that such relationships are soeties strict. We then generalize these results to the DIC proble (Th. 2, Cor. 2 in Section IV-B), and show their efficacy by another exaple. Finally in Section IV-C, we present a separate result. That is, we identify a 9-essage CIC proble, which is, to the best of our knowledge, the sallest CIC proble in ters of the nuber of essages ever identified, for which non-shannon-type inequalities are necessary to derive a tighter converse on the capacity region. Notation: For positives integers a, b, [a] denotes the set {, 2,, a}, and [a : b] denotes the set {a, a,, b}. If a > b, [a : b] =. For a set S, S denotes its cardinality. II. SYSTEM MODEL Assue that there are n essages, x i {0, } ti, i [n]. For brevity, when we say essage i, we ean essage x i. Let X i be the rando variable corresponding to x i. We assue that X,..., X n are independent and uniforly distributed. For any S [n], set S c. = [n] \ S, xs = (x i, i S), and X S = (X i, i S). By convention x = X =. There are n receivers, where receiver i [n] wishes to obtain x i and knows x Ai as side inforation for soe A i [n] \ {i}. The set of the interfering essages at receiver i (i.e., neither wanted nor known) is denoted by B i = (A i {i}) c. To avoid redundancy, in the rest of this section, we focus on describing the reaining syste odel for the DIC proble, in which there are 2 n servers, each of which contains a unique nonepty subset of the n essages. The server indexed by J N contains essages x J, where N =. {J [n] : J }. Every server is connected to all receivers via its own noiseless broadcast channel with finite link capacity C J 0. Let y J {0, } r J be the output of server J to be broadcast, which is a function of x J, and Y J be the rando variable corresponding to y J. For any DIC proble, we define a (t, r) = ((t i, i [n]), (r J, J N)) distributed index code by See [], [2, Section 5.3] for soe previously identified CIC probles, for which non-shannon-type inequalities are necessary.

2 2 n encoders, one for each server J N, such that φ J : j J {0, }tj {0, } r J aps the essages x J in server J to an r J -bit sequence y J, and n decoders, one for each receiver i [n], such that ψ i : J N {0, }r J k A i {0, } t k {0, } ti aps the received sequences (y J, J N) and the side inforation x Ai to ˆx i. We say that a rate capacity tuple (R, C) = ((R i, i [n]), (C J, J N)) is achievable if for every ɛ > 0, there exist a (t, r) code and a positive integer r such that the essage rates ti r, i [n] and broadcast rates r J r, J N satisfy R i t i r, i [n], C J r J, J N, () r and that P{( ˆX,..., ˆX n ) (X,..., X n )} ɛ. For a given link capacity tuple C, the capacity region C (C) is the closure of the set of all rate tuples R such that (R, C) is achievable. The syetric capacity is defined as C sy (C) = ax{r sy : (R sy,, R sy ) C (C)}. (2) Clearly, the CIC proble can be seen as a special case of the DIC proble with C J = (noralized) for J = [n] and C J = 0 otherwise, and hence the centralized index code, the achievable rate tuple R, the capacity region C, and the syetric capacity C sy can be defined accordingly. Any CIC or DIC proble can be represented by a sequence (i j A i ), i [n]. For exaple, for A =, A 2 = {3}, and A 3 = {2}, we write ( ), (2 3), (3 2). Any proble can also be represented by a side inforation graph G with n vertices, in which vertex i represents essage i, and a directed edge (i, j) represents that i A j. For any nonepty S [n], G S denotes the subgraph of G induced by S. III. PRELIMINARIES We briefly review the MAIS bound [6] and the internal conflict bound [9], [0]. The forer is stated below. Proposition (MAIS bound, [6]): Given an arbitrary CIC proble (i A i ), i [n], if R sy is achievable, then in S [n]:g S is acyclic S. For the internal conflict bound, we first re-state the definition of the alignent chain [9] in our notation as follows. Definition (Alignent Chain, [9]): For any CIC proble (i A i ), i [n], essages i 0, i,, i and k,, k constitute an alignent chain of length denoted as k i k 0 2 k i i2 i, (3) if the conditions listed below are satisfied, ) i 0 B i or i B i0 ; 2) for any j [], we have B kj {i j, i j }. For any CIC proble, if such an alignent chain of length exists, then we say that there is an internal conflict between essages i 0 and i, leading to an upper bound [9] on the achievable syetric rate as 2. It has been shown in [9] that a given CIC proble is half-rate-infeasible, i.e. C sy < 2, if and only if there exists at least one alignent chain. We state the internal conflict bound as follows. Proposition 2 (Internal conflict bound, [0]): Given an arbitrary half-rate-infeasible CIC proble (i A i ), i [n], if 2 R sy is achievable, then, where denotes the iniu length of the proble s alignent chains. Definition depends on neither the nuber of the servers, nor the essage distribution aong the servers, and thus also works for the DIC proble. However, the internal conflict bound has never been extended to the ulti-server scenario. Also, the syetric capacity for the DIC probles with no alignent chains reains unclear. In this paper we only focus on the, naley, internally conflicted CIC and DIC probles, where there exist at least one alignent chain. IV. MAIN RESULTS A. Iproved Necessary Conditions for CIC Definition 2 (Disjoint Weighted Alignent Chain): For any internally conflicted CIC proble (i A i ), i [n], we have the following disjoint weighted alignent chain i 0 k,w k,2 k, i k 2,w2 k 2,2 k 2, i2 k 3,w3 k 3,2 k 3, k,w k,2 k, i, (4) siply denoted as i 0 i, where I =. {i j : j [0 : ]} and K =. {k j,l : j [], l [w j ]} are groups of essages, if the conditions listed below are satisfied, ) i 0 B i or i B i0 ; 2) B kj,l {k j,, k j,2, k j,l }, j [], l [w j ]; 3) for every j [], l [w j ], there exist two integers p j,l [0 : j ], q j,l [j : ], such that a) i pj,l B kj,l, and i qj,l B kj,l ; b) for any l < l 2 [w j ], we have j = p j, p j,l p j,l2, and j = q j, q j,l q j,l2 ;. c) for any j [], set G j = [pj,wj : q j,wj ] and M =. {j [] : G j 2}, then for any j j 2 M, G j G j2 =. Reark : For a disjoint weighted alignent chain, there are edges. The edge j [] is between essages i j and i j, and of weight w j. Any alignent chain can be seen as a disjoint weighted alignent chain with w j =, j []. k,w k,w k, k 2,w2 k 2,w2 k 2, k 4,w4 k 4,w4 k 4, i 0 i i 2 i 3 i 4 i k,w k,w k, Figure. A visualization exaple of the disjoint weighted alignent chain in Definition 2. For any edge j M = {j [] : G j 2}, the arrows outgoing fro its associated k j,l essages are purple. For any edge j [] \ M, the arrows outgoing fro its associated k j,l essages are blue. The arrows of the sae color do not intersect.

3 To visualize Definition 2, see Figure. For better understanding, for soe edges, we draw dashed straight arrows fro their associated k j,l to the corresponding i pj,l and i qj,l. The dashed arrows are blue if the corresponding edge j satisfies that p j,wj = q j,wj, and purple otherwise. Conditions 3b and 3c in Definition 2 ensure that the dashed arrows of the sae color can never cross. We have the following theore. Theore : Given an arbitrary internally conflicted CIC proble (i A i ), i [n], if R sy is achievable, then for any disjoint weighted alignent chain i 0 i we have I K = j [] w. (5) j The proof is presented in Section V-A. Fixing p j,l = j, q j,l = j for any j [], l [w j ], the disjoint weighted alignent chain reduces to the singleton weighted alignent chain defined as follows. Definition 3 (Singleton Weighted Alignent Chain): For any internally conflicted CIC proble (i A i ), i [n], we have the following singleton weighted alignent chain i 0 k,w k,2 k, i k 2,w2 k 2,2 k 2, i2 k 3,w3 k 3,2 k 3, k,w k,2 k, i, (6) siply denoted as i 0 s i, if the conditions listed below are satisfied, ) i 0 B i or i B i0 ; 2) B kj,l {k j,,, k j,l, i j, i j }, j [], l [w j ]. The corollary below follows directly fro Theore. Corollary : Given an arbitrary internally conflicted CIC proble (i A i ), i [n], if R sy is achievable, then for any singleton weighted alignent chain i 0 s i we have I K = j [] w. (7) j For any internally conflicted CIC proble (i A i ), i [n], let R DW, R SW, R, and R MAIS denote the upper bounds given by Theore, Corollary, the internal conflict bound in Proposition 2, and the MAIS bound in Proposition, respectively. The following proposition characterizes the relationships between these upper bounds. Proposition 3: R DW R SW R, and R SW R MAIS. Proof: As any alignent chain can be seen as a singleton weighted alignent chain, and any singleton weighted alignent chain can be seen as a disjoint weighted alignent chain, it is clear that R DW R SW R. It reains to copare R SW and R MAIS. Set s = R MAIS, then there exist s essages such that their corresponding vertex induced subgraph of the proble s side inforation graph is acyclic. In other words, there ust exist an ordering of these s essages, siply denoted as {i, i 2,, i s }, such that B ij {i,, i j }, j [s]. (8) Therefore, it can be verified that we have the following singleton weighted alignent chain, i i s i 3 i2, (9) and thus by Corollary, we have R SW s = R MAIS. The relationships in Proposition 3 can be strict soeties. Exaple : Consider the 6-essage CIC proble ( 2, 3, 4, 6), (2 4, 5, 6), (3, 2, 4, 5, 6) (4, 2, 6), (5 2, 3, 4, 6), (6 ). For this proble, R = R MAIS = 3. However, we have the following singleton weighted alignent chain, , and thus by Corollary we have 2 34 = 2 7, which atches the coposite coding lower bound [8] on the syetric capacity. Therefore, for this proble we have C sy = R DW = R SW = 2 7 < R = R MAIS = 3. Exaple 2: Consider the 0-essage CIC proble ( 3, 4, 5, 6, 7, 8, 9, 0), (2 3, 4, 5, 6, 7, 8, 9, 0) (3, 2, 4, 5, 6, 7, 8, 9, 0), (4, 2, 3, 5, 6, 7, 8, 9, 0), (5, 3, 6, 7, 8, 9, 0), (6 2, 4, 5, 7, 8, 9, 0), (7, 2, 5, 6, 8, 9, 0), (8, 3, 6, 7, 9, 0), (9 2, 3, 5, 7, 8, 0), (0, 2, 5, 6, 8, 9). For this proble, R = R MAIS = R SW = 3. However, we have the following disjoint weighted alignent chain, , and thus by Theore we have 3 46 = 3 0, which atches the coposite coding lower bound on the syetric capacity. Therefore, for this proble we have C sy = R DW = 3 0 < R SW = R = R MAIS = 3. Note that the above disjoint weighted alignent chain is not a singleton weighted alignent chain according to Definition 3 as B 9 = {, 4, 6} {, 3, 6}. The new bounds can be also be useful for soe probles with large n, for which the PM bound is coputationally infeasible in practice due to its forbidding coplexity. Exaple 3: Consider the 7-essage CIC proble as follows, denoted by (i B i ), i [n] rather than (i A i ), i [n] for liited space, ( 6), (2 7, 8), (3 8,, 7), (4 ), (5 ), (6 ), (7, 2), (8, 2, 3, 4, 7), (9 2, 3), (0, 4, 9), ( 3, 4, 8), (2 5, 6), (3 4, 5), (4 4, 6, 7, 3, 5, 7), (5 ), (6 5, 6, 2), (7 8). For this proble, R = R MAIS = R SW = 3. However, we have the following disjoint weighted alignent chain, ,

4 and thus by Theore we have R DW 5 60 = 5 6 < 3, which atches the coposite coding lower bound on C sy. Note that the above chain is not a singleton weighted alignent chain. For exaple, B 0 = {, 4, 9} {2, 3, 9}. B. Iproved Necessary Conditions for DIC Definitions 2 and 3 also work for the DIC proble, and hence Theore and Corollary can be extended as follows. Recall that M = {j [] : G j 2} is defined in Definition 2. Define M. = []\( G j). Also, for sipler notation, we use K j,l to denote the essage sequence k j,l, l L for soe j [], L [w j ]. Theore 2: Given any internally conflicted DIC proble (i A i ), i [n], if R sy is achievable, then for any disjoint weighted alignent chain i 0 i we have where ( I K l [q j,s :q j,s] ( s [2:w j] l [p j,s:p j,s ] J T 3, J T 5 T = {i pj,wj, i qj,wj, K j,[wj]}, C J ) j M M J T, J T 2 J T 3, J T 4 C J C J ), (0) T 2 = {i pj,wj, i, K j,[wj]}, T 3 = {i l, i l, K l,[wl ]}, T 4 = {i pj,s, i l, K l,[wl ]}, T 5 = {i qj,s, i l, K l,[wl ]}. The proof, which is based on the proof of Theore will not be presented due to the liited space. Corollary 2: Given any internally conflicted DIC proble (i A i ), i [n], if R sy is achievable, then for any singleton weighted alignent chain i 0 s i we have I K j [] J {i j,i j,k j,[wj ]} =, J {i j,i,k j,[wj ]} = C J. () Given that the RHS of (0) siplifies to the RHS of () when p j,l = j, q j,l = j for any j [], l [w j ], we can see that Corollary 2 follows directly fro Theore 2. Exaple 4: Consider the following DIC proble with n = 5 essages and equal link capacities C J =, J N, ( 2, 3, 4, 5), (2, 3, 4, 5), (3 2, 4, 5), (4 3, 5), (5, 4). For this proble, there exists a singleton weighted alignent 4 5 chain as 2 3, and thus by Corollary 2, 3 2 ( J {,2,4} =, J {,3,4} = J {2,3,5} = ) = 54 5, which achieves the syetric capacity of the proble. C. A 9-Message CIC Proble Where Non-Shannon-Type Inequalities Are Necessary Here we identify a 9-essage CIC proble, which is the CIC proble with the sallest nuber of essages n identified so far, for which non-shannon-type inequalities are necessary to derive a tighter converse on the capacity region. Exaple 5: Consider the following 9-essage CIC proble, denoted by (i B i ), i [n], ( 2), (2, 5, 8), (3 ), (4 ), (5 2, 4, 8), (6, 3), (7 3, 4), (8 2, 3, 5), (9, 4, 6). The PM bound 2 gives i [n] R i 9/6. However, applying Zhang-Yeung non-shannon-type inequalities [3] to the proble, the upper bound can be further tightened to i [n] R i (2) Note that for this proble we are focusing on the achievable su-rate, rather than the syetric rate for which Shannontype inequalities are capable of giving tight upper bounds. Detailed proof for (2) is presented in Section V-B. V. PROOFS. For any CIC proble (i A i ), i [n], set g(s) = r H(Y [n] X S c), S [n]. It can be verified that the set function g(s) is non-decreasing and subodular. It can also be verifeid that g( ) = 0, and that g(s), S [n]. For sipler notation, we soeties use g(i, i S) to denote g(s). According to the additional decoding constraints fro [2], we have the following lea. Lea : For any i [n], we have R i g(b) = g(b {i}), B B i. (3) Particularly, when B =, g(b) = 0, and thus R i = g(i). A. Proof of Theore Given any i 0 i according to Definition 2, divide the edges into two groups, M and [] \ M. As M = {j [] : G j 2}, we have [] \ M = {j [] : G j = }. According to the Conditions 3b and 3c in Definition 2, for any j M, l l 2 [2 : w j ], [p j,l : p j,l ] [p j,l2 : p j,l2 ] =, (4) [q j,l : q j,l ] [q j,l2 : q j,l2 ] =, (5) and for any j j 2 M, G j G j2 =, which eans that [p j,w j : q j,w j ] [p j2,w j2 : q j2,w j2 ] =. (6) According to the Condition in Definition 2, {i 0 } B i or {i } B i0. Then according to Lea, we have R i0 R i = g(i 0, i ). (7) 2 It is shown in [2] that the PM bound, including the additional decoding constraints, is as tight as the bound utilizing all Shannon-type inequalities.

5 Consider any j [] \ M, as G j =, according to the Conditions 3b and 3c in Definition 2, we have p j,w j = = p j, = j, q j,w j = = q j, = j, and thus B kj,l {K j,[l ], i j, i j }, l [w j ]. Then by Lea and the fact that g(s), S [n], we have g(i j, i j ). (8) l [w j ] Consider any j M. For any s [2 : w j ], we have where the inequality follows fro the subodularity of g(s). According to (4) and (5) and the onotonicity of g(s), suing up (22) for all s [2 : w j ] leads to l [p j,wj :q j,wj ] s [w j ] l [2:w j] R il g(i pj,s, i qj,s ) j [p j,wj :q j,wj ]\{j} g(i j, i j ) g(i pj,l, i qj,l, i pj,l, i qj,l ). (23) g(k j,[s ], i pj,s, i qj,s ) g(i pj,s, i qj,s, i pj,s, i qj,s ) g(i pj,s, i qj,s ) g(k j,[s ], i pj,s, i qj,s ) = g(k j,[s 2], i pj,s, i qj,s ) R kj,s, (9) where the inequality follows fro the subodularity and the onotonicity of the set function g(s), and the equality follows fro Lea with B kj,s {K j,[s 2], i pj,s, i qj,s } by the Conditions 2 and 3a in Definition 2. Suing up (9) for all s [2 : w j ] yields g(k j,[wj ], i pj,wj, i qj,wj ) g(i pj,s, i qj,s, i pj,s, i qj,s ) s [w j] g(i pj,s, i qj,s ) s [2:w j] s [w j ] R kj,s. (20) According to the Condition 3c in Definiton 2, for any j [p j,wj : q j,wj ]\{j}, we have j []\M, since otherwise G j 2, and G j G j. Hence by (8) we have j [p j,wj :q j,wj ]\{j} ( l [w j ] j [p j,wj :q j,wj ]\{j} g(i j, i j ) ) = q j,wj p j,wj. (24) Therefore, suing up (2), (23), and (24) gives g(i pj,wj, i qj,wj ) j [p j,wj :q j,wj ] l [w j ] l [p j,wj :q j,wj ] R il q j,wj p j,wj. (25) Thus, we have s [w j] g(i pj,s, i qj,s ) s [2:w j] s [w j] R kj,s g(i pj,s, i qj,s, i pj,s, i qj,s ) g(k j,[wj ], i pj,wj, i qj,wj ) R kj,wj g(i pj,s, i qj,s, i pj,s, i qj,s ). (2) s [2:w j] where the first inequality follows fro (20) and the last inequality follows fro Lea with B kj,wj {K j,[wj ], i pj,wj, i qj,wj } and that g(s), S [n]. For any s [2 : w j ], we have ( g(i j, i j ) g(i pj,l, i qj,l ) ) j [p j,s:q j,s] = ( j [p j,s :q j,s ] j [p j,s:p j,s ] g(i pj,s, i qj,s ) l [:s ] g(i j, i j ) g(i j, i j ) j [q j,s :q j,s] g(i pj,s,, i pj,s, i qj,s,, i qj,s ) R il l [p j,s:p j,s ] l [q j,s :q j,s ] l [:s 2] g(i j, i j ) g(i pj,l, i qj,l ) ) R il, (22) Finally, suing up (25) for all j M and (8) for all j M = [] \ ( G j), we have ( j [p j,wj :q j,wj ] l [w j ] g(i pj,wj, i qj,wj ) ) j M ( l [w j ] l [p j,wj :q j,wj ] g(i j, i j ) ) R il (q j,wj p j,wj ). (26) j M To further bound the LHS of (26), recall that we have for any j M, p j,w j = j, q j,w j = j, and hence g(i pj,wj, i qj,wj ) j M g(i j, i j ) l ( M {pj,w j,qj,w j })\{0,} R il g( {i pj,wj, i qj,wj }) M l M {pj,w j,qj,w j } R il, (27) where the first inequality follows fro the subodularity of g(s), and Lea, and the second inequality follows fro the onotonicity of g(s), and (7), and that there ust exist soe j, j 2 M M such that p j,w j = 0, q j2,w j2 =.

6 By (6) and (27), the LHS of (26) can be bounded as, LHS j [] R kj,l l [w j] l M {pj,w j,qj,w j } R il = K R sy l [0:] l [p j,wj :q j,wj ] R il R il = ( K I )R sy. (28) For the RHS of (26), we have RHS = (q j,wj p j,wj ) [] \ ( [p j,wj : q j,wj ]) =. (29) Given (26), (28), and (29), we can conclude that K I. (30) B. Proof of (2) For the proble described in Exaple 5, we show that (2) holds by showing that and that g(, 2, 3) g(, 2, 4) 6R 6R 2 3R 3 3R 4 4R 5 4R 6 2R 7 4R 8 4R 9. (3) 4 g(, 2, 3) g(, 2, 4) 2R 2R 2 5R 3 5R 4 4R 5 4R 6 6R 7 4R 8 4R 9. (32) in the following. We first show that (3) holds as follows. First of all, it can be verified the following inequalities hold g(, 3, 4) g(, 4) g(, 3) R 6 R 9, (33) g(2, 3, 4) g(2, 4) g(2, 3) R 5 R 8, (34) g(, 3, 5) g(3, 5) g(, 5) R 2 R 8, (35) g(, 4, 8) g(4, 8) g(, 8) R 2 R 5, (36) g(, 2, 3) g(2, 3) g(, 5) R 2 R 8, (37) g(, 2, 4) g(2, 4) g(, 8) R 2 R 5, (38) aong which we only give detailed derivations for the first one, (33), as shown in the following, while all the others can be obtained via siilar steps. Consider g(, 3, 4) R 9 g(, 3, 4) g(, 4, 6, 9) R 9 g(, 3, 4) g(, 4, 6) g(, 4) g(, 3, 6) = g(, 4) g(, 3) R 6, (39) where the first step follows fro the fact that g(s), S [n], and the third step follows fro the onotonicity and the subodularity of g(s), and the second and the fourth steps follow fro Lea with {, 4, 6} B 9 and {, 3} B 6, respectively. The inequality (39) directly leads to (33). Next, it can be verified that according to Lea, as well as the subodularity and the onotonicity of g(s), we have the following inequalities, g(2, 3) g(, 3) g(2, 4) g(, 4) R 3 R 4 g(, 2, 3, 4) R R 2, (40) g(, 2, 3, 4) g(, 3, 4, 5, 8) g(, 3, 4) R 2 R 5 R 8, (4) g(, 3, 4, 5) g(, 3, 4, 8) g(, 3, 4) g(, 3, 4, 5, 8), (42) 2g(3, 4) g(3, 5) g(4, 8) R 3 R 4 g(3, 4, 5) g(3, 4, 8), (43) g(3, 4, 5) g(, 4, 5) g(3, 4, 8) g(, 3, 8) R 4 R 5 g(, 3, 4, 5) R 3 R 8 g(, 3, 4, 8), (44) g(, 3) g(, 5) g(, 4) g(, 8) g(, 3, 5) g(, 4, 8) 2R, (45) 3 R 2R 6 2R 9 g(, 3) g(, 4), (46) 2 2R 7 2g(3, 4). (47) By (35) and (36), and the subodularity of g(s), we have g(, 3, 5) g(, 4) g(, 4, 8) g(, 3) g(3, 5) g(, 4, 5) g(4, 8) g(, 3, 8) By (33), we have 2R 2R 2 R 5 R 8 2. (48) 2 2g(, 3, 4) 2(g(, 4) g(, 3) R 6 R 9 ). (49) Suing up (40)-(49), we have 9 g(2, 3) g(2, 4) g(, 5) g(, 8) 6R 4R 2 3R 3 3R 4 3R 5 4R 6 2R 7 3R 8 4R 9. (50) Then, adding (50), (37) and (38), we obtain (3). It reains to show that (32) holds, and for that purpose we will use the Zhang-Yeung non-shannon-type inforation inequality [3], stated as follows, 3H(A, C) 3H(A, D) 3H(C, D) H(B, C) H(B, D) 2H(C) 2H(D) H(A, B) H(A) H(B, C, D) 4H(A, C, D), (5) where A, B, C, D each denotes an arbitrary subset of the set of all rando variables for the proble, {Y [n], X, X 2,, X n }. Set A = {Y [n] } X {,2,4} c, B = {Y [n] } X {,2,3} c, C = {Y [n] } X {2,3,4} c, D = {Y [n] } X {,3,4} c. (52) By (5) and essage independence, as well as the definition of the set function g(s), we have 3g(2, 4) 3g(, 4) 3g(3, 4) g(2, 3) g(, 3) 2g(2, 3, 4) 2g(, 3, 4) g(, 2) g(, 2, 4) g(3) 4g(4). (53)

7 Switch the value of A and B in (52), while keep C and D unchanged. By (5) and essage independence, as well as the definition of the set function g(s), we have 3g(2, 3) 3g(, 3) 3g(3, 4) g(2, 4) g(, 4) 2g(2, 3, 4) 2g(, 3, 4) g(, 2) g(, 2, 3) g(4) 4g(3). (54) Adding (53) and (54), we obtain 4g(2, 3) 4g(, 3) 6g(3, 4) 4g(2, 4) 4g(, 4) 4g(2, 3, 4) 4g(, 3, 4) 2g(, 2) 5g(3) 5g(4) g(, 2, 3) g(, 2, 4) 4g(2, 3, 4) 4g(, 3, 4) 2R 2R 2 5R 3 5R 4 g(, 2, 3) g(, 2, 4), (55) where the second inequality follows fro Lea with {} B 2. Therefore, we have 6 6g(3, 4, 7) = 6R 7 6g(3, 4) 6R 7 2R 2R 2 5R 3 5R 4 4 ( g(2, 3, 4) g(2, 4) g(2, 3) ) 4 ( g(, 3, 4) g(, 4) g(, 3) ) g(, 2, 3) g(, 2, 4) 6R 7 2R 2R 2 5R 3 5R 4 4(R 5 R 8 ) 4(R 6 R 9 ) g(, 2, 3) g(, 2, 4), (56) where the first step follows fro that g(s), S [n], and the second step follows fro Lea with {3, 4} B 7, and the thrid step follows fro (55), and the last step follows fro (33) and (34). We can see that (56) directly leads to (32). Now that both (3) and (32) have been shown, by adding the together and dividing by 8 on both sides, we can conclude that R i 25 8, (57) i [n] which copletes the proof. REFERENCES [] Y. Birk and T. Kol, Infored-source coding-on-deand (ISCOD) over broadcast channels, in IEEE INFOCOM, Mar. 998, pp [2] F. Arbabjolfaei, Y.-H. Ki et al., Fundaentals of index coding, Foundations and Trends R in Counications and Inforation Theory, vol. 4, no. 3-4, pp , 208. [3] L. Ong, C. K. Ho, and F. Li, The single-uniprior index-coding proble: The single-sender case and the ulti-sender extension, IEEE Trans. Inf. Theory, vol. 62, no. 6, pp , Jun [4] Y. Liu, P. Sadeghi, F. Arbabjolfaei, and Y.-H. Ki, Capacity theores for distributed index coding, arxiv preprint arxiv: , 208. [5] M. Li, L. Ong, and S. J. Johnson, Cooperative ulti-sender index coding, IEEE Trans. Inf. Theory, 208. [6] Z. Bar-Yossef, Y. Birk, T. Jayra, and T. Kol, Index coding with side inforation, IEEE Trans. Inf. Theory, vol. 57, pp , 20. [7] A. Blasiak, R. Kleinberg, and E. Lubetzky, Lexicographic products and the power of non-linear network coding, in Proc. Syp. on Foundations of Coputer Science (FOCS), 20, pp [8] F. Arbabjolfaei, B. Bandeer, Y.-H. Ki, E. Sasoglu, and L. Wang, On the capacity region for index coding, in Proc. IEEE Int. Syp. on Inforation Theory (ISIT), 203, pp [9] H. Maleki, V. R. Cadabe, and S. A. Jafar, Index coding: An interference alignent perspective, IEEE Trans. Inf. Theory, vol. 60, no. 9, pp , 204. [0] S. A. Jafar, Topological interference anageent through index coding, IEEE Trans. Inf. Theory, vol. 60, no., pp , 204. [] H. Sun and S. A. Jafar, Index coding capacity: How far can one go with only shannon inequalities? IEEE Trans. Inf. Theory, vol. 6, no. 6, pp , 205. [2] Y. Liu, Y.-H. Ki, B. Vellabi, and P. Sadeghi, On the capacity region for secure index coding, in Proc. IEEE Inforation Theory Workshop (ITW), Guanzhou, China, Nov [3] Z. Zhang and R. W. Yeung, On characterization of entropy function via inforation inequalities, IEEE Trans. Inf. Theory, vol. 44, no. 4, pp , 998.