An Achievable Rate Region for the Two-Way Multiple-Relay Channel

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by  Jonathan Ponniah

An Achievable Rate Region for the Two-Way
Multiple Relay Channel
Jonathan Ponniah Liang-Liang Xie
Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering
Texas A&M University University of Waterloo
Abstract—An achievable rate-region forthetwo-way multiple- all the other sources (the TWMRC is a special case of this
relaychannelisproposedusingdecode-forwardblockMarkovian channel).Inadecode-forwardschemethemessagesgenerated
coding. We identify a fundamental tension between the informa-
by each source are successively decoded and forwarded by
tion flow in both directions that leads to an intractable number
each relay in the channel before arriving at any destination.
of decode-forward schemes and achievable rate regions, none of
whichareuniversallybetterthantheothers.Weintroduceanew The order in which the nodes forward the message from a
6
concept in decode-forward coding called ranking, and discover particularsourcedefinesapath.LetS denotethesetofsource
1
0 thateachoftheserateregionsaredifferentrealizationsofasingle nodes. For each s∈S, let the vector p¯s denote the fixed path
expression that depends on the rank assignment. This discovery
2 makes it possible to characterize the complete achievable rate assigned to source s and let p¯ := {p¯s : s ∈ S}. The first
element of p¯ is source node s, and each subsequent element
l regionthatincludesalloftheinterestingdecode-forwardschemes s
u and corresponding rate regions. is the next node on the path. Let ps,k denote the kth element
J
of p¯ .
s
4 I. INTRODUCTION
2 Example 1. Set M := 5 and S := {1,5}. Suppose the mes-
The two-way multiple relay channel (TWMRC) is implicit
sages w and w follow the paths 1→3→2→4→5 and
inalmostallformsofmoderncommunication.Atthemostfun- 1 5
]
5 → 2 → 3 → 4 → 1 respectively. Then p¯ = (1,3,2,4,5),
T damentallevel,itmodelsthesimultaneous,interactive,bidirec- 1
p¯ =(5,2,3,4,1), p =3, and p¯:={p¯ ,p¯ }.
I tionalexchangeofinformationbetweentwosource/destination 5 1,2 1 5
.
s pairs with the assistance of intermediate relay nodes; the For any fixed p¯, let P (s) denote the set of nodes that
c i
[ type of exchange that occurs over the internet for instance, precede node i on the path p¯s. Suppose node i is the kth
where every single received data packet is acknowledged.We node on the path p¯s. That is, i = ps,k. Then Pi(s) := {ps,j :
2
consider the most generalform of the TWMRC which allows 1 ≤ j ≤ k − 1}. For any non-empty subset S ⊆ S, let
v
9 thetransmissionsofeverynodetoinfluencethereceivedsignal Pi(S)= [ Pi(s) and P˜i(S):={k ∈/ Pi(S)}.
2 of each individual node. s∈S
2 This paper extends a history of previous work in network
Example 2. Set M := 5 and S := {1,5}. Set p¯ :=
6 information theory beginning with the two-way channel [1], 1
0 followed by the one-way relay channel [2], the one-way (1,3,2,4,5) and p¯5 := (5,2,3,4,1). Then P2({1}) = {1,3},
1. multiple-relay channel [3], the one-way multiple-access relay P2({5}) = {5}, and P2({1,5}) = {1,3,5}. Similarly,
P˜ ({1})={2,4,5},P˜ ({5})={1,2,3,4},andP˜ ({1,5})=
0 channel [4], the two-way, one-relay channel [5], and the two- 2 2 2
6 {2,4}.
way, two-relay channel [6].
1
The channel model of interest is the discrete memoryless Set |S| := N for some N ≤ M, and S := {s ,...,s },
: 1 N
v channel consisting of M nodes labeled from 1,...,M. The wheresj ∈{1,...,M} foreachj =1,...,N. Foranyp¯and
i
X input-output dynamics are expressed as follows: non-empty S ⊆S define the constraint:
ar (X1×...×XM,p(y1,...,yM|x1,...,xM),Y1×...×YM). RS <I(XPi(S);Yi|XP˜i(S)), (1)
That is, at every time instant t = 1,2,..., the out- where R denotesthe sum R , and X denotesthe set
S Pj∈S j S
puts y (t),...,y (t) received by the M nodes respec- {X : j ∈ S}. Fix p¯and let R (p¯) be the set of rate vectors
1 M j i
tively only depend on the inputs x (t),...,x (t) trans- R¯ :=(R ,...,R ) that satisfy (1) for all non-empty S ⊆ S
1 M 1 N
mitted by the M nodes at the same time according to if i∈/ S and all non-empty S ⊆S\i if i∈S.
p(y (t),...,y (t)|x (t),...,x (t)). If node i is a relay, A rate vector R¯ := (R ,...,R ) for a multi-source,
1 M 1 M 1 N
the input x (t) into the channel at time t depends only on multi-relay, all-way channel with N sources is achievable
i
the symbols received in the previous time instants, so that by definition, if there exists an encoding/decoding scheme
x (t)=F (y (t−1),y (t−2),...) for all t, where F can that allows source node s for each j ∈ {1,...,N} to
i i,t i i i,t j
be any causal function. send information at rate R to all the other sources with an
j
Consider the multi-source, multi-relay, all-way channel in arbitrarily small probability of error. An outer-bound on the
which each source is interested in the messages generated by region of achievable rate vectors that can be recovered using
decode-forward schemes in this channel is given by: that decides, at each relay, the extent to which the OWMRC
is simulated in one direction at the expense of the other
M
direction. Each decision prevents the decode-forward scheme
C := R (p¯). (2)
d [\ i from recovering some of the rate pairs in C . This tension
p¯ i=1 d
generates many different decode-forward schemes and rate
To observethe differencebetween(2) andthe cut-setouter-
regions, all of which cumulatively fail to recover C and
d
bound, replace (1) with the following constraint:
none of which include the others. Furthermore, the regions
R <I(X ;Y |X ). (3) recovered by each of these decode-forward schemes share no
S Pi(S) P˜i(S) P˜i(S)
obviouspattern.Asaresult,itbecomesintractabletoexplicitly
Assume multi-way communication so that N ≥2. For any p¯,
characterize the rate region that includes all possible decode-
let Rˆ (p¯) be the set of rate vectors R¯ := (R ,...,R ) that
i 1 N forward schemes for an arbitrary number of nodes.
satisfy (3) for all non-empty S ⊂ S if i ∈/ S and all non-
The main contribution of this paper is the discovery that
empty S ⊆ S \i if i ∈ S. A key difference between Rˆ (p¯)
i the rate region corresponding to any decode-forward scheme
and R (p¯) is that the former requires S to be a strict subset
i that attempts to recover C in the TWMRC is a particular
d
of S for N ≥2 since there is no cut that puts all the sources
realizationof a single expression.This expressiondependson
on the same side in multi-way communication. The cut-set
the rank assignment, where the rank assigned to each node
outer-bound corresponds to the following region:
is determined by the decode-forward scheme that attempts to
M recoverCd. This propertymakes it possible to characterizeall
C := Rˆ (p¯). (4) the interesting decode-forward schemes by describing the set
\\ i
p¯ i=1 of rankings instead.
This paper focuses on the decode-forward relay scheme.
The regions (4) and (2) coincide if N = 1 and there is
Another important but fundamentally different relay scheme
one path p¯over which the channel is physically degraded. In
originally proposed in [2] is the compress-forward scheme,
theone-waymulti-relaychannel(OWMRC),thetwo-wayone-
which has also been successfully extended to more general
relaychannel,andthethree-waybroadcastchannel[5],C can
d
networks in [7] and [8]. It is well known that neither decode-
be achieved for all joint-distributions in the first case and all
forward nor compress-forward is absolutely better than the
product distributions in the latter two cases. The key feature
other, and their relative superiority depends on the network
ofthechannelthatisexploitedin[3]istheunidirectionalflow
topology in general [9]. However, for the two-way traffic
of information.Node i can remove the interference generated
by the nodesin P˜(S) becausethese nodesbeing downstream consideredinthepaper,especiallywhenrelaynodesareevenly
i
placed in between, it is arguably clear that decode-forward
of node i, transmit messages already decoded by i.
performs better.
It turns out that any attempt to recover the rate vectors in
The rest of this paper is organized as follows: Section
C formulti-waychannelswithtwoormorerelaysencounters
d
II describes all valid rankings, Section III states the main
a fundamental tension between the information flow in one
result which is the complete achievable rate region, Section
direction and the information flow in the opposite direction.
IV proves the main result and section V concludes the paper.
This tension is illustrated in the two-way two-relay channel.
Example 3. Thetwo-waytwo-relaychannelconsistsofnodes II. RANKING
{1,2,3,4}. Define the set of source nodes as S := {1,4} Given a multi-source multi-relay all-way channel with M
and consider the paths p¯ :=(1,2,3,4) and p¯ :=(4,3,2,1). nodes, a rank index is a number between 1...M uniquely
1 4
In order to decode a message w from node 1 at the rate assigned to each node. The nodes are also labeled from
1
R <I(X ;Y |X X X ),node2needstoknowthemessage 1,...,M but the rank indices are distinct from the labels.
1 1 2 2 3 4
w simultaneously transmitted by node 4. But w is new If node i is assigned rank k, then rank(i) = k. A rank
4 4
information and node 2 does not know it a priori. Hence we assignment r¯ is a one-to-one mapping of rank indices to
have the following requirement: labels and is represented by an M-dimensional vector where
(i) Node 3 must first decode and forward w4 before node 2 r¯= (r1,r2,...,rM) and ri = rank(i). The rank indices are
decodes w . orderedby the binaryrelations“>”,“<”, “=”, “≥”,and “≤”.
1
These relations retain their usual meaning in the sense that
However, the reverse situation occurs when node 3 tries to
M >M −1>···>2>1 and 1<2<···<M −1<M.
decode w from node 4 at rate R < I(X ;Y |X X X ).
4 4 4 3 1 2 3 There is no ordering defined on the labels.
Then we have the following requirement
The analysis in the sequel will be limited to the TWMRC
(ii) Node 2 must first decode and forward w before node 3
1 withS :={1,M}andpathvectorp¯:={p¯ ,p¯ }wherep¯ :=
1 M 1
decodes w .
4 (1,2,...,M − 1,M) and p¯ := (M,M − 1,...,2,1). A
M
It is impossible to simultaneously satisfy (i) and (ii); either path-rank-assignmentpair (p¯,r¯) is valid by definition if there
(ii) is satisfied at the expense of (i) or (ii) is satisfied at the is only one local minimum (with respect to the rank indices)
expense of (i). overbothpathsp¯ andp¯ .Moreprecisely,foreachs∈S,let
1 M
i :=argminrank(p ).Thatis,theithelementofp¯ hasthe
AnyattempttorecoverCdleadstoadecode-forwardscheme s k s,k s s
lowestrank.Then(p¯,r¯)is validbydefinitionifitsatisfies the {3}, A ({1}) = {1}, A ({5}) = {}, A ({1,5}) = {1},
4 4 4
following conditions for all s∈S: rank(p )>rank(p ) B ({1}) = {2,3}, B ({5}) = {5}, B ({1,5}) = {2,3,5},
s,i s,i+1 4 4 4
for all i < i and rank(p ) > rank(p ) for all i > i . B˜ ({1}) = {4,5}, B˜ ({5}) = {2,3,4}, and B˜ ({1,5}) =
s s,i s,i−1 s 4 4 4
The notation v¯ = (p¯,r¯) will be used to denote a valid path- {4}.
rank-assignment pair (p¯,r¯).
For any non-empty S ⊆S and v¯∈V define the constraint:
Example 4. The following are examples of rank as-
R < I(X ;Y |X )+I(X ;Y |X ). (5)
signments r¯ that correspond to a valid (p¯,r¯) when S X j i L(j) Bi(S) i B˜i(S)
M = 8: (1,2,3,4,5,6,7,8), (8,7,6,5,4,3,2,1), (8,6,4,2,1,3,5,7), j∈Ai(S)
(8,7,6,4,3,1,2,5),(7,6,4,3,1,2,5,8). Let R (v¯) be the set of rate pairs (R ,R ) that satisfy
i 1 M
(5) for all non-empty S ⊆ S if i ∈/ S and satisfy (1) for all
LetV denotethesetofallvalidpath-rank-assignmentpairs.
non-emptyS ⊆S\iifi∈S. We havethefollowingtheorem.
For a fixed v¯ ∈ V and each s ∈ S, let u(i,s) denote the
“one-hop” predecessor upstream of i on the path p¯s. More Theorem 1. Foranyproductdistribution p(x1)···p(xM) the
precisely,supposeps,k =i forsomek. Thenu(i,s)=ps,k−1. following set of rate pairs is achievable:
Let U(i) = {u(i,s) : s ∈ S} denote the set of all one-hop
M
predecessor nodes upstream of node i. Furthermore, at each
R (v¯)
relay i∈/ S define the reference node with respect to i as the [ \ i
v¯∈V i=1
highest ranked one-hop predecessor upstream to node i over
all paths. That is, ref(i):=arg max rank(j). IV. PROOF OF THEOREM1
j∈U(i)
The proof is based on the block-markov, decode-forward
Example 5. Set M := 5 and paths p¯1 := (1,2,3,4,5) framework in which the transmissions are divided into B
and p¯5 := (5,4,3,2,1). Define the rank assignment r¯ := blocksof T channeluses. Let R1 and RM denote the rates at
(5,3,2,1,4). Then u(2,1) = 1, u(2,5) = 3 and U(2) = which nodes 1 and M transmit information to each other.
{1,3}. It follows that ref(2) = 1. Similarly, u(3,1) = 2,
u(3,5)=4, and U(3)={2,4}. It follows that ref(3)=2. A. Codebook Generation
The valid path-rank-assignmentscapture all of the ways in For node 1, independentlygenerate 2TR1 i.i.d T-sequences
which the tension between two opposing information flows is x¯ := (x ,...,x ) in XT according to p(x ). Index
1 1,1 1,T 1 1
resolved in a decode-forward scheme when p¯ := {p¯1,p¯M}. them as x¯1(w1), w1 ∈ {1,2,...,2TR1}. For node i =
The definition of a valid path-rank-assignment thus far has 2,...,M − 1, independently generate 2T(R1+RM) i.i.d T-
been limited to the paths p¯1 := (1,2,3,...,M − 1,M) sequences x¯i := (xi,1,...,xi,T) in XiT according to p(xi).
and p¯M := (M,M −1,...,3,2,1). The symmetry of these Index them as x¯i(wi), wi ∈ {1,2,...,2T(R1+RM)}. For
paths simplifies the characterization of the valid path-rank- node M, independently generate 2TRM i.i.d T-sequences
assignments. In future work, we will define the valid path- x¯ := (x ,...,x ) in XT according to p(x ). Index
M M,1 M,T M M
rank-assignments over arbitrary paths. them as x¯ (w ), w ∈{1,2,...,2TRM}.
M M M
III. MAIN RESULT B. Encoding
Consider the TWMRC with M nodes and S := {1,M}. In each block b ∈ 1,...,B, nodes 1 and M generate
For a fixed v¯ ∈ V and any non-empty S ⊆ S define the message indices w (b) ∈ {1 ...,2TR1} and w (b) ∈
1 M
higher orthant set (the orthant set “above”) Ai(S), as the set {1 ...,2TRM} and transmit the T-sequences x¯1(w1(b)) and
ofnodesinPi(S)ofrankhigherthanthereferencenoderef(i). x¯M(wM(b)) respectively. Simultaneously, each relay node i
Similarly,definethelowerorthantset(theorthantset“below”) chooses a message index w (b) ∈ {1,...,2T(R1+RM)} and
i
Bi(S), as the set of nodes in Pi(S) of rank lower than or transmits the T-sequence x¯i(wi(b)). The index wi(b) corre-
equal to the reference node. Finally, define the lower orthant spondsto auniquemessagepair(w (b−d ),w (b−d ))
1 i,1 M i,M
set B˜i(S) as the set of nodes in P˜i(S) of rank lower than or where di,1 and di,M, referred to as the encoding delays, are
equaltothereferencenode.Theorthantsetscanbeexpressed strictly positive integers. The encoding scheme specifies the
as follows: encoding delays at the relay nodes. At the end of each block
b, every relay node i decodes the message pair (w (b −
A (S):={j ∈P (S)|rank(j)>rank(ref(i))}, 1
i i d˜ ),w (b−d˜ ))wherethedecodingdelays,d˜ andd˜
i,1 M i,M i,1 i,M
B (S):={j ∈P (S)|rank(j)≤rank(ref(i))},
i i arestrictlypositiveintegers.Thedecodingschemespecifiesthe
B˜i(S):={j ∈P˜i(S)|rank(j)≤rank(ref(i))}. decoding delays at the relay nodes. The relay cannot encode
anymessagepairthatithasnotalreadydecoded,sod >d˜ .
Let L(i) denote the set of nodes of strictly lower rank than i. i,s i,s
Apart from this causality constraint, the messages decoded in
That is, L(i):={j |rank(j)<rank(i)}.
one block need not determine the messages encoded in the
Example 6. Set M := 5, p¯ := (1,2,3,4,5), p¯ := next block; there may be many decoding schemes causally
1 5
(5,4,3,2,1),andr¯:=(5,3,1,2,4).Then ref(4)=5,L(4)= consistent with a fixed encoding scheme.
Thisdefinitionofp¯impliesthatd >d foreachrelay
2 2 i,M i+1,M
node i. This channel will be called a biased multiple-access
relaychannel(BMARC)sincetherelaysonlyhelponesource
andnottheother.TheencodingschemeisillustratedinFigure
w4(b−d3,4) w1(b) w4(b−d3,4) 1 for a BMARC of size M =4. Let R2(p¯) be the set of rate
3 1 3 pairs (R1,RM) that satisfy (1) for all non-empty S ⊆ S for
the BMARC with path p¯:={p¯ ,p¯ }.
1 M
Lemma1. Givenanyproductdistributionp(x )···p(x )the
1 M
w1(b) w4(b) w4(b) rate pairs in R2(p¯) are achievable for the BMARC.
1 4 4 Proof: First we show that node 2 can decode (w (b +
1
(i) (ii) d ),w (b)) in block b+d if (R ,R ) satisfies:
i,M M 3,M 1 M
R <I(X ;Y |X ...X ) (8)
1 1 2 2 i
Fig. 1: An encoding scheme and two decoding schemes that
R <I(X ...X ;Y |X ) (9)
cumulatively recover R (p¯), where p¯ =(1,2) and M 3 i−1 2 2
2 1
p¯4 :=(4,3,2). In block b nodes 1, 3, and 4 encode w1(b), +I(Xi...XM;Y2|X1X2X3...Xi−1)
w4(b−d3,4), and w4(b) respectively. In decoding scheme (i), R1+RM <I(X−2;Y2|X2), (10)
w (b) and w (b) are jointly decoded in block b+d . In
1 4 3,4
where X := {X : 1 ≤ j ≤ M,j 6= 2}. Consider the
decoding scheme (ii) w (b+d ) and w (b) are jointly −2 j
1 3,4 4
decoding of w (b+d ). Since information flows from M
decoded in block b+d . 1 i,M
3,4
to node 2, and w (b) is transmitted by node i in block
M
b+d , nodes 2,...,i transmit messages already decoded
i,M
Every valid path-rank-assignment generates an encoding by node 2 and nodes i + 1,...,M transmit messages that
scheme and a corresponding set of causally-consistent decod- are new. Hence, the mutual information in (8) is conditioned
ing schemes. For each s ∈ S, let s˜:= S \s. For any v¯ ∈ V on nodes X2,...,Xi. Next consider the decoding of wM(b).
andeach s∈S andi∈{1,...,M}\s˜, definef asfollows: The messages transmitted by node 1 and nodes i,...,M
i,s
during the blocks in which w (b) is transmitted by nodes
M
 X fk,s˜ if i=ref(ps,i+1), 3,...,i−1, have not been decoded by node 2. Hence, the
fi,s :=k∈Bi(s˜) (6) first mutual information in (8) is conditioned only on X2. On
1 otherwise theotherhand,themessagestransmittedbynodes1andnodes
 2,...,i−1 have been decoded by node 2 during the blocks
Foreachs∈S andi∈{2,...,M−1},defined asfollows:
i,s in which nodes i,...,M transmit w (b). Hence, the second
M
mutual information in (9) is conditioned on X ,...,X . It
d := f . (7) 1 i−1
i,s X j,s follows that (8)-(10) is achievable.
j∈Pi(s)
Next we show that for every (R ,R ) in R (p¯) there is
1 M 2
Example 7. Let M = 4, p¯1 := (1,2,3,4), p¯4 := (4,3,2,1), somei,3≤i≤M suchthat(R1,RM)satisfies(8)-(10).The
andr¯:=(4,2,1,3).Thenf1,1 =f4,4+f3,4+f2,4,f2,1 =f3,4, proof is by induction. Consider i=M. If (R1,RM) satisfies
f3,1 = 1, f4,4 = f2,1+f3,1, f3,4 = 1, f2,4 = 1. Expanding (8)-(10), then we are done. Suppose otherwise. Since (8) and
gives f1,1 = 4, f2,1 = 1, and f4,4 = 2. Therefore d2,1 = (10) are boundariesof R2(p¯) at i=M it follows that RM >
f1,1 = 4, d3,1 = f1,1 + f2,1 = 5, d3,4 = f4,4 = 2, and I(X3,...XM−1;Y2|X2)+I(XM;Y2|X1...XM−1). Butthis
d2,4 =f4,4+f3,4 =3. together with (10) implies that R1 <I(X1;Y2|X2...XM−1)
which satisfies (8) for i=M −1.
C. Decoding and the Analysis of the Probability of Error Next, consider any 3 < i < M. If (R ,R ) satisfies (8)-
1 M
It remains to show that there exists a set of causally- (10), then we are done. Suppose otherwise. By the inductive
consistent decoding delay pairs {(d˜ ,d˜ )} for each i = hypothesis, R satisfies (8). Furthermore (10) is a boundary
i,1 i,M 1
2,...,M−1thatallownodeitorecoveranyratepairinR (v¯). of R (p¯). It follows that R > I(X ...X ;Y |X ) +
i 2 M 3 i−1 2 2
First,weproveapreliminarylemma.Defineamultiple-access I(X ...X ;Y |X ...X ). But this together with (10)
i M 2 1 i−1
relay channel consisting of M nodes, where node 1 and node implies that R < I(X ;Y |X ...X ) which satisfies (8)
1 1 2 2 i−1
M aresources,node2isadestination,andnodes3...,M−1 for i−1. The argumentterminatesat i=3 since (9) and (10)
are relay nodes. Assume block-Markovian encoding; in each are boundaries of R (p¯) at i=3. Thus the lemma is proved.
2
block b of T channel uses, node i ∈ {1,M} generates the
messagew (b)∈{1,...,2TRi}.Furthermore,supposeagenie Figure 1 illustrates the proof of Lemma 1 for M = 4.
i
reveals the message w (b−d ) to relay node i just before Finally, we show for any v¯ ∈ V, the region M R (v¯) is
M i,M Ti=1 i
blockb,whered isafixedconstantforeachi.Hence,relay achievable. The proof is by induction. Consider, the two-way
i,M
node i encodes w (b−d ) in block b. The path vector p¯ one-relay channel where node 1 and 3 are the sources and
M i,M
is defined as p¯ :=(1,2) and p¯ :=(M,M −1,...,4,3,2). node 2 is the relay. The paths are defined as p¯ := (1,2,3)
1 M 1
and p¯ :=(3,2,1). In [5] the region 3 R (p¯) is shown to 1 remain unchanged; they do not include X . These
3 Ti=1 i M
beachievable.Itisstraightforwardtocheckthattherearefour contributions are expressed by I(X ;Y |X ) +
X j i L(j)
validrankingsforthischannel,(3,2,1),(1,2,3),(3,1,2),and
j∈Ai(S)\M
(2,1,3), and that the region in Theorem 1 is the same as the I(XBi(S);Yi|XB˜i(S))forallnon-emptyS ⊆S.Theright-side
region in [5]. extension forces relay node i to decode w (b) before w (b).
1 M
The first step of the induction is to perform a left or right As a result, the contribution of node M as seen by node i is
extension of the two-way one-relay channel to create a two- I(X ;Y |X ...X ) or equivalently, I(X ;Y |X ).
M i 1 M−1 M i L(M)
way two-relay channel. Without loss of generality, choose a Hence relay node i ∈ {2,...,M −1} can recover any rate
right-side extension where node 3 becomes a relay and node pair in the region defined by R < I(X ;Y |X )+
S X j i L(j)
4 is added as a source and given rank 4. The new paths are
j∈Ai(S)
p¯1 := (1,2,3,4) and p¯4 := (4,3,2,1). By the design of (6) I(XBi(S);Yi|XB˜i(S)) for all non-empty S ⊆S. To finish the
and (7), node 3 waits to receive the message from node 1 inductive step, observe that node 1 and node M each see a
before encoding the message simultaneously transmitted by one-way multiple-relay channel which is a simple BMARC.
node 4. Since node 3 knows all of the source 1 messages to Thereforeit followsfrom Lemma 1 that node i∈{1,M}can
itsright,andallofthesource4messagestoitsleft,thechannel decode at any rate in R (p¯) as defined by (1).
i
it sees is the BMARC of Lemma 1 (or a reflection of it). It The definition of a valid path-rank-assignment guarantees
follows from Lemma 1 that node 3 can recover all the rate that we can always start with the three nodes of lowest rank
pairs in R (p¯), which is equivalent to the region defined by andreachthegeneraltwo-wayM-relaychannelbyasequence
3
R < I(X ;Y |X ) for all non-empty S ⊆ S. Note of left and right extensions with nodes of successively higher
S B3(S) 3 B˜3(S)
that A (S)={} for all non-empty S ⊆S. rank. Thus Theorem 1 is proved.
3
Since node 2 does not know the transmissions from node
V. CONCLUDING REMARKS
4 a priori, the previous mutual informations that describe
the contributions of nodes 1 and 3 remain unchanged; they We showed that the rate regions of all interesting decode-
do not include X . These contributions are expressed by forward schemes are different realizations of a single expres-
4
I(X ;Y |X ) for all non-empty S ⊆ S. The right- sionthatdependsonarankassignment.Thisdiscoverymakes
sideBe2x(tSe)nsio2n fBo˜2rc(Se)s node 2 to decode w (b) before w (b). it possible to characterizethe completeachievablerate region.
1 4
As a result, the contribution of node 4 as seen by node It remains to be seen whether some version of Theorem 1 is
also true for multi-source, multi-relay, all-way channels.
2 is I(X ;Y |X X X ) or equivalently, I(X ;Y |X ).
4 2 1 2 3 4 2 L(4)
Hence relay node i ∈ {2,3} can recover any rate pair
VI. ACKNOWLEDGMENTS
in the region defined by R < I(X ;Y |X ) +
S X j i L(j) The authors would like to thank Xiugang Wu for helpful
j∈Ai(S) discussions on random binning and network coding. This
I(X ;Y |X ) for all non-empty S ⊆S. To finish the
Bi(S) i B˜i(S) material is based upon work partially supported by NSF
first inductive step, observe that node 1 and node 4 each see
ContractCNS-1302182,AFOSR ContractFA9550-13-1-0008,
a one-waymultiple-relaychannel,whichisa simpleBMARC.
and NSF Science & Technology Center Grant CCF-0939370.
Therefore it follows from Lemma 1 that node i ∈ {1,4} can
decode at any rate in Ri(p¯) as defined by (1). Thus the first REFERENCES
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