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in this paper , we consider a single - source multicast discrete memoryless relay network in which the source wants to send the same message reliably to multiple destinations with the help of one or more relays . a model of relay networks
was introduced by van der meulen in  @xcite .
however , the single - letter capacity characterization has been open even for three - node relay networks , i.e. , relay networks having a source , a relay , and a destination . in their seminal paper
@xcite , cover and el gamal developed two fundamental coding strategies for three - node relay networks .
one of them is decode - and - forward ( df ) , where the relay decodes the message and forwards it to the destination , which was shown to be optimal for physically degraded channels @xcite .
df was generalized for multiple relays in  @xcite . in another strategy ,
compress - and - forward ( cf ) , the relay compresses its received block and sends the compressed information to the destination .
cf was shown to achieve the capacity for some classes of relay networks  @xcite .
recently , cf was generalized to noisy network coding in @xcite for multiple relays , which includes many previous results on relay networks @xcite as special cases .
a potentially better strategy is to decode as much as possible and compress the residual information , i.e. , a combination of df and cf  @xcite .
indeed such a strategy was shown to be optimal by kang and ulukus for a certain class of diamond networks in  @xcite , which consists of a source , a noisy relay , a noiseless relay that receives exactly what the source sends , and a destination that has orthogonal finite - capacity links from relays .
for this class of diamond networks , it was shown that a combination of df and cf at the noisy relay is optimal and the cut - set bound is in general loose  @xcite .    in this paper , we show the optimality of a combination of df and cf for a new class of single - source multicast relay networks with an arbitrary number of nodes , which includes the class of diamond networks in @xcite as a special case . in this class , which we call multicast tree networks ,
a network has a tree topology in which the root node is the source and each parent node in the graph has at most one noisy child node and any number of noiseless child nodes .
we note that the achievability and converse for diamond networks in @xcite can not be directly generalized to those for our multicast tree networks .
first , the codebook constructions and relay operations of the coding scheme in @xcite for diamond networks , which has a single destination , vary according to the link capacities from relays to the destination .
this can not be used for multicast tree networks since they have arbitrarily many destinations .
next , it would not be easy to generalize the converse proof technique in @xcite for diamond networks , which have only four nodes in three levels , for our multicast tree networks , which have arbitrarily many nodes in arbitrarily high levels .
therefore , for these two reasons , we need new techniques .
the key technical contributions in the achievability and converse in this paper are as follows :    * achievability : for the generalization to multicast tree networks , we construct a _ robust _ coding scheme where codebook constructions and relay operations are independent for each node and do not depend on the network topology .
such a robustness of the coding scheme makes the generalization from a single destination to multiple destinations possible . *
converse : to get a very simple min - cut expression , we use a novel technique of iteratively manipulating inequalities , i.e. , we recursively reduce a number of inequalities into one using the tree topology .
the organization of this paper is as follows .
the model of a class of multicast tree networks is presented in section [ sec : model ] . in section [ sec : main ] , we present lower and upper bounds on the capacity of the class of multicast tree networks and show a condition for these two bounds to coincide . in section [ sec : achievability ] , we derive the lower bound by presenting a coding scheme where each noisy relay employs a combination of df and cf and each noiseless relay performs a random binning . in section [ sec : ub ] , the upper bound is shown using a recursion exploiting the tree topology . in section [ sec : diamond ] , we present an equivalent capacity expression for diamond networks that shows that without loss of optimality we can construct the coding scheme such that what is compressed after decoding at a noisy relay is a noisy observation of almost uncoded information . the conclusion of this paper is given in section  [ sec : conclusion ] .
the following notations will be used in the paper . for
two integers @xmath0 and @xmath1 , @xmath2 $ ] denotes the set @xmath3 , @xmath4 denotes a row vector @xmath5 , and @xmath6 denotes @xmath7 .
@xmath8 for a set @xmath9 denotes a row vector @xmath10 . according to the context
, @xmath11 sometimes denotes the single - element set @xmath12 for notational convenience .    in this paper ,
we follow the notion of @xmath13-robustly typical sequence introduced in  @xcite .
let @xmath14 denote the number of occurrences of @xmath15 in the sequence @xmath16 . then
, @xmath16 is said to be @xmath13-robustly typical ( or just typical ) for @xmath17 if for every @xmath15 , @xmath18 the set of all @xmath13-robustly typical @xmath16 is denoted as @xmath19 , which is shortly denoted as @xmath20 .
similarly , let @xmath21 denote the number of occurrences of @xmath22 in the sequence @xmath23 .
the sequence @xmath24 is said to be @xmath13-robustly typical ( or just typical ) if @xmath25 for every @xmath26 .
the set of all @xmath13-robustly typical @xmath23 is denoted by @xmath27 or @xmath20 in short .
a single - source multicast discrete memoryless relay network of @xmath28 nodes @xmath29 consists of alphabets @xmath30 for @xmath31 $ ] and a collection of conditional probability mass functions @xmath32 where @xmath33 and @xmath34 for @xmath31 $ ] .
let @xmath35 denote the number of destinations .
let @xmath36 and @xmath37 denote the source and the set of nodes that forms the @xmath38-th destination , respectively , and let @xmath39 for @xmath40 $ ] .
we note that @xmath37 for @xmath40 $ ] are not necessarily disjoint .
let @xmath41}d_d$ ] .    a @xmath42 code for a single - source multicast discrete memoryless relay network of @xmath28 nodes
consists of a message set @xmath43 $ ] , a source encoder that assigns a codeword @xmath44 to each message @xmath45 , a set of relay encoders , where encoder @xmath46\setminus d$ ] assigns a symbol @xmath47 to every received sequence @xmath48 for @xmath49 $ ] , and a set of decoders , where decoder @xmath50 $ ] assigns an estimate @xmath51 to each received sequence @xmath52 .
the message @xmath53 is chosen uniformly from the set @xmath54 .
the average probability of error for a @xmath55 code is given as @xmath56 \right\}.\end{aligned}\ ] ] a rate @xmath57 is said to be _ achievable _ if there exists a sequence of @xmath55 codes such that @xmath58 as @xmath59 .
the capacity is the supremum of all achievable rates .
a single - source multicast discrete memoryless relay network is called a multicast tree network if the probability distribution has the form of @xmath60}p\left(y_k|x_{p_k}\right ) % \label{eqn : joint_tree}\end{aligned}\ ] ] where @xmath61 is called the _ parent node _ of node @xmath11 and @xmath11 is called a _ child node _ of node @xmath61 .
a child node is considered to be one level lower than its parent node .
a node without a parent node is called the _ root node _ and a node that has no child node is called a _
leaf node_. let @xmath62 for @xmath63 $ ] denote the set of leaf nodes that branches out from node @xmath11 .
@xmath64 , let @xmath65 for @xmath63 $ ] denote the _ subtree _ of @xmath64 that consists of node @xmath11 and all of its descendants in @xmath64 . in this paper ,
our goal is to present lower and upper bounds on the capacity of a class of multicast tree networks and to find some tightness conditions of those two bounds . in this class of multicast tree networks ,
the source node is the root node , @xmath66 for @xmath67 $ ] , and each parent node has at most one noisy child node and any number of noiseless child nodes , i.e. , @xmath68 if @xmath11 is a noiseless child node of node @xmath61 . without loss of generality , we assume that @xmath69 .
let @xmath70 for @xmath67 $ ] .
let @xmath71 and @xmath72 for @xmath63 $ ] denote the noisy child node and the set of noiseless child nodes of node @xmath11 , respectively . let @xmath73 for @xmath63 $ ] denote the set of child nodes of node @xmath11 , i.e. , @xmath74 .
from now on , we only consider this class of multicast tree networks .
see fig .
[ fig : tree_ex ] .
a practical example of our multicast tree networks is depicted in fig .
[ fig : sensor ] , which represents a sensor network where a sensor node wants to send a message to the gateway nodes at the boundary connected with infinite - capacity wired links . in this example , each relay node has outgoing links to its neighbor relays such that one of the links is arbitrarily noisy and the others are noiseless . motivation for assuming noiseless links comes from a practical scenario where a transmitter is using a fixed modulation scheme tuned for the worst link and thus the transmission from the transmitter to the other receivers with better channel qualities looks almost noiseless
let us present lower and upper bounds on the capacity of multicast tree networks .
[ eqn : cardinality ] latexmath:[\ ] ]    now , note that the right - hand side of ( [ eqn : multicast_cov ] ) is clearly upper - bounded by @xmath292}c(1 , d)$ ] .
hence , the lower and upper bounds in theorem [ thm : tree_multicast ] coincide .
consider @xmath63 $ ] .
we have @xmath293 where @xmath211 is from csiszr sum identity  @xcite , which proves ( [ eqn : cov_basic1 ] ) .
we have @xmath294 note that combining ( [ eqn : basic_second_proof ] ) with ( [ eqn : cov_basic1 ] ) proves ( [ eqn : cov_basic2 ] ) .
we have @xmath295 where @xmath211 is from the following markov chains : @xmath296 which proves ( [ eqn : cov_basic3 ] ) .
for @xmath63 $ ] , the inequality ( [ eqn : cov_rec_1 ] ) holds trivially .    for @xmath187
, we have @xmath297 where @xmath211 is from the markov chain @xmath298 @xmath249 is from the markov chain @xmath299 for @xmath105 , @xmath300 is because @xmath301 for @xmath231 , and @xmath302 is from the following markov chain @xmath303 for @xmath304 . note that ( [ eqn : cov_rec_131 ] ) holds since @xmath304 is not a leaf node from the definition of @xmath305 . thus , ( [ eqn : cov_rec_2 ] ) is proved .    for @xmath188
, we get @xmath306 where @xmath211 is from the markov chain @xmath307 @xmath249 is from the markov chains ( [ eqn : cov_rec_111 ] ) and ( [ eqn : cov_rec_121 ] ) , @xmath300 is because @xmath301 for @xmath231 , and @xmath302 is from the markov chains ( [ eqn : cov_rec_131 ] ) and @xmath308 note that ( [ eqn : cov_rec_131 ] ) and ( [ eqn : markov ] ) hold since @xmath309 and @xmath304 are not leaf nodes from the definition of @xmath86 .
thus , ( [ eqn : cov_rec_3 ] ) is proved .
let us note that the constraint on @xmath310 in ( [ eqn : cap_org ] ) can be easily verified to be redundant .
fix @xmath310 and @xmath311 .
let @xmath312 and @xmath313 denote ( [ eqn : cap_org ] ) without the constraint on @xmath310 and ( [ eqn : cap_alter ] ) , respectively .
it is trivial to show @xmath314 . to show @xmath315 , it is enough to show that for all @xmath316 such that @xmath317 and @xmath318 , where @xmath319 , there exists @xmath320 that satisfies      now , consider a joint distribution of @xmath316 such that @xmath317 and @xmath318 .
let @xmath322 denote a bernoulli random variable with parameter @xmath323 $ ] .
let @xmath324 and @xmath325 denote the triplets of random variables given as @xmath326 we will show the existence of @xmath320 that satisfies ( [ eqn : another ] ) separately for the cases of @xmath327 and @xmath328 .
first , consider the case of @xmath327 .
let @xmath329 , @xmath330 , and @xmath331 .
note that @xmath332 is a continuous function of @xmath333 and becomes @xmath334 and @xmath335 when @xmath336 and @xmath337 , respectively . from the intermediate value theorem
, there exists @xmath323 $ ] such that @xmath338 .
furthermore , ( [ eqn : another2 ] ) and ( [ eqn : another3 ] ) are satisfied from @xmath339 and @xmath340 respectively .
next , consider the case of @xmath328 .
let @xmath341 , @xmath342 , and @xmath343 .
note that @xmath332 is a continuous function of @xmath333 and becomes @xmath344 and @xmath335 when @xmath336 and @xmath337 , respectively . from the intermediate value theorem
, there exists @xmath323 $ ] such that @xmath345 .
furthermore , ( [ eqn : another2 ] ) and ( [ eqn : another3 ] ) are satisfied from @xmath346 and @xmath347 respectively .
in this paper , we characterize the capacity of a new class of single - source multicast discrete memoryless relay networks having a tree topology in which the root node is the source and each parent node in the graph has at most one noisy child node and any number of noiseless child nodes . 
 this class of multicast tree networks includes the class of diamond networks studied by kang and ulukus as a special case , where they showed that the capacity can be strictly lower than the cut - set bound . for achievablity 
 , a novel coding scheme is constructed where each noisy relay employs a combination of decode - and - forward ( df ) and compress - and - forward ( cf ) and each noiseless relay performs a random binning such that codebook constructions and relay operations are independent for each node and do not depend on the network topology . for converse , a new technique of iteratively manipulating inequalities exploiting the tree topology 
 is used .    relay network , compress - and - forward , decode - and - forward , diamond network , multicast tree network