IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 3, MARCH 2014
1023
Bounding the Advantage of Multicast Network
Coding in General Network Models
Xunrui Yin, Yan Wang, Student Member, IEEE, Zongpeng Li, Senior Member, IEEE, Xin Wang, Member, IEEE,
Jin Zhao, Member, IEEE, Xiangyang Xue, Member, IEEE
Abstract—Network coding encourages information flow mixing
in a network. It helps increase the throughput and reduce
the cost of data transmission, especially for one-to-many multicast applications. An interesting problem is to understand
and quantify the coding advantage and cost advantage, i.e., the
potential benefits of network coding, as compared to routing, in
terms of increasing throughput and reducing transmission cost,
respectively. Two classic network models were considered in previous studies: directed networks and undirected networks. This
work further focuses on two types of parameterized networks,
including bidirected networks and hyper-networks, generalizing
the directed and the undirected network models, respectively. We
prove upper- and lower-bounds on multicast coding advantage
and cost advantage in these models.
Index Terms—Network coding, multicast throughput, fractional solution, packing number.
I. I NTRODUCTION
N
ETWORK coding [1] encourages information flows to
be encoded within a data network, besides merely being
forwarded and replicated. Such a departure from the classic
store-and-forward principle has proven effective in increasing
the network capacity. Higher end-to-end throughput, particularly for multicast data transmission, is witnessed in a number
of network scenarios [1], [2], [3]. Multicast represents an
increasingly more important class of applications on the Internet, encompassing traditional and emerging one-to-many data
dissemination applications, such as software patch distribution,
live media streaming and video conferencing.
A fundamental problem in network coding is to quantify the
benefits of network coding over routing, known as the coding
advantage, measured as the ratio of the achievable throughput
with network coding over that with routing. Without network
coding, a multicast routing solution is based on a multicast
tree, or packing a set of multicast trees [3], [4].
In the directed network setting where each link has a predefined direction, there exists a combination network pattern
Manuscript received April 28, 2013; revised October 25, 2013. The editor
coordinating the review of this paper and approving it for publication was P.
Popovski.
X. Yin and Z. Li are with the Department of Computer Science, University
of Calgary (e-mail: {xunyin, zongpeng}@ucalgary.ca).
Y. Wang, X. Wang, J. Zhao, and X. Xue are with the School of
Computer Science, Fudan University (e-mail: {11110240029, xinw, jzhao,
xyxue}@fudan.edu.cn). Y. Wang is also with the School of Software, East
China Jiao Tong University.
This paper was presented in part at IEEE INFOCOM 2012, when X. Yin
was with Fudan University. This work is supported in part by the Natural
Sciences and Engineering Research Council of Canada (NSERC).
Digital Object Identifier 10.1109/TCOMM.2014.011614.130316
3
4
2
2
1
2
5
4
u
4
3
v
3
u
1
3
2
1
2
1
3
v
Fig. 1. An undirected network (left) and a bidirected network (right). Link
capacities are labeled besides the links.
where the coding advantage is unbounded as the network
size grows [2]. However, in the undirected network setting,
where capacity at each link can be shared flexibly between
the two directions, a contrasting result was proved: the coding
advantage is upper-bounded by a constant of 2 [4], [5], [6].
Directed and undirected graphs are classic subjects of study
in theoretical computer science. While simple and easy to
apply, they do not faithfully depict the wireline or wireless
network topologies in practice. For example, large coding
advantages in the directed setting are observed in contrived,
extremely asymmetric topologies that favors network coding
over tree packing, with links existing in one direction only
between neighboring nodes. This is apparently different from
the picture of the Internet, where pair-wise router interconnections are mostly bidirectional, i.e., if a router A can transmit
to a neighbor router B, so can B transmit to A [7].
This work studies the coding advantage in two types of
parameterized networks with richer modeling power. The first
is the bidirected network model, parameterized with α, the
highest ratio of opposite link capacities between neighboring
nodes. When α = 1, we have a (completely) balanced
network, fairly close to the Internet core/backbone [7], [8].
When α = ∞, we arrive at the directed network model.
Results obtained for the bidirected model directly carry over
to special cases including these two extreme ones. Note that
bidirectional transmissions are also supported in an undirected
network. The key difference between undirected links and
bidirected links is that opposite links between two neighbor
nodes share the total available capacity in a flexible way in the
former, and in a pre-defined way in the latter. For example,
in Fig. 1, the flow rate f (u, v) = f (v, u) = 1.5 is feasible in
the undirected network but not the bidirected network, since
f (u, v)+f (v, u) does not exceed the capacity of the undirected
link c({u, v}) = 3 but f (v, u) is larger the capacity of the
directed link c(v, u) = 1. Another motivation for studying
bidirected networks is that, interestingly, it further provides a
tool for analyzing the coding advantage in hyper-networks.
In the hyper-network model, a hyper-link connects two or
c 2014 IEEE
0090-6778/14$31.00 1024
more nodes; a transmission from one node reaches all. A
hyper-network H is parameterized with β, the max number of
nodes a hyper-link may connect. An undirected network is a
special case with β = 2. An Ethernet bus or a multicast switch
can be viewed as a hyper-link, in that one transmission reaches
all nodes simultaneously. Furthermore, the topology of wireless networks are usually described by directed hypergraphs
[9]. We note that, in the study of minimum-energy multicasting
where interference is irrelevant, the wireless network can be
modeled as a directed hyper-network.
Our main problem of study is: how large can the coding
advantage be, in bidirected networks and hyper-networks?
We explore how the coding advantage is related to the link
symmetry of a network, and how an (approximately) balanced
network behaves differently from a directed or undirected
network. For hyper-networks, we aim to prove the first upperbound on the coding advantage.
Combining the technique of link splitting in Eulerian graphs
with Edmonds’ Theorem on spanning tree packing, we prove
that, for a node-balanced multicast network where each node
has symmetrical transmit and receive capacities, the coding
advantage is always 1. This implies that the coding advantage
is 1 for link-balanced networks, improving the existing upperbound of 4 [10]. We show that in either link-balanced or nodebalanced networks, the max multicast throughput is always
feasible without any information processing (replication or
encoding) at relay nodes. If one assumes Internet routers have
symmetrical inbound and outbound capacities available, then
we can conclude that in-network processing (IP-multicast or
network coding) is unnecessary, and end system multicast suffices. The tight upper-bound of 1 further implies that multicast
tree packing, NP-hard in general, becomes polynomial-time
computable in balanced networks.
An α-balanced bidirected network relaxes link symmetry
from 1 to α ≥ 1, capturing a larger class of networks in
practice. We prove that the coding advantage here is at most
α, improving the known upper-bound√of 2(α + 1) [10]. We
further prove a first lower-bound Ω( α) on the maximum
coding advantage in α-balanced networks. These two bounds
for α-balanced networks unify previous bounds for balanced
networks and directed networks that can be arbitrarily imbalanced, revealing a connection between the asymmetry of a
network and its coding advantage.
For hyper-networks, Li et al. [10] proposed the following
open problem: does a constant upper-bound for multicast
coding advantage exist in hyper-networks? We provide a
negative answer, by proving a lower-bound of Ω(log(β))
on the maximum coding advantage, through generalizing the
3-layer combination network [11] into the hyper-network
paradigm. We further prove an upper-bound of β, through a
network transformation technique relating hyper-networks to
balanced networks. The proof reveals an interesting connection
between the two seemingly distinct network models. The result
implies the well-known upper-bound of 2 for multicast coding
advantage in undirected networks [4], [5], [6].
Throughput advantage and cost advantage of network coding follow a loose primal-dual relation [5], [11]. The latter is
more general, and depends on not only link capacities but link
costs. For a bidirected network with both balanced link capac-
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 3, MARCH 2014
ities and balanced link costs, we prove that the cost advantage
is at most 2. When the ratio of cost between a pair of nodes is
between 1 and α , we prove a generalized upper-bound of 2α ,
unifying the unbounded cost advantage in directed networks
and the upper-bound of 2 in undirected networks. For hypernetworks, we prove that the cost advantage is at most β, which
directly implies the upper-bound 2 in undirected networks [5].
In the rest of the paper, we review previous research in
Sec. II, and define models and notations in Sec. III. Coding
advantage is analyzed for bidirected networks in Sec. IV,
and for hyper-networks in Sec. V. Cost advantage is studied
in Sec. VI. We extend the results in bidirected networks to
bidirected hyper-networks in Sec. VII. In Sec. VIII and Sec. IX
discuss realworld relevance and conclude the paper.
II. P REVIOUS R ESEARCH
Li et al. studied the coding advantage in undirected networks [4], [10]. Applying graph theory techniques, they
first prove that the coding advantage is upper-bounded by
a constant of 2. The same bound is later proved using
linear programming techniques [5] and different graph theory
techniques [6]. This work was partly inspired by the work of Li
et al.. In particular, applying graph splitting for handling relay
nodes still plays a role in our analysis, although in directed
not undirected fashion. Li et al. further proposed as an open
question, whether a constant upper-bound exists for multicast
in hyper-networks [10]. We resolve this open problem, and
generalize the upper-bound of 2 for undirected networks to
an upper-bound of β for hyper-networks. In Peer-to-Peer and
overlay networks, the coding advantage was studied by Chiu
et al. [12], and by Shao and Li [13]. Recently, Xu et al. studied
the benefit of network coding for group communications, in
undirected networks with fairness constraints [14].
The benefit of network coding in wireless networks was first
studied by Liu et al. [15], and then extended by KeshavarzHaddad and Riedi [16]. They consider wireless interference
and geometric properties in modeling a wireless network,
and prove that both coding and cost advantages are upper
bounded by constant factors. Karande et al. [17] consider the
case of multiple-multicast in wireless Ad-hoc networks and
prove order bounds on the throughput improvement. Yang et
al. studied the coding advantage of physical layer network
coding in two-way relaying systems [18].
For multicast throughput, we adopt the standard definition
of symmetric throughput in the literature, i.e., all receivers are
required to receive and recover information flows transmitted
from the source at an equal rate. It is also possible to relax
the symmetry requirement, by allowing different receivers
to receive at different rates. Under such a model, coding
advantage is studied by Chekuri et al. [19], for average
throughput across all receivers.
III. N ETWORK M ODELS AND N OTATIONS
Undirected Networks. An undirected network is a capacitated
graph G(V, E, c), with node set V , edge set E, and link
E
. Here Z+ is the set of positive
capacity vector c ∈ Z+
integers. An undirected link uv with total capacity c(uv) can
YIN et al.: BOUNDING THE ADVANTAGE OF MULTICAST NETWORK CODING IN GENERAL NETWORK MODELS
→
u
c(uv)
→
z
c({u, v, w})
c(vu)
u
→
u
z
w
v
c(uv)
v
→
c(vu)
(a) Bidirected Link
1025
u
v
Fig. 3.
(b) Hyper-link
Fig. 2. Communication links in (a) bidirected networks and (b) hypernetworks. In bidirected networks, we use a double-headed arrow to represent
a pair of opposite links for simplicity.
→
→
support a flow f (uv) from u to v and a flow f (vu) from v
→
→
to u simultaneously, as long as f (uv) + f (vu) ≤ c(uv).
Directed Networks. A directed network is a directed graph
D(V, A, c), with node set V , link set A, and link capacity
→
A
. A directed link uv can transmit a flow from
vector c ∈ Z+
→
→
u to v only, subject to f (uv) ≤ c(uv).
Bidirected Networks. In a bidirected network, node connection is always bidirectional (Fig. 2 (a)): ∀u, v ∈ V ,
→
→
uv ∈ A implies vu ∈ A. Given a bidirected network
B(V, A, c), we define the link capacity balance parameter
→
c(uv)
→
as α maxuv∈A
→ , and refer to B(V, A, c) as an αc(vu)
balanced network. A special case is α = 1, corresponding
to a (completely) balanced network.
As a classic technique in graph theory, a link with integral
capacity can be replaced with multiple unit-capacity links
between the same pair of nodes. Such a transformation results
in a directed multi-graph, and does not change the network
connectivity between any pair of nodes.
Hyper-Networks. A generalization of the undirected network
model is the hyper-network model, where each hyper-link
connects two or more nodes (Fig. 2 (b)). Specifically, we
model a hyper-network with a hypergraph H(V, E), where E
is the set of hyper-edges. Each hyper-edge e ∈ E ‘covers’ a set
of two or more nodes. The size of a hyper-edge is the number
of nodes it connects. We denote the maximum hyper-edge size
as β. An undirected network can be viewed as a special type
of hyper-network with β = 2.
When one node transmits an information flow through a
hyper-link, the flow reaches all the other nodes adjacent to
the hyper-link simultaneously. Let f (e, v), e ∈ E, v ∈ e denote
flow transmitted from v through hyper-link e. The total flow
on a hyper-link at a given time is upper-bound by the hyperlink capacity. For example, if e covers three nodes u, v, w, we
have f (e, u) + f (e, v) + f (e, w) ≤ c(e).
IV. C ODING A DVANTAGE IN B IDIRECTED N ETWORKS
A. Optimal Multicast in Bidirected Networks
Multicast models the dissemination of information from a
common source to a set of receivers within the same network.
Given a (bi)directed network D(V, A, c), let s ∈ V be the
multicast source, and T ⊂ V \{s} be the set of multicast
receivers. Note that unicast (one-to-one) and broadcast (oneto-all) can be viewed as special cases of multicast, with
|T | = 1 and |T | = |V | − 1, respectively. A multicast rate
(throughput) r is achieved if each receiver can receive and
recover information flows from the source at rate r.
v
u
v
Directed splitting operation at z.
Let λD (u, v) be the max-flow rate from u to v, which
equals the maximum number of link-disjoint paths from u
to v in the unit-capacitated multigraph model. By the maxflow min-cut theorem in network flows, λD (u, v) equals the
capacity of the min cut between u and v. The celebrated result
[1] on multicast rate feasibility extends the max-flow min-cut
theorem from one-to-one unicast to one-to-many multicast:
with network coding, a multicast rate r is feasible in a directed
network off it is feasible as a unicast rate to each receiver
t ∈ T independently. Therefore, the maximum multicast rate
with network coding is Rnc (D, s, T ) = mint∈T λD (s, t).
Parameters (D, s, T ) are dropped when clear from context.
Without network coding, i.e., with routing and replication,
a multicast solution can be decomposed into a set of multicast
trees, since each information flow propagates along a tree
rooted at the multicast source, covering all multicast receivers.
In this context, achieving the maximum multicast throughput
is equivalent to the combinatorial problem of Steiner tree
packing [4]. Let T denote the set of all possible multicast
trees. For each tree τ ∈ T , let rτ be the information flow
rate to transmit along τ . A tree packing solution is feasible if
→
for each link uv ∈ A, the total flow rate of trees containing
→
→
uv is bounded by the link capacity c(uv). The maximum
throughput achieved by routing, denoted by Rtree (D, s, T ),
is the maximum aggregated flow rates of a feasible multicast
tree packing, which can be formulated into a linear program:
maximize
subject to :
τ ∈T
→
rτ
→
rτ ≤ c(uv),
rτ ≥ 0,
τ :uv∈τ
→
∀ uv ∈ A
∀τ ∈ T
By definition, Rtree ≤ Rnc for any (D, s, T ), since a
multicast tree packing solution is a special case of a network
coding solution. The coding advantage is Rnc /Rtree , the
ratio of maximum throughput with network coding over that
without coding, which is at least 1.
B. Upper-bound for Completely Balanced Networks
We introduce two lemmas from graph theory for studying
the coding advantage in completely balanced networks. The
first is about the splitting operation illustrated in Fig. 3.
In particular, a directed splitting at a node z refers to the
→ →
→
replacement of two links uz, zv with a new link uv.
Lemma 1. [20] Let D = (V + z, A) be a directed multigraph
with symmetrical nodal in- and out- degrees. For each link
→
→
uz ∈ A, there exists a link zv ∈ A, such that after splitting off
→ →
uz, zv, the link connectivity i.e., the number of directed linkdisjoint paths, between every pair of nodes in V is unchanged.
Lemma 2. [21] In a directed multigraph D = (V, A), the
maximum number of arc-disjoint spanning trees rooted at s ∈
V is mint∈V −s λD (s, t).
1026
Theorem 1. For multicasting in a completely balanced network B, Rtree = Rnc . Furthermore, the optimal multicast
throughput can be achieved with integral link flow rates, and
without coding or replication at relay nodes.
Proof: We consider the equivalent directed multigraph,
where a link with capacity larger than 1 is replaced with
parallel links each with unit capacity. We prove that there
exist Rnc link-disjoint multicast trees, each of which may
contribute a multicast throughput of 1.
A completely balanced network satisfies the nodesymmetric condition of Lemma 1. After each splitting operation, the node symmetric property remains true, although
link symmetry may no longer hold. Consequently, we can
completely isolate a node z by repeatedly applying splitting
operations at z, while preserving the max flow rate between
any two other nodes.
We apply complete splitting to relay nodes in V − s − T sequentially, and denote the resulting digraph as D. According to
Lemma 1, Rnc (B) = mint∈T λB (s, t) = mint∈T λD (s, t) =
Rnc (D), i.e., the optimal throughput achieved by network
coding is not affected. We do not explicitly distinguish these
two notations in the rest of the proof.
By Lemma 2, network coding cannot improve broadcast
throughput. D has no relay nodes and is a broadcast network,
hence there must exist Rnc broadcast trees in D.
The splitting operations at the relay nodes indicate how to
forward the received information. No replication or encoding
is required at the nodes being split off. We can achieve
the optimal throughput Rnc by applying reverse splitting
operations on broadcast trees in D, and using the resulting
multicast trees for transmission. As a result, we only need to
replicate information at the source and receiver nodes.
Polynomial Time Algorithm for Steiner Tree Packing.
Steiner Tree Packing in general graphs is a well known NPhard problem. Interestingly, for completely balanced networks,
a polynomial time tree packing algorithm can be extracted
from the proof of Theorem 1:
Step 1: Split off all relay nodes to obtain a directed graph D;
Step 2: Apply the polynomial time algorithm of Wu et al. [22]
to compute a spanning tree packing in D;
Step 3: Translate spanning trees in D to Steiner trees in B,
by reversing the split operations.
For step 1, brute-force search for splittable link pairs takes
O(|T ||E||V |3 ) time. Step 2 can be done in time O(R3nc |T ||E|)
[22]. Step 3 can be done in time O(|E|) with the split
operations in step 1 stored in memory. Thus the overall
complexity is O(|T ||E||V |3 + R3nc |T ||E|).
Wu et al. [23] designed heuristic algorithms for Steiner
tree packing in a set of six-ISP topologies that are all linkbalanced. Our algorithm outperforms their heuristic algorithms
by guaranteeing optimal solutions.
Discussions. The “link-balanced” condition of the theorem can
be relaxed to “node-balanced”, which guarantees the existence
of the desired directed splitting operation. We emphasize that
the optimal throughput can be achieved by packing multicast
trees with integer flow rates, while in general, fractional
packing outperforms integral packing [5]. For example, in the
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 3, MARCH 2014
classic butterfly network (directed) [1], [24], fractional packing
can achieve a multicast rate 1.5, while integral packing can
only achieve a rate 1.
While access networks at the edge of the Internet may not be
balanced, the core of the Internet is rather close to a balanced
network [25]. Then by Theorem 1, in such type of balanced
networks, optimal multicast is feasible without IP-multicast
or network coding requirements at routers in the middle of
the network. Furthermore, the fact that fractional flows are
unnecessary for achieving the maximum multicast throughput
helps reduce the overhead in traffic splitting and management
for achieving optimal multicast.
C. Upper-bound for α-Balanced Networks
Networks in practice may exhibit approximate rather than
absolute symmetry in opposite link capacities, and can be characterized using the α-balanced network model. The coding
advantage in α-balanced networks was previously analyzed
by Li et al. [10] and was proved to be upper-bounded by
2(α + 1). We apply Theorem 1 to improve this upper-bound
to α.
Theorem 2. The coding advantage for multicast in an αbalanced network is upper-bounded by α.
Proof: Let B = (V, A, c) be an α-balanced network,
let B1 = (V, A, c
) denote the completely balanced graph
induced from B by truncating the larger link capacity to the
smaller one, between each pair of neighboring nodes. We have
Rtree (B) ≥ Rtree (B1 ).
Let αB1 = (V, A, αc
) be the digraph induced from B1
by scaling each link’s capacity by α. Each link’s capacity
in αB1 is no less than in B, hence Rnc (αB1 ) ≥ Rnc (B),
and Rtree (B1 ) = Rnc (B1 ) = α1 Rnc (αB1 ), where the first
equality is due to Theorem 1, since B1 is a completely
balanced network, and the second equality holds because the
maximum throughput scales linearly with the edge capacities.
We conclude that Rtree (B) ≥ α1 Rnc (B).
D. Lower-bound for α-Balanced Networks
Can the improved upper-bound of α in Theorem 2 be further
tightened? Apparently, the bound is tight for the case of α = 1.
For larger values of α, we show that there
√ exist α-balanced
networks with coding advantage of Ω( α). The upper- and
lower- bounds we prove are therefore √
approximately tight
against each other, within a factor of O( α).
Our approach to show the lower-bound of the maximum
coding advantage is to construct bidirected multicast networks
with large coding advantages. We manipulate a class of
directed networks with known coding advantage into the bidirected domain. The directed networks chosen are ZK(p, N )
networks introduced by Chekuri et al. [19] in their investigation of the coding advantage for average multicast throughput.
The ZK(p, N ) networks turn out to be more powerful than
directed combinational network, in terms of providing higher
coding advantage with less terminal nodes.
A ZK(p, N ) network consists of a source s, N receivers
t1 , t2 , · · · , tN, and three layers of relay nodes. The first layer,
N
layer A, has p−1
nodes, which can be indexed as AU , with
U ⊂ {1, 2, · · · , N } and |U | = p − 1. The next two layers
YIN et al.: BOUNDING THE ADVANTAGE OF MULTICAST NETWORK CODING IN GENERAL NETWORK MODELS
s
1
Combining the above inequalities (1–4), we have that the
maximum rate for fractional routing
2
3
Layer A
4
Rtree ≤
Layer B
12
13
14
23
24
t2
t3
p−1
Rtree
1
2(p − 1)
≤
+ +
Rnc
N −p+1
p
α
t4
Fig. 4. The α-balanced network ZKα (p = 2, N = 4). Each downward
link has capacity α, each upward link has capacity 1.
N B and C each has p nodes, which can be indexed as BW
and CW respectively, with W ⊂ {1, 2, · · · , N } and |W | = p.
These nodes are connected in the following way: for any nodes
AU , BW , CW , there is a link from s to AU , a link from AU to
BW if U ⊂ W , a link from BW to CW , and a link from CW
to ti if i ∈ W . So far each link is unidirectional. To construct
the α-balanced network ZKα (p, N ) from ZK(p, N ), we add
an opposite link of unit capacity for each link in ZK(p, N ),
and set all the original links’ capacity to α. Fig. 4 illustrates
the ZKα (p = 2, N = 4) network.
Theorem
3. The coding advantage in ZKα (p, N ) is of order
√
Ω( α).
Proof: For each rooted Steiner tree τ , let aτ denote the
number of layer A nodes in τ , cτ denote the number of
layer C nodes with its predecessor in layer B, and dτ denote
the number of layer C nodes with its predecessor being a
receiver. For a fractional tree packing T = {τ1 , τ2 , · · · , τ|T | }
with tree flow rates r1 , r2 , · · · , r|T | , consider the aggregated
link capacity constraints at nodes counted in aτ , cτ , and dτ
respectively,
rτ aτ
≤
rτ cτ
≤
r τ dτ
≤
τ ∈T
τ ∈T
τ ∈T
N
α
p−1
N
α
p
N
p
p
N
+p
p
(1)
(2)
(3)
The maximum number of receivers directly covered by
nodes counted in cτ is aτ (p−1)+cτ . This is because the nodes
with the same grand parent in layer A have p − 1 receivers
in common. Grouping them according to their grand parent in
layer A, each group will cover p − 1 common receivers with
each member covering one special receiver at most. For each
node in dτ , it can cover at most p − 1 receivers. Therefore,
(p − 1) + cτ + dτ (p − 1) ≥ N .
to cover all the receivers, aτ
Multiplying both sides with τ ∈T rτ ,
N
τ ∈T
rτ ≤
τ ∈T
rτ aτ (p − 1) + cτ + dτ (p − 1)
1
N
N
N
α(p − 1)
+α
+ 2p(p − 1)
N
p−1
p
p
−1
. Therefore,
Furthermore, Rnc = α Np−1
34
Layer C
t1
1027
(4)
√
√
For large α that α and α are both integers, let p−1 = α
Rtree
1
3
and N = α, then Rnc < √α−1 + √α .
V. C ODING A DVANTAGE IN H YPER -N ETWORKS
Wireless networks represent a promising paradigm for application of network coding. Due to the broadcast nature of a
wireless transmission, an encoded packet can be delivered to
multiple neighbors in a single transmission, helping more than
one of them in a bandwidth efficient manner. Correspondingly,
a hyper-network model for wireless networks was looked at
by Li et al. [10], but bounding the coding advantage there
is left open. In this section, we prove the first upper-bound
for the coding advantage in hyper-networks.Hyper-networks
capture the broadcast nature but not interference of wireless
transmissions, hence results in this section do not directly
extend to real-world wireless networks.
A. Optimal Multicast in Hyper-networks
Let H = (V, E) be an (undirected) hypergraph. In this section, we use e to denote a hyper-edge in E. A path connecting
s and t is a sequence of hyper-edges (e1 , e2 , · · · , en ), such
that s ∈ e1 , t ∈ en , ei ∩ ei+1 = ∅, i = 1, 2, · · · , n − 1, and
ei = ej , ∀i = j. The edge connectivity λH (s, t) is defined as
the maximum number of hyper-edge disjoint paths between
s and t. Assume each hyper-edge has unit capacity, then
the maximum throughput from s to t is upper bounded by
λH (s, t), i.e., Rnc (H, s, T ) ≤ mint∈T λH (s, t).
An orientation of a hypergraph is obtained by assigning
a direction to each hyper-edge e, via identifying a tail for e,
indicating which node is sending messages through this hyperlink.
For s ∈ V, T ⊂ V \{s}, define an s-T hyper-tree as a set
of hyper-edges τ that has an orientation containing at least
one directed path from s to t, ∀t ∈ T . For routing in the
hyper-network model, the trace of a message from s to all
receivers forms an s-T hyper-tree. Therefore, the maximum
throughput achieved by routing is the maximum packing of
s-T hyper-trees.
Let T denote the set of all possible s-T hyper-trees. Similar
to the bidirected network case, Rtree (H, s, T ) is the optimal
value of the following linear program:
maximize
subject to :
rτ
r
≤
c(e),
τ :e∈τ τ
rτ ≥ 0,
τ ∈T
∀e ∈ E
∀τ ∈ T
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w
w
u
e1
e2
x
v e1
v e2
u
y
C. Lower-bound for Hyper-networks
x
y
Fig. 5. A hyper-network (left) and its corresponding completely balanced
bipartite network (right).
B. Upper-bound for Hyper-networks
Theorem 4. In an undirected hypergraph H = (V, E),
the coding advantage Rnc /Rtree is upper bounded by the
maximum hyper-edge size β.
Proof: As the maximum throughput Rnc is upper
bounded by λH (s, T ) = mint∈T λH (s, t), it is sufficient to
prove that Rtree ≥ β1 λH (s, T ).
As illustrated in Fig. 5, given the hypergraph H = (V, E),
we construct a completely balanced bipartite digraph B(V ∪
VE , A), where V is the original node set in H, and VE =
{ve | ∀e ∈ E}. Connect v ∈ V and ve ∈ VE if v is an end
→
→
node of e. In other words, A = {vve , ve v | ∀v ∈ e}. For
any pair of nodes u, v ∈ V , λB (u, v) ≥ λH (u, v), because
for any n hyper-edge disjoint s-t paths P1 , P2 , · · · , Pn in H,
we can find n edge disjoint s-t paths P1 , P2 , · · · , Pn in B by
the following method: let Pi = (e1 , e2 , · · · , en ), we choose
a sequence of intermediate nodes vi ∈ ei ∩ ei+1 , so the path
can be rewritten as s = v0 , e1 , v1 , e2 , · · · , vn−1 , en , vn = t.
Then we construct Pi as (v0 , ve1 , v1 , · · · , ven , vn ). Because
ei appears only once in P1 , P2 , · · · , Pn , P1 , P2 , · · · , Pn are
edge disjoint.
By Theorem 1, in the completely balanced network B, there
is a rooted Steiner tree packing
τ1 , τ2 , · · · , τm with weights
r1 , r2 , · · · , rm , such that m
j=1 rj = Rtree (B) = Rnc (B) =
λB (s, T ) ≥ λH (s, T ). We complete the proof by constructing
m hyper multicast trees τ1 , τ2 , · · · , τm
in H with weights
r1 /β, r2 /β, · · · , rm /β. Let τj = {e ∈ E | ve ∈ τj }. The
orientation and s-t directed path in τj can be determined
referring to the s-t path in τj , so τj is an s-T hyper-tree.
For any hyper-edge e ∈ E,
j:e∈τj
=3
rj /β =1
→
v∈N(ve ) j:vv
e ∈τj
rj /β
→
j:ve ∈τj
rj /β =2
j:∃!v∈V,vve ∈τj
rj /β ≤4
1/β ≤5 1
v∈N(ve )
where N (ve ) is ve ’s neighbor set in B. In the above
derivation, =1 is due to the construction of hyper-tree τj ,
and =2 holds because there is a unique parent v for the
intermediate node ve , as τj is an s rooted tree. In other words,
→
ve ∈ τj iff there is a unique link vve ∈ τj . The right side
of =3 rewrites the sum by enumerating the neighbors of ve .
≤4 holds because τ1 , τ2 , · · · , τm with weights r1 , r2 , · · · , rm
form a feasible tree packing. ≤5 holds because each hyperedge covers β nodes at most, and the neighbors of ve in B
are the same nodes covered by e in H.
with weights
Therefore, the hyper-trees τ1 , τ2 , · · · , τm
constitute
a
feasible
hyper-tree
packing
r1 /β, r2 /β, · · · , rm /β
m
1
1
in H. Recall that
r
/β
=
λ
≥
λ
,
we
have
j
B
H
j=1
β
β
Rtree (H) ≥ β1 λH (s, T ) ≥ β1 Rnc (H).
It is postulated that the upper-bound of 2 for coding
advantage in undirected networks is not tight [10], and the
largest value known is 8/7. Closing the gap between 2 and
8/7 has remained an important open problem. For hypernetworks, we prove that the maximum coding advantage is of
order Ω(log β), through transforming combination networks
[11] into hyper-networks.
Theorem 5. The maximum coding advantage among hypernetworks of maximum hyper-edge size β is of order Ω(log β).
Proof: We construct a hypergraph HC(n, k) from the
undirected combination network C(n, k), by viewing the
star topology centered at each relay node
as a hyper-edge.
HC(n, k) contains one source node, nk receiver nodes, and
n hyper-edges each of unit capacity. Label the receivers with
the chosen set M ⊂ {1, 2, · · · , n}, |M | = k, hyper-edge
ei (i = 1, 2, · · · , n) connects the source and all the receivers
M if i ∈ M . Each receiver is adjacent
to k hyper-edges, and
each hyper-edge has size β = n−1
k−1 + 1.
Let T denote the set of all possible s-T hyper-trees, and
{rτ |τ ∈ T } be the optimal hyper-tree packing. For any hypertree τ ∈ T which connects s to all the receivers, τ has at
least n − k hyper-edges, because if there are k hyper-edges
not contained in τ , the corresponding receiver only connecting
to them is uncovered.Sum all the inequalities of hyper-edge
capacity
constraints τ :ei ∈τ rτ ≤ 1, i = 1, 2, · · · , n, we
hyper-edges
have τ ∈T rτ |τ | ≤ n, where |τ | is the number of in τ and is at least n − k. Therefore, Rtree = τ ∈T rτ ≤
n
n−k . We also claim Rnc = k, since each receiver is directly
connected to the source by k hyper-edges.
Let n = 2k,
+
1
<
22k , we have
Rnc /Rtree ≥ k/2. Since β = n−1
k−1
1
Rnc /Rtree > 4 log β.
VI. C OST A DVANTAGE
Besides throughput improvements, bandwidth and cost saving is another important benefit of network coding. With the
help of network coding, less overall bandwidth consumption
is required to achieve a target multicast rate. More generally,
we assume a link cost vector w ∈ QA
+ , indicating the cost
to transmit a unit flow through a link. Here Q+ is the set
of positive rational numbers. For a multicast solution with an
→
→
underlaying flow of rate f (uv) at each link uv, total cost is
→
→
→
w(uv)f (uv).
uv∈A
A. Upper-bound for Bidirected Networks
We start with uncapacitated networks that have sufficient
bandwidth supply at each link, such that link capacities are not
a limiting factor of optimal multicast cost. Given an uncapacitated bidirected graph D(V, A) and a link cost vector w, let
Ctree (D, w) and Cnc (D, w) be the min cost to achieve a unit
multicast rate with routing and network coding respectively.
The cost advantage is defined as Ctree (D, w)/Cnc (D, w).
With link capacity limits removed, an undirected network
and a bidirected network differ only in that link weights in
opposite directions are always equal in the former but not the
latter. Let α denote the maximum ratio of link weights of the
YIN et al.: BOUNDING THE ADVANTAGE OF MULTICAST NETWORK CODING IN GENERAL NETWORK MODELS
1029
→
w(uv)
→
two directions, i.e., α maxuv∈A
→ . The weight vector
w(vu)
w is called α -balanced.
When α = 1, we are essentially considering the maximum
cost advantage in an undirected network, which was proved
to be equal to the maximum coding advantage of undirected
networks [5], [11], and therefore, is upper bounded by 2:
Theorem 6. In an uncapacitated bidirected network with
symmetric link weights, the cost advantage is at most 2.
For the general case α ≥ 1, we have the following result
that echoes the the coding advantage.
Theorem 7. In an uncapacitated bidirected network (B, w),
the cost advantage is at most 2α .
Proof: Let w
denote weight vector derived from
w by truncating the larger weight to the smaller one, so
that w
is symmetric. As w is α -balanced, α w
≥
w ≥ w
. Because the min cost is monotonic to
the weight vector, we have Cnc (B, w) ≥ Cnc (B, w
)
and Ctree (B, α w
) ≥ Ctree (B, w). As w
is symmetric, Cnc (B, w
) ≥ 12 Ctree (B, w
) according to theorem 6. Combining these inequalities with the fact that
1
1
2 Ctree (B, w
) = 2α Ctree (B, α w
), we conclude that
1
Cnc (B, w) ≥ 2α Ctree (B, w).
If the network is not bidirected, the cost advantage is
unbounded, as can be demonstrated in directed combination
networks C(n, k) with the following setting of link weights:
set the link weight to 1 for each link from the source to an
intermediate node, and set the link weight to → 0 for other
links. As each multicast tree contains at least n − k + 1 links
from the source to the intermediate nodes, Ctree ≥ n − k + 1.
But with network coding, we can achieve a unit multicast
rate with each link carrying a flow of rate 1/k, which means
Cnc ≤ n/k. Thus letting n = 2k → ∞, we can see the cost
advantage is unbounded. This can be viewed as a special case
of α = ∞ (link costs are arbitrarily unbalanced).
It is interesting to note that unlike the duality between the
cost advantage and coding advantage in undirected networks,
we have coding advantage of 1 for completely balanced
networks, but maximum cost advantage lower bounded by 9/8
[11] with completely balanced link weights.
B. Lower-bound for Bidirected Networks
Similar to the case of√coding advantage, we now derive a
lower bound of order Ω( α ) for the maximum cost advantage
of bidirected networks with α -balanced link weights.
√
Theorem 8. A lower-bound of order Ω( α ) exists for the max
cost advantage in an α -weight-balanced bidirected network.
with
Proof: For a large integer
√ α , we construct a network
cost advantage of order Ω( α ) by assigning an α -balanced
link weight vector w to the bidirected network ZK(p, N ),
where N = p(p − 1) = α . Assign weights to the downward
links: 1) w(e) = N/p for a link from the source to layer A; 2)
w(e) = 1/p for a link from layer A to layer B; 3) w(e) = 1
for a link from layer B to layer C; 4) w(e) = 1/p for a link
from layer C to receivers. Each upward link weight is N times
the backward link’s cost. The network is α = N balanced.
A network coding solution with unit
each
rate on N
−1flow
with cost p−1 ·
downward link achieves multicast rate Np−1
N
N
N
N
1
1
p + p p · p + p + p p · p . Therefore, the min-cost to
N
achieve a unit multicast rate Cnc ≤ N −p+1
· Np + 3 · Np
For any multicast tree τ , let aτ denote the number of layer
A nodes in τ , cτ the number of layer C nodes with predecessor
in layer B, and dτ the number of layer C nodes with a receiver
predecessor. Considering the total cost of links entering these
nodes, the cost of τ is no less than aτ Np + cτ + dτ Np =
aτ (p − 1) + cτ + dτ (p − 1) ≥ N , where the last inequality
is due to (4). Hence Ctree ≥ N , and for sufficiently large N ,
N
1
3
5
√5 .
N −p+1 · p + p ≤ p ≈
α
C. Cost Advantage in Hyper-networks
For a hyper-network H, assign each hyper-link a weight
w(e) indicating the cost to transmit a unit flow there, then
the cost of a multicast solution is defined in the same way as
in the bidirected model. Inspired by Agarwal and Charikar’s
work [5] on the relation between coding advantage and cost
advantage in undirected networks, we prove that the maximum
cost advantage in hyper-networks equals the maximum coding
advantage, suggesting that these two quantities can be studied
under a unified framework.
Theorem 9. Given a hyper-network H = (V, E), the maximum coding advantage of any link capacity function c equals
the maximum cost advantage of any link weight function w:
max
c
Rnc (H, c)
Ctree (H, w)
= max
w
Rtree (H, c)
Cnc (H, w)
Proof: We carry out the prove in two steps. First, we
Ctree (H,w)
nc (H,c)
prove that maxc RRtree
(H,c) ≤ maxw Cnc (H,w) by finding a
weight function w for any given capacity c, such that the cost
advantage with w is no less than the coding advantage with c.
Without loss of generality, we may assume Rnc (H, c) = 1 and
consider the dual of the optimal hyper-tree packing problem:
minimize
subject to :
c(e)y(e)
e∈E
y(e)
≥ 1,
e∈τ
y(e) ≥ 0,
∀τ ∈ T
∀e ∈ E
In fact, the optimal solution y of this dual problem gives
the desired weight function. As Rnc (H, c) = 1, there is a
network coding
solution achieving unit multicast rate with cost
no more than e∈E c(e)y(e), which equals Rtree (H, c) according to the strong duality property. Therefore, Cnc (H, y) ≤
Rtree (H, c). On the other hand, the constraints of the dual
problem promise that each hyper-tree has cost at least 1, i.e.,
Ctree (H, y) ≥ 1.
Ctree (H,w)
nc (H,c)
Second, we prove that maxc RRtree
(H,c) ≥ maxw Cnc (H,w) .
Without loss of generality, assume Ctree (H, w) = 1. Consider
a network coding solution achieving the min cost Cnc (H, w).
Let c(e) be the sum of information flows through this link.
Obviously, Rnc (H, c) ≥ 1. As the cost of each hyper-tree is
at least 1, the optimal tree
packing number Rtree (H, c) is no
more than the total cost e∈E c(e)w(e) = Cnc (H, w), which
completes the proof.
Combining the above theorem with our previous results on
the coding advantage in section V, we can derive:
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s
1
1
2
2
coding solution of cost 6. The cost advantage therefore is no
less than 7/6 in this example.
Combining all the results on completely balanced networks
in this paper, we may conclude that the benefits of network
coding in this scenario is marginal, not only in terms of
maximum throughput, but also in terms of computational
complexity. We can find a routing solution in polynomial time,
which achieves the optimal multicast rate with cost at most
twice of the optimal cost with network coding.
t2
t1
Fig. 6. A capacitated bidirected network, where both link capacity and link
weight are balanced. Each link has unit capacity. Four pairs of links have
labeled weights; other links have negligible weights. Target multicast rate is
2. A network coding solution using all the downward links has cost 6. A
brute-force search shows that the min cost of a routing solution has cost 7,
shown with color links. The cost advantage is at least 7/6.
Corollary 1. In undirected hyper-networks, the cost advantage is upper bounded by the maximum hyper-link size β, and
lower bounded by 14 log β.
D. Cost Advantage in Capacitated Bidirected Networks
So far in this section, we have assumed that links are
over-provisioned in bandwidth. We now remove this assumption, and consider cost advantage in capacitated networks, which further depends on the required multicast rate.
Given a (bi)directed graph D(V, A), a link capacity vector c, a weight vector w, and a required multicast rate r
which can be achieved by both routing and network coding,
we define the cost advantage of a capacitated network as
(D, c, w, r)/Cnc
(D, c, w, r), where Ctree
(D, c, w, r) and
Ctree
Cnc (D, c, w, r) are the min cost of routing solution and
network coding solution for achieving multicast rate r, respectively. The uncapacitated scenario can be viewed as a special
→
→
case of the capacitated scenario, with r ≤ c(uv), ∀ uv ∈ A.
For any network where the coding advantage is larger than
1, we can add a very high cost link from the source to each
receiver, and set the required multicast rate accordingly, so
that the routing solution has to resort to the high cost links,
while a network coding solution does not. The cost advantage
then becomes arbitrarily large in such capacitated networks.
Theorem 10. In a completely balanced network (B, c, w), the
cost advantage is upper bounded by 1 + α for any achievable
rate r.
→
Proof: Let f (uv) be themin cost flow for network
→
→
→
= uv∈A
w(uv)f (uv). Since the
coding solution, so that Cnc
network is completely balanced, there is a completely balanced
→
→
subgraph (B , c ) from f with capacity c (uv) = c (vu) =
→
→
max{f (uv), f (vu)}. By Theorem 1, there is a rout
→
→
→
ing solution in (B , c ). Thus Ctree
≤
w(uv)c (uv
uv∈A
→
→
→
w(uv)(f (uv) + f (vu)) ≤
→
→
α
w(
vu)f
(vu) ≤ (1 + α )Cnc
.
vu∈A
) ≤
→
uv∈A
→
→
uv∈A
→
→
w(uv)f (uv) +
For an example where the cost advantage is strictly larger
than 1, consider the completely balanced network in Fig. 6,
where each link has unit capacity, the target multicast throughput is 2, and the weight vector is symmetric (labeled beside the
links). By enumerating all possible routing solutions, we find
that the min cost with routing is 7, while there is a network
VII. E XTENSION TO B IDIRECTED H YPER - NETWORKS
So far, we have considered undirected hyper-networks
which generalize the undirected network in terms of link size.
For directed hyper-networks, since a directed network can be
regarded as a directed hyper-network with link size β = 2,
existing results show that coding advantage can be arbitrarily
large even in directed hyper-networks with small link size [2].
“Bidirectional transmission” also exists in broadcast links,
i.e., if node A can broadcast messages to node B, node B is
able to broadcast to A as well. This motivates us to study
the “bidirected hyper-networks”. In particular, let (e, v), v ∈ e
denote a directed hyper-link that transmits messages from v
to the set of nodes e\v. A directed hyper-network H(V, A) is
bidirected if ∃(e, u) ∈ A : v ∈ e implies ∃(e , v) ∈ A : u ∈ e .
Coding Advantage Given a bidirected hyper-network
H(V, A) with link capacity c, we define the link capacity
balance parameter α as
(e ,v)∈A,u∈e c(e , v)
α
max
u,v∈V,∃(e,v)∈A,u∈e
(e ,u)∈A,v∈e c(e , u)
Note that there may be different hyper-links that connect u to
v. Thus we take the sum capacity of these links as the direct
transmission capacity from u to v in the definition. We refer
to H(V, A, c) as an α-balanced bidirected hyper-network.
Theorem 11. The coding advantage in α-balanced bidirected
hyper-networks is upper bounded by α(β − 1), where β is the
maximum link size.
Proof: Given the hyper-network H(V, A) with link capacity c, we construct an α-balanced bidirected network
B(V, A , c ) by replacing each hyper-link (e, v) ∈ A with
|e| − 1 directed links (v, u), ∀u ∈ e\{v}, whose capacities
are set to c(e, v). By the definition of α-balanced bidirected
hyper-network, the constructed network B is α-balanced.
For any s, t ∈ V , the maximum (s, t)-flow in B is no less
than the corresponding max-flow in H because any (s, t)-flow
in H is also an (s, t) flow in B. Thus, Rnc (B) ≥ Rnc (H). By
Theorem 2, Rtree (B) ≥ α1 Rnc (B). To complete the proof, it
1
Rtree (B).
is sufficient to show that Rtree (H) ≥ β−1
Let τ1 , τ2 , · · · , τm with weights r1 , r2 , · · · , rm be a feasible
tree packing in B, we obtain a hyper-tree τi from τi by
replacing each link (u, v) ∈ τi with
the hyper-link (e, u)
from which (u, v) is introduced. As j:(u,v)∈τj rj ≤ c(e, u)
and (e, u) is replaced with |e| − 1 links in the construction
of B, the hyper-tree packing τ1 , τ2 , · · · , τm
with weights
r1 /(β − 1), r2 /(β − 1), · · · , rm /(β − 1), is a feasible tree
1
Rtree (B).
packing in H. Therefore, Rtree (H) ≥ β−1
YIN et al.: BOUNDING THE ADVANTAGE OF MULTICAST NETWORK CODING IN GENERAL NETWORK MODELS
15
A. Cost Advantage
Our upper−bound
Prev. upper−bound [10]
Coding advantage
Let w(e, u) denote the weight of a directed hyper-link
(e, u). As multiple hyper-links may connect u to v, we use
the minimum link weight to define the link weight balance
parameter α :
α min(e ,v)∈A,u∈e w(e , v)
u,v∈V,∃(e,v)∈A,u∈e min(e ,u)∈A,v∈e w(e , u)
10
backbone
networks
5
max
Theorem 12. In a bidirected hyper-network with α -balanced
link weights, the cost advantage is upper bounded by 2α (β −
1), where β is the maximum link size.
Proof: Let H(V, A) denote the bidirected hyper-network
with α -balanced link weights w. Construct a bidirected network B(V, A) as A = {(u, v)|∃(e, u) ∈ A, v ∈ e}, and let
the link weight w (u, v) = min∃(e,u)∈A,v∈e w(e, u). By the
definition of α , B is α -balanced.
By Theorem 7, we have Ctree (B) ≤ 2α Cnc (B). Each
multicast tree in B can be directly converted into a hypertree in H with the same cost by replacing each directed link
(u, v) with the hyper-link (e, u), v ∈ e that has the minimum
weight w (u, v). Thus, Ctree (H) ≤ Ctree (B). To complete the
proof, it is sufficient to show that Cnc (B) ≤ (β − 1)Cnc (H).
For a network coding solution in H, let f (e, u) denote
the flow rate on each hyper-link (e, u). We can use |e| − 1
flows f (u, v) = f (e, u), v ∈ e\{u} to mimic the broadcast
transmission, yielding a network coding solution in B with
cost at most (β − 1)Cnc (H).
VIII. C ONNECTIONS TO R EAL -W ORLD N ETWORKS
A. Bidirected Networks
Most of today’s wireline networks operate in full-duplex
mode. The two opposite transmissions have independent link
capacities. Consequently, these networks can be modeled as
bidirected networks. The Internet can be roughly divided into
access networks and backbone networks. Access networks
connect subscribers to their ISPs. As most consumers have
a larger download demand than upload demand, the access
links are often imbalanced. There are many types of wireline
access technologies. For example, α is usually 3 ∼ 12 for
ADSL and VDSL links [26]. For an access network with
optical fiber or cable modem, the capacities of up-stream and
down-stream links depend on the ISP’s sale plans. According
to the empirical study on residential broadband plans over
29 countries [27], α in broadband plans from 25 countries
ranges from 2 to 21, with a median value of 8.53. A recent
measurement on the broadband performance in US also reveals
that α lies between 2 and 16 [28]. Ethernet connections are
commonly found in companies and schools. For an Ethernet
with twisted-pair or fiber optic links in conjunction with fullduplexing switches, its α = 1. We note that if the Ethernet is
connected by coaxial cables or hubs that work in half-duplex
mode, it should be modeled as an undirected hyper-network.
Backbone networks inter-connect local access networks.
Measurement studies shows that the core of the Internet has
completely balanced link capacities. For example, the six
ISP topologies in the Rocketfuel project are all completely
balanced [23]. Internet structure studies often apply a single
value for specifying link capacities in both directions [7]. If
1031
access networks
1
Fig. 7.
5
α
10
15
Upper-bounds of coding advantage in real-world networks.
the background traffic is considered, the available bandwidths
for a multicast session may be imbalanced. Fraleigh et al.
[25] report that the closer to the Internet backbone, the more
symmetric the communication traffic is. For the links they
examined (OC-48), the opposite traffic ratio is always between
1:1 and 5:1. John and Tafvelin [8] tap a backbone link
(OC-192), and report that no significant difference between
opposite link flow volumes can be observed. Fig. 7 summaries
the implication of the coding advantage upper-bounds in realworld networks.
B. Hyper-Networks
Wireless networks have the salient feature of supporting
broadcast transmissions. Both directed and undirected hypernetwork models model such broadcast nature but not wireless
interference, and hence cannot be used to study the coding
advantage in wireless networks. However, the directed hypernetwork model can be used to study the cost advantage that
refers to the minimum cost (e.g. energy) to multicast a unitlength message, where interference is irrelevant. Wu et al.
proved that the minimum energy multicast formulation is
equivalent to a cost minimization with linear pricing [29],
and there are a number of works using directed hypergraphs
to represent the network topology [9]. We note that many
wireless networks are bidirected, which can be modeled by the
bidirected hyper-network. The maximum size of a broadcast
link varies in a wide range. For ad hoc wireless networks, the
broadcast range is often restricted, resulting in a small β. For
example, β ≤ 14 in the testbeds of wireless network coding
application [30]. For cellular networks, a base station may
serve thousands of terminals.
IX. C ONCLUSION
We have focused on the benefits of network coding in
two types of parameterized networks throughout this work,
including bidirected networks and hyper-networks. Compared
to simple directed and undirected network models, these
networks are more powerful and flexible for characterizing
real-world networks. We proved a number of upper-bounds on
the potential benefits of network coding, in terms of improving
multicast throughput and saving multicast cost.
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Xunrui Yin received his B.S. and Ph.D. degrees
in computer science from Fudan University in 2007
and 2013, respectively. He is currently a Postdoctoral fellow at University of Calgary, Canada.
His research interests include network coding, network topology design and coding theory.
Yan Wang received her Master’s degree in computer
science from Nanchang University in 2006. She
is currently working towards her Ph.D. at Fudan
University, China. She is an Assistant Professor
in East China Jiaotong University since 2006. She
is a recipient of the 2013 Google China Anita
Borg Memorial Scholarship. Her research interests
include distributed storage systems, network coding
and data center networks.
Zongpeng Li received his B.E. degree in Computer
Science and Technology from Tsinghua University
(Beijing) in 1999, his M.S. degree in Computer
Science from University of Toronto in 2001, and
his Ph.D. degree in Electrical and Computer Engineering from University of Toronto in 2005. Since
2005, he has been with the Department of Computer
Science at the University of Calgary. In 2011-2012,
Zongpeng was a visitor at the Institute of Network
Coding in CUHK. His research interests are in
computer networks and network coding.
Xin Wang received his BS Degree in Information
Theory and MS Degree in Communication and
Electronic Systems from Xidian University, China,
in 1994 and 1997, respectively. He received his PhD
Degree in Computer Science from Shizuoka University, Japan, in 2002. He is currently a professor
in Fudan University, Shanghai, China. His research
interests include quality of network service, nextgeneration network architecture, mobile Internet and
network coding.
Jin Zhao received the B.Eng. degree in computer
communications from Nanjing University of Posts
and Telecommunications, China, in 2001, and the
Ph.D. degree in computer science from Nanjing
University, China, in 2006. He has been with Fudan University since 2006,where he is currently
an Associate Professor. He stayed at University of
Goettingen, Germany for 3 months as a visiting
scholar in 2010. His research interests include future
Internet and network coding theory. He is a member
of IEEE and ACM.
Xiangyang Xue received the B.S., M.S., and Ph.D.
degrees in communication engineering from Xidian
University, Xian, China, in 1989, 1992, and 1995,
respectively. He joined the Department of Computer Science, Fudan University, Shanghai, China,
in 1995. Since 2000, he has been a Full Professor.
His current research interests include multimedia
analysis, retrieval and filtering. He has authored
more than 100 research papers in these fields.