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ISIT2007, Nice, France, June 24 - June 29, 200O
Network Coding in Wireless Networks with Random Access
Danail Traskovt, Desmond S. Lun§, Ralf Koettert, and Muriel Medardt
§Broad Institute
+LIDS
tCoordinated Science Laboratory
MIT and Harvard
MIT
University of Illinois at Urbana-Champaign
Urbana, IL
Cambridge, MA
Cambridge, MA
Email: {traskov2, koetter}@uiuc.edu
Email: [email protected] Email: [email protected]
The joint resource provision and allocation strategy we will
investigate is random, linear, intra-session network coding
combined with a slotted-ALOHA random medium access
scheme. To be more precise, network coding takes place only
within the set of packets of a particular user (i.e. intra-session)
and is applied by forming linear combinations of packets with
random coefficients at intermediate nodes of the subgraph.
This scheme is easy to implement, it achieves the capacity for
a single connection [3], and has a simple flow optimization
I. INTRODUCTION
formulation [4]. Given multiple connections, it is reasonable
The problem we consider is to establish a set of multicast to ask why we restrict ourselves from using network coding
or unicast connections in a wireless network using network combining packets of different users (i.e. inter-session). There
coding. The utility of network coding in such a setup has exists a variety of techniques for inter-session network coding,
been demonstrated e.g. in [1], [2]. There, the authors develop such as [5]-[8], but these are quite complex and therefore siga mathematical framework for computing a network coding nificantly further down the road of practical implementation. In
subgraph that can support the desired connections and provide contrast COPE [9] is a practical and successfully implemented
simulation results showing significant advantages of a network network coding protocol. However, our focus is on developing
coded approach. As they point out, to include the interference mathematical models in terms of optimization of flows and
constraints of the wireless network in the optimization model the opportunistic and heuristic nature of COPE is difficult to
may pose quite a challenge. In fact, it is the structure of capture in such a framework. Although we use solely intrathe medium access constraints of the network that essentially session network coding, it is important to keep in mind that, in
determines the difficulty of the resulting optimization problem. general, inter-session network coding is necessary to approach
Conceptually, one can view the overall problem, of es- the capacity of the network and it is reasonable to expect
tablishing a set of connections in a wireless network with further gains by applying inter-session coding.
At the medium access layer of the network, we use a
interference, as two coupled subproblems. Resource provision
(or scheduling) deals with the supply of network resources random slotted-ALOHA-like mechanism. The network is assuch as interference-free sets of links, whereas the concern of sumed to operate in time slots and, for every slot, a node is
resource allocation is how to assign the available resources assigned some transmission attempt probability. We assume
to different users. For resource provision one can consider a that in every time slot nodes decide to transmit with these
scheduled approach which effectively eliminates interference fixed probabilities but otherwise independently of each other.
or, on the other hand, an ALOHA-like random medium ac- Random access is an important medium access technique used
cess mechanism. For the resource allocation problem possible in various applications (see e.g. [10]). One main advantage lies
in the simplicity, since just deciding, whether a vector of link
choices are routing as opposed to network coding.
The authors in [1], [2] focus on the resource allocation flows can be supported by a collision-free transmission schedaspect, assuming that nodes in the network do not experience ule is an NP-complete problem [11]. Furthermore, if nodes
interference. The wireless network is regarded as essentially are assumed to be mobile, frequent changes in the network
interference free, due to an underlying schedule. However, to topology have to be taken into account and a low-complexity
bring network coding one step closer to practical implementa- random access approach is expected to cope better with the
tion, it is necessary to investigate the interplay between these updates due to mobility. Furthermore, our subgraph selection
by no means independent problems. The focus of our work procedure can be combined with a scheduled approach in
is to combine the network coding subgraph optimization with a fairly straightforward fashion. The problem of cross-layer
network coding and random access is also treated in [12],
the resource provision aspect of the overall problem.
[13].
This work was supported by the Control-Based Mobile Ad-Hoc Networking
As it turns out, the problem of finding an optimal network
(CBMANET) Program under DARPA subcontract no. 060786, by the National
coding
subgraph subject to the random access interference
Science Foundation under grant no. CCR-0325673, and by AFOSR.
The authors would like to thank R. Srikant his valuable input.
constraints is too difficult to be tractable in practical networks
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1-4244-1429-6/07/$25.00 (.)2007 IEEE
Abstract- We consider the problem of applying network
coding in wireless networks with random medium access. To
optimize the network coding subgraph and the transmission
attempt probabilities jointly is a tractable problem only for
rather small networks. Therefore, we suggest a suboptimal, yet
practical and decentralized algorithm to combine network coding
with random access. We illustrate the performance gains of our
approach with simulations.
ISIT2007, Nice, France, June 24 - June 29, 2007
'v
in its memory and upon an opportunity transmits a linear
combination of the stored packets with coefficients drawn
uniformly from a finite field. Coding is restricted to packets
belonging to the same user (intra-session).
Fig. 1. An example of a hypergraph. Here, the node set is JA
and the hyperarc set is A = {(1, {2, 3}), (2, 3)}.
{1, 2, 3}
III. THE JOINT NETWORK CODING AND RANDOM ACCESS
OPTIMIZATION PROBLEM
We consider C multicast sessions, where a multicast session
(sc, Tc, Rc), c C 1, .... C is characterized by its source sc, its
of moderate size. This is largely due to the non-convexity destination set
TC C AV and rate Rc. Every node t e TC reof the medium access rate region. Furthermore, in [14], [15] ceives information at the common rate
Rc. We seek a network
the authors have looked at the problem of communicating in
coding subgraph that is able to support these requirements
multiple access networks from an information theoretic per- and that is subject to the multiple access constraints of the
spective and reach similar conclusions regarding the difficulty network. This can be formulated as an optimization problem
of the general problem. We therefore suggest a relaxation of and translates into the following constraint set [4]
the original problem, which is suboptimal, yet practically imc
plementable. Our algorithm can be carried out in a distributed
V (i, J) C A,
fashion, a highly desirable property in practical networks, and
(2)
E ii < zij,
c=l
therefore can serve as the baseline of a protocol.
The remainder of this paper is organized as follows: In
< Yij iJK,
(3)
S
Section II, we introduce the wireless network model and the
jeK
network coding scheme. In Section III, we formulate the joint
V (i,J) c A, Kc J, tCTc, C= 1,... ,C,
network coding and random access problem and discuss its
complexity. Section IV suggests a relaxation of this problem
Z
j (RC
t = SC,
(t, c)
combined with an iterative procedure to verify feasibility and
vJKiJCA
v(t,c)
i = tc,
assign transmission attempt probabilities. In Section V, we
1{jt,iAc
E
J
jEEJ
i
I(i,J)EE,4
(i,
I)EA,iGI
11
0
else,
evaluate the performance gains of our algorithm by means
of simulations. Section VI concludes the paper.
(4)
icAJ\f\t}, t C T,v c= I,....,C,
t'j)
II. NETWORK MODEL
The wireless network (following the model from [4]) is
represented by a hypergraph X = (JV, A), where AV stands
for the set of nodes and A is the set of hyperarcs. A hyperarc,
our approach to model a broadcast channel, is a pair (i, J),
where i is a node and J is a nonempty subset of the node set
AV. An example of a hypergraph is depicted in Fig. 1.
If node i injects a packet on hyperarc J it is received by
some subset K C J, possibly K being the empty set 0. Let
Ai (T) be the counting process describing packet injections on
hyperarc J and AiJK (T) be the counting processes accounting
for the packets received precisely by the subsets K. Obviously,
we have EKCJ AiJK(T) = Aij(T). We assume that for the
injection processes time averages exist, i.e. limT"Q Ai(T)
exists with probability 1, is finite, and equals Zij. Similarly,
= ZiJK- With these assumptions
we define limTO AiJK(T)
T
Zij = EKCJ ZiJK is the average packet injection rate on
hyperarc J. In the remainder of the paper we will assume that
ZiJK is proportional to ziJ and define
PiJK
ZiJK
ZiJ
to be the probability that a packet injected on J is received
precisely by the subset K. We call the rate vector (ZiJ)(i,J)C,
the network coding subgraph.
Once the subgraph is computed our network coding technique of choice will be the well known random network coding
[3], [4], some more practical aspects of which can be found in
[16]. Roughly, that means that a node stores received packet
SCj >, 0,
(ziJ) C Z,
V (i, J) C A, j C J,
V
(i, J) C A,
(5)
(6)
where we define
biJK
PiJLE
{LCJILnK 0}
Here, biJK stands for the probability that a packet injected on
hyperarc (i, J) is received by at least one node in K.
As discussed Zij is the packet injection rate on hyperarc
(i, J), and y(c) is the rate allocated for multicast session c on
(t,c)
this hyperarc. The variable xij
represents the flow between
node i, the source of a hyperarc, to a node j e J in the
destination set of the hyperarc. We will therefore, slightly
abusing terminology, refer to a variable of this type as a
link flow. Constraint (3) defines the feasible rate region of
link flow allocations for a hyperarc (i, J). Constraints (2)-(5)
relate the network coding subgraph (ziJ) and the link flows
(t,c)
xij. The network coding subgraph is restricted within some
constraint set Z which is dictated by the medium access layer
of the network. It is the structure of this set that essentially
determines the difficulty of the optimization problem.1
We consider a random access mechanism, where in every
time slot a node makes a transmission attempt determined by
a Bernoulli random variable. Nodes make the transmission
ths yprac.Th
vribl
'As we will show in the next section the constraints (2)-(5) alone if
combined with a convex cost function define a fairly tractable convex
optimization problem.
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ISIT2007, Nice, France, June 24 - June 29, 200O
decisions independent of each other with the same probability
over different slots. We assume that the time granularity of
the system is sufficiently high and do not use a backoff
mechanism, such as binary exponential backoff. We make no
assumptions on the statistical properties of the traffic in the
network except that average rates exist. This is without loss of
generality, since in [17] it is shown that bursty traffic does not
decrease the capacity region of the multiple access channel.
To formulate the constraint set under a random medium
access policy, let qijj be the probability that node i places
a packet on hyperarc J which is intended to be received by
node j e J. In the following we will make the assumption that
every node has precisely one outgoing hyperarc, which is not
a major restriction and serves mainly to streamline notation.
We will therefore write qij suppressing the hyperarc index.
Let
(7)
Qi = Eqij
jCJ
be the probability that node i transmits in a slot.
If node i transmits a packet on its outgoing hyperarc J then
it will be received by node j C J if and only if
1) it is not lost due to an erasure,
2) node j is not transmitting itself, and
3) none of the nodes with outgoing hyperarcs containing j
transmit.
As a consequence of this interference model for every
hyperarc (i, J) the link throughputs fij, for all (i,j), j
,
must be smaller than the capacity
=
IV. A RELAXATION OF THE JOINT OPTIMIZATION PROBLEM
The joint optimization problem is very difficult and intractable for all but rather small network instances. We therefore suggest a simple relaxation that is easily implementable
not only with a manageable amount of complexity but also
in a distributed fashion. To that end, we drop the random
access constraints and solve the remaining optimization problem which is the selection of the network coding subgraph.
Once we have computed the required link throughputs fij to
support the given demands, we can use the iterative procedure
from [10] to either find transmission attempt probabilities
qij supporting the desired link throughputs or to rule out
their feasibility. In case we encounter an infeasible set of
demands, we reduce the rates (according to some heuristic
rule), recompute the required link throughputs and check again
for feasibility.
Consider to that end the following linear program, where
for brevity we have omitted the qualifying conditions:
qij(I Qj)
1
(1 - Qk),
(8)
kKjK\i
where Ki C AV is the set of all nodes whose hyperarcs contain
node j. The link throughput fij between node i and j is
cij
constraint set which, to the best of our knowledge, cannot be
overcome with convexifying transformations. In an apparently
similar setting, the authors in [18] have successfully applied a
logarithmic transformation to exploit the fact that the RHS of
(10) is log-concave (i.e. its logarithm is concave). However,
they consider joint medium access and end-to-end rate control
only. We on the other hand are also interested in finding a
subgraph to code over. In this case, this and related well known
transformations fail to yield a convex problem formulation.
minimize A ZiJ
(i, J)GA
-
subject to
c
fi;
=
E: max x(tJjc):
C=l
(9)
E
c=l
where the maximum is taken over the link flows required
to serve the different sinks of one multicast session and the
summation implies that the traffic demands due to different
multicast sessions contribute additively to the overall network
flow2. This is due to the fact that we code only within one
session and different multicast connections share the network
without network coding cooperation across them.
The medium access constraint (6) thus becomes
C
S maxxt'c)
c=l
< q.
(I QH)II (1 -Qk)
kgKj\i
This is an intriguing closed form expression for the problem
of jointly selecting a network coding subgraph and assigning
optimal transmission attempt probabilities. Unfortunately, it
is very difficult to solve for general networks except for
very small ones. This is due to the non-convexity of the
2Strictly speaking, (9) as it stands holds in the case where the network does
not experience erasures. However, to take these into account is straightforward
as we merely have to scale the link flows with a proper set of coefficients
depending on the loss probabilities.
(1 1)
x i< < zij,
c
jeK
S S
{JI(i,J)EA4
jEJ
(t,c)
< Yi(j) biJK,
ti'c)
()i
E
Z(
t,
c)
zjtSiC
{iI (j,I)EA,iGl
(t,
(12)
t-j
(RC
10
-°
R
t
=
sc,
i
=
tc,
else,
(13)
(14)
The cost function we have chosen to minimize is the total
number of transmissions necessary to support a given rate,
however any convex separable function of the network coding
subgraph3 can easily be accommodated in our framework.
This optimization problem has a decentralized solution. As
shown in [19], for a single session, the multicast subgraph
selection can be carried out in a distributed fashion. Under our
assumptions on the cost function and the relaxed constraint set
it is readily seen that minimizing
Zij is equivalent
to minimizing jC7 Z.j)
Yi((c Furthermore, the different
Z(iJ)c,
3That means a function of the form f(z) =(i,J) v4 fij(zij). where
the fij (.) are increasing and convex functions.
2728
ISIT2007, Nice, France, June 24 - June 29, 2007
Lt
multicast sessions are not coupled through any constraints,
which implies that a solution to the subgraph optimization
problem can be obtained by solving the different sessions oneby-one and then adding the resulting link flows. Clearly, in this
case the distributed algorithm from [19] applies.
After having computed the network coding subgraph, we
can determine the required link throughputs fij using (9).
To find transmission attempt probabilities qij yielding the
desired link throughputs we can use the iterative procedure
from [10], which consists of the following steps:
0.222^
.2
Path coding/scheduling
Our algorithm
Path coding/random access
Link-by-link retransm./random access
...- .-...-
0.2
0.18
o 0.1
0.16.c .........
co:
0.12
...
...
...
..
...............
0.14
01
qij [O]
1) Initialize:
=
0.
0.08
15
2) Perform the iteration:
Qi [n]
Pij [n]
=
=
1: qij [n],
(I1- Qj [n] )
kg Kj \i
(]-Qk [n]),
(15)
20
25
30
Number of nodes
35
40
Fig. 2. Maximal rate of a unicast connection as a function of the number
of nodes. The number of neighbors is upper bounded by 5.
(16)
0.24
(17)
Pij Inl]
If the iteration converges, then it is guaranteed that the
transmission attempt probabilities qij support the required
link throughputs. If, on the other hand, at some point any
of the Qi[n] exceeds 1 the link throughputs are infeasible.
Then we have to reduce the end-to-end rates, compute a new
subgraph and repeat the feasibility iteration loop. Note that
at every step of this iteration all required variable updates
can be performed with information gathered within a one
hop neighborhood, implying that also this component of our
algorithm is decentralized and, combined with the above
subgraph optimization, yields a fully distributed approach.
Our algorithm is suboptimal since we separated the original
problem into a subgraph selection problem and a multiple
access problem and solved them more or less independently.
However, simulation results illustrate that our approach shows
significant gains. This is the focus of the next section.
Path coding/scheduling
Our.. algorithm
. ..
Path coding/random access
:Link-by-link retransm./random access
-
0.22
02
0.18
o , 16
'r 0.14
EU a 0.12.-
..
..
0.1
0.08
..
15
...
20
..
25
30
Number of nodes
35
40
Fig. 3. Maximal rate of a unicast connection as a function of the number
of nodes. The number of neighbors is upper bounded by 6.
In Fig. 2 and Fig. 3, we are interested in the maximal rate
of a single point-to-point connection. The source is taken to be
the node with the smallest x-coordinate and the destination the
V. SIMULATION RESULTS
one with the highest x-coordinate. The difference between the
To get an idea of the performance gains of our approach two figures is merely the limit on the number of neighbors,
we conduct simulations over random network topologies. We with Fig. 3 having slightly richer connectivity. We simulate
assume that a number of nodes are uniformly scattered over a the following schemes:
square region with unit node density. Two nodes are in radio
1) Path coding and scheduling: The shortest path4 from
range if their distance is below some threshold, the radius
source to sink is chosen and nodes on the path employ
of connectivity, which we take to be 1.8. Transmissions are
a random linear code (e.g. the coding scheme from [1])
subject to erasures which may be due to distance attenuation
to compensate for erased packets. This is a form of
and Rayleigh fading. When a node transmits another node in
network coding. Alternatively, one could employ a FECradio range at distance d will receive the packet correctly if
code on every link and let intermediate nodes decode and
-yd2 > /3, where 'yis a unit mean exponential variable and
re-encode packets. However, such an approach seems
Q = 1/4 is our chosen SNR threshold, otherwise the packet
prohibitive due to large delays especially on longer
is lost completely. We also take into account interference
paths. An optimal collision-free schedule is used which,
by assuming nodes to operate half-duplex (i.e. they cannot
for the special case of a single connection on a path
transmit and receive at the same time) and furthermore if in
rather than a subgraph, is not a difficult task.
a given time slot a node is in radio range of more than one
4See also [20] for an interesting approach using multiple paths.
transmitting neighbor then it cannot recover any of the packets.
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ISIT2007, Nice, France, June 24 - June 29, 200
VI. CONCLUSION
We have suggested a combined network coding and random
access strategy which shows significant performance gains in
simulations. It can be implemented in a distributed fashion
which is a basic requirement if it is to be used as the basis
of a protocol. What is not yet clear in particular is the
delay characteristics of packets in actual network operation.
This is an important aspect in networking and requires more
investigation.
0.13 F
0.121
0.1 12
0.11
Ir
0.1
0
E 0.09
E
cEi
0.08
5 0.07-
0.06
L Path coding/scheduling
0.05
*Our algorithm
Path coding/random
0.04
15
20
25
30
Number of nodes
REFERENCES
access
35
40
Fig. 4. Maximal common rate of four simultaneous unicast connection as a
function of the number of nodes. The number of neighbors is upper bounded
by 6.
2) Subgraph coding and random access heuristic: This
is our algorithm as described in Sections III and IV.
Note, that in contrast to the three other techniques, we
code over a subgraph rather than over a path.
3) Path coding and random access: The shortest path
between source and destination is chosen and the same
coding scheme from the first approach is used on the
path. However, instead of scheduling we use random
access with optimal transmission attempt probabilities.
4) Link-by-link retransmission and random access: The
shortest path between source and destination is chosen
and nodes are required to acknowledge received packets.
If a packet or an acknowledgement is lost a retransmission is triggered, however we do not explicitly use
negative acknowledgements (NACK's).
It is important to keep in mind that approaches (1)-(3)
network coding, while approach (4) does not. Therefore,
although approach (3) operates on a path rather than on a
subgraph it capitalizes already significantly on network coding
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subgraph rather than a path to code over and therefore utilizes
the overhearing of packets by other neighbors, a phenomenon
called the wireless broadcast advantage [19]. We observe, that
our suboptimal network coding algorithm is able to close much
of the gap between random access and a scheduled approach.
The gains are more pronounced in Fig. 3 where the network
use
connectivity
is
richer.
Another experiment we conducted is to use the same network topology and setup simultaneously four unicast connections sharing the network. Fig. 4 shows the maximal common
rate at which they can transmit and, again, the curves illustrate
the gains due to network coding. Together, these experiments
give a compelling reason to expect considerable advantages
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results in [2], [21], [22], it is reasonable to investigate further
the application of network coding in wireless networks.
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