On the geometry of wireless network multicast in 2-D

Abstract

We provide a geometric solution to the problem of optimal relay positioning
to maximize the multicast rate for low-SNR networks. The networks we consider,
consist of a single source, multiple receivers and the only intermediate and
locatable node as the relay. We construct network the hypergraph of the system
nodes from the underlying information theoretic model of low-SNR regime that
operates using superposition coding and FDMA in conjunction (which we call the
"achievable hypergraph model"). We make the following contributions. 1) We show
that the problem of optimal relay positioning maximizing the multicast rate can
be completely decoupled from the flow optimization by noticing and exploiting
geometric properties of multicast flow. 2) All the flow maximizing the
multicast rate is sent over at most two paths, in succession. The relay
position is dependent only on one path (out of the two), irrespective of the
number of receiver nodes in the system. Subsequently, we propose simple and
efficient geometric algorithms to compute the optimal relay position. 3)
Finally, we show that in our model at the optimal relay position, the
difference between the maximized multicast rate and the cut-set bound is
minimum. We solve the problem for all (Ps,Pr) pairs of source and relay
transmit powers and the path loss exponent \alpha greater than 2.

Full text

Page 1

arXiv:1106.0027v1 [cs.IT] 31 May 2011
On the geometry of wireless network multicast in
2-D
Mohit Thakur
Institute for communications engineering,
Technische Universit¨ at M¨ unchen,
80290, M¨ unchen, Germany.
Email: mohit.thakur@tum.de
Nadia Fawaz
Technicolor Research Center,
Palo Alto, CA, USA.
Email: nfawaz@mit.edu
Muriel M´ edard
Research Laboratory for Electronics,
Massachusetts Institute of Technology,
Cambridge, MA, USA.
Email: medard@mit.edu
Abstract—We provide a geometric solution to the problem of
optimal relay positioning to maximize the multicast rate for low-
SNR networks. The network we consider consists of a single
source, multiple receivers and the only intermediate and locatable
node as the relay. We construct network the hypergraph of the
system nodes from the underlying information theoretic model
of low-SNR regime that operates using superposition coding and
FDMA in conjunction (which we call the “achievable hypergraph
model”). We make the following contributions.
1) We show that the problem of optimal relay positioning
maximizing the multicast rate can be completely decoupled
from the flow optimization by noticing and exploiting
geometric properties of multicast flow.
2) All the flow maximizing the multicast rate is sent over at
most two paths, in succession. The relay position depends
on only one path (out of the two), irrespective of the number
of receiver nodes in the system. Subsequently, we propose
simple and efficient geometric algorithms to compute the
optimal relay position.
3) Finally, we show that in our model at the optimal relay
position, the difference between the maximized multicast
rate and the cut-set bound is minimum.
We solve the problem for all (Ps,Pr) pairs of source and relay
transmit powers and the path loss exponent α ≥ 2.
Index Terms—Low-SNR, broadcast relay channel, geometry.
I. INTRODUCTION
We primarily consider the problem of optimal relay posi-
tioning in order to maximize the multicast rate in low-SNR
networks consisting of a single source s, a set of multiple
receivers T and an arbitrarily locatable relay r, on a 2-
D Euclidean plane. In [1], the authors previously addressed
this problem under a heavy and complex network flow opti-
mization framework. They showed that optimizing the relay
position can lead to a strong gain in the multicast rate.
In [2] the authors introduced equivalent hypergraph models
for the low-SNR Broadcast (BC) and Multiple Access channels
(MAC). The authors then derived an achievable hypergraph
model for the broadcast relay channel (BRC), obtained by
concatenating the equivalent BC and MAC hypergraphs. This
concatenated model follows from constraining the source and
relay to transmit using the optimal schemes for the low-SNR
BC and MAC: superposition coding and frequency division,
respectively. In this paper, building on this model, we solve
geometrically the problem of optimal relay positioning under
the pretext of multicast rate maximization, which is much
simpler and efficient than the solution proposed in [1].
Most importantly, we establish the fact that for a given
low-SNR BRC hypergraph G(N,A), the multicast rate is
maximized by sending all the flow through at most two paths
in succession, independently of the number of destination
nodes. This is a consequence of simply maximizing the
multicast min-cut. The dependency of the multicast min-cut on
the relay position is essentially through a single path (out of
the two), and this motivates a simple geometric interpretation
and formulation of the problem. It should be noted that, the
“optimal relay position” refers to the position that maximizes
the multicast rate over a given achievable hypergraph, but in
general the achievable hypergraph model is not necessarily
optimal in terms of meeting the cut-set bound for low-SNR
networks. On the other hand, the achievable hypergraph model
performs closely to the peaky binning scheme in the case
of a single destination [3], and enjoys an important practi-
cal advantage of being easily scalable to more complicated
topologies. Finally, under our model the difference between the
maximum multicast rate and the cut-set bound is minimized
at the optimal relay position.
In the proposed geometric approach, we decouple the prob-
lem of rate maximization from the problem of computing the
optimal relay position. This substantially reduces the complex-
ity (compared to the flow optimization based framework in
[1]) and also provides a great deal of insight in understanding
the nature of such network planning problems. Finally, we
show that at the optimal position the difference between the
maximum multicast rate and the cut-set bound is minimized
under the achievable hypergraph model.
The paper is organized as follows. We introduce the low-
SNR achievable hypergraph model of the BRC in section II.
Then we prove certain geometric properties of multicast in
section III. The computation of optimal relay position is
divided in two parts, section IV for Ps= Pr and section V
for Ps?= Pr. Finally, we conclude in section VI.
II. LOW-SNR SYSTEM AND HYPERGRAPH MODEL
A. System model and notations
The network topology is given by a hypergraph G(N,A),
where N = {s,r,T}, and all nodes except r are fixed on the 2-

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s
t1
t2
P
h1
h2
N0
N0
(a) BC
s
t1
t2
Rc
R1
R2
(b)
equivalent hypergraph
WidebandBC
h1
h2
s1
s2
t
P1
P2
N0
(c) MAC
s1
s2
t
h2
1
P1
N0
h2
2
P2
N0
(d) Wideband
equivalent hypergraph
MAC
s
r
t1
t2
Ps
Pr
N0
N0
h1s
h2s
hrs
h1r h2r
(e) BRC
s
r
t1
t2
r0
r1
r2
r3
r4
(f)
achievable hypergraph
WidebandBRC
Fig. 1.
The BRC rates: r0=
Wideband Multiple User Channels. The BC rates: R1= (1 − β)h2
β0Ps
D2
D2
1
P
N01]h2
2,+∞[(h2
µ2Pr
D2
1), R2= (1 − β)P
rt2N0. Here, h gives the path loss and Dij the distance from i to j.
N01[0,h2
2[(h2
1), Rc = β min{h2
1,h2
2}P
N0.
srN0, r1=
β1Ps
st1N0, r2=
β2Ps
D2
st2N0,r3=
µ1Pr
D2
rt1N0,r4=
D Euclidean plane. T = {t1,..,tn} denotes the set of n = |T|
receivers ordered in increasing distance from s. C represents
the convex hull of {s,T}. The multicast rate from s to T is
defined as RsT? min
s to receiver t ∈ T. Ps and Pr= γPsare the total transmit
powers of s and r, respectively, and γ > 0 is their ratio. Duv
denotes the Euclidean distance between nodes u and v, and
α ≥ 2 the path loss exponent. For a subset Q ⊆ N\r, define
LQ(C) as the point in C, that minimizes the maximum over
the distances between itself and each node in Q, i.e.
?
The value of objective function of the output of Program (A)
is denoted as DQ.
t∈T(Rst), where Rstis the total rate from
LQ(C) ? argmin
r∈C
max
j∈{Q}(Drj)
?
.
(A)
B. Low-SNR BC, MAC and BRC hypergraph models
In [1], [2], it was shown that concatenating the low-SNR
BC (superposition coding) and MAC (FDMA) equivalent
hypergraph models results in an achievable hypergraph model
for the low-SNR BRC. The rate region of this model is
included in the capacity region of the low-SNR broadcast
relay channel. In fact, even though superposition coding and
FDMA are independently capacity achieving for the low-SNR
AWGN BC and MAC channels respectively, their combination
in general is not capacity achieving for the low-SNR relay
channel, and a fortiori for the low-SNR BRC [3].
In this section, we briefly recall the equivalent hypergraph
models for the low-SNR BC and MAC, and the achievable
hypergraph model for the BRC [1]. Note that in the low-SNR
regime, BC and MAC are not limited by interference.
1) Low-SNR BC equivalent hypergraph: Superposition
coding is known to achieve the capacity region of the AWGN
BC. In the low-SNR regime, the rates achieved by superpo-
sition coding boil down to the time-sharing region [4]–[6].
For a given topology with |T| = n receivers, the hypergraph
will contain at most n hyperarcs with non-zero capacities [1].
Figures 1(a) and 1(b) illustrate the two-destination case.
2) Low-SNR MAC equivalent hypergraph: In the low-SNR
regime, interference becomes negligible with respect to the
noise [1], [2], and all sources can achieve their point-to-
point capacity to the common destination, like with frequency
division multiple access (FDMA). In the general wideband
MAC with n sources, the hypergraph model consists of n
hyperarcs of size 1 from each source si, i ∈ {1,..,n} to
the destination with non-zero capacity. Figures 1(c) and 1(d)
illustrate the two-source case.
3) Low-SNR BRC achievable hypergraph: We can obtain
an achievable hypergraph model of the low-SNR BRC by sim-
ply concatenating the BC and MAC equivalent hypergraphs,
as shown in Figures 1(e) and 1(f) for the two-destination case.
As mentioned before, this achievable hypergraph model is
suboptimal in general for the BRC, but the ability to scale
easily to larger and complex networks is one of its biggest
strength.
III. GEOMETRIC PROPERTIES OF MULTICAST
In this section, we derive the geometric properties of the
optimal relay position maximizing the multicast rate for the
BRC. We first focus on the single destination case of the BRC:
the relay channel, in Section III-A. Then, these preliminary ob-
servations and properties are extended for the general problem
with an arbitrary number of destinations, in Section III-B.
A. Single destination: low-SNR relay channel
Consider the simple network in Figure 2 (a), with a fixed
source s, a fixed receiver t and an arbitrarily positionable relay
r, where the multicast rate Rstfrom s to t is to be maximized.
Naturally, Rst depends on the position of r. The achievable
hypergraph in Figure 2 (a) can be broken into two subgraphs,
shown in Figures 2 (b) and (c), which are essentially the two
disjoint paths from s to t.
Our claim is that the optimal position of the relay maximiz-
ing the multicast rate from s to t lies on the line segment s−t
joining s and t, and at this optimal position all the flow Rstis
sent through a single path consisting of two hyperarcs, namely
{(s,r),(r,t)} shown in Figure 2 (c). This holds true for any
given pair of power constraints (Ps,Pr) ≻ 0 and for any path
loss exponent α ≥ 2. We prove this claim in Lemmas 1 and
2 hereafter.
We first recall the following lemma from [1].
Lemma 1 (Lemma 1 [1]): The optimal position of r maxi-
mizing RsT lies inside the convex hull C.
Here, Lemma 1 simply implies that the optimal position of
r lies on the segment s − t.
The rates over the three hyperarcs {(s,r),(r,t),(s,rt)} =
A are given by,
Rsr=
Dα
Dα
Psr+ Psrt≤ Ps, Prt≤ Pr,
Psr
srN0, Rrt=
Prt
rtN0, Rsrt=
Psrt
Dα
stN0,
(1)
(2)

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s
s
s
s
s
s
r
r
r
r
r
(b)
r
(a)
(c)
(d)(e)
(f)
t
t
t
t
t
t
Fig. 2.
(b) and (c), respectively. (d): Optimal position of r for Ps= Pr and α = 2,
which is at the perpendicular bisector (red) of line segment s − t. (e): Left
bias for Ps< Pr. (f): Right bias for Ps> Pr.
(a): One receiver case decomposed into two subgraphs from s to t,
where N0 is the noise power spectral density. Note that the
multicast rate is given by Rst= Rsrt+ min(Rsr,Rrt).
Lemma 2: The optimal location of r on the segment s − t
for a simple BRC with γ ∈ (0,∞) and α ≥ 2 that maximizes
the multicast rate Rstsatisfies,
D∗
sr=
Dst
1 +
α√γ, D∗
rt=
α√γDst
1 +
α√γ,
(3)
and the optimal (maximized) multicast rate is given by,
R∗
st=
Ps
sr)αN0
(D∗
=
γPs
rt)αN0
(D∗
(4)
where all the flow R∗
In Lemma 2 the starred entities refer the optimal values and
for the proof the reader is referred to Appendix A.
Lemma 2 essentially gives the position of r in terms of
how far it is from s and r on the segment s − t. Also, it
provides the maximized multicast rate R∗
this position. It can be easily seen that the relay position only
affects the rate over the path {(s,r),(r,t)}. Since the min-cut
of the path {(s,r),(r,t)} is strictly larger than the min-cut
of the path {(s,rt)}, i.e. the rate that can be sent for a unit
power over the former path is strictly larger than the latter path
(Rsrt< min(Rsr,Rrt)), the rate over the path {(s,r),(r,t)}
should be maximized first by simply maximizing its min-
cut min(Rsr,Rrt) before allocating any power to the path
{(s,rt)}. The min-cut min(Rsr,Rrt) is maximized at the
position on the segment s − t such that rates over the two
hyperarcs of the path {(s,r),(r,t)} become equal, and all
the flow from s to t is transmitted over this path only. The
maximized multicast flow R∗
rates of either of the two hyperarcs.
Several important conclusions can be drawn from Lemma 2.
The multicast flow optimization can be separated from the
determination of the optimal relay position that maximizes the
multicast flow. Even if the aim is not to maximize the multicast
flow (for instance by simply choosing not to use all the source
and relay powers), Lemma 2 still gives the most suitable relay
position for any feasible multicast rate Rst≤ R∗
time, the algorithmic style intuitive proof arguments in the
previous paragraph indicate that upon computing the optimal
relay position, the multicast rate maximization problem could
be casted as a straightforward linear program resulting in a
stis sent over the path {(s,r),(r,t)}.
stthat is achieved at
stis then simply given by the
st. At the same
simple power allocation scheme maximizing the multicast rate.
This fact will prove handy for the general case with arbitrary
number of destinations. On the other hand, we observe the
dependency of the optimal relay position on the constants α
and γ. If γ = 1 i.e. Ps = Pr, the optimal relay position is
always at the mid-point of the segment s−t for any value of
α ≥ 2. When γ ?= 1, there will be a natural bias on the optimal
position of r either towards s or t, depending on the value of
γ. This bias will also depend on the value of α. Figure 2(e)
and 2(f) show the bias effect.
B. Multiple destinations
In this subsection, we extend the simple geometric insights
developed in Section III-A for a single destination to the
general case of an arbitrary number of destinations |T| = n.
Let us first note the following. For a given hypergraph
G(N,A), and a fixed position of r, we have at most (n+1)+
(n) hyperarcs in the system, i.e. |A| = 2n + 1. The former
(n + 1) are source hyperarcs, emanating from s to the nodes
in N\s and the latter n are the relay hyperarcs, emanating
from r to all T. Also, for any given position of r there always
exist at least two paths that will span all the receiver set T,
namely {(s,T)} (or {s,t1..tn}) and {(s,T1),(r,T2)} (where
r ∈ T1and T1∪ T2= {r,T}).
Now, consider that each hyperarc (i,J) ∈ A is associated
with a continuous function fiJ(P+
is a monotonically increasing in the transmit node’s power Pi
and monotonically decreasing in the distance DiJ, where DiJ
is the Euclidean distance between the transmit node i and the
farthest receiver node j ∈ J (from i) spanned by the hyperarc.
Then the following theorem holds true.
Theorem 1: Given a hypergraph G(N,A) and the associ-
ated rate functions fiJ(P+
perarc in A, at the optimal position maximizing the multicast
rate RsT one of the two multicast flow characteristics holds:
(i) all the optimal flow R∗
{(s,T1,(r,T2)} and {(s,T)}, in succession.
(ii) all the optimal flow R∗
the two paths {(s,T)} and {(s,T1),(r,T2)}.
For the proof of Theorem 1, refer to Appendix B.
Theorem 1 partially generalizes Lemma 2. We say partially,
because on one hand, Theorem 1 establishes the important
multicast flow characteristics at the optimal relay position,
but it does not provide a simple numerical result that de-
termines the optimal relay location (like Lemma 2). Note
that, for a given relay position there could be multiple paths
from s, through r, to all T, but in the Theorem 1 by path
{(s,T1),(r,T2)} we mean the path from s, through r, to all
T that has the highest min-cut among all the paths from s,
throughr, to all T. Intuitively, Theorem 1 states that only those
paths will contain the multicast flow from s to the receiver set
T that serve all T, namely {(s,T)} and {(s,T1),(r,T2)}. All
other path that serve proper subsets of T will carry no flow as
they do not contribute to the multicast flow and among all the
paths serving all T through r, only the path with the highest
i,D−
iJ) : R2−→ R, that
i,D−
iJ) : R2−→ R for each hy-
sTgoes through at most two paths
sTcan be arbitrarily split between

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min-cut will carry the multicast flow. This fact is a simple yet
fundamental consequence of the definition of multicast.
Theorem 1 reveals a lot about the nature of multicast flow
over a hypergraph. The dependence of relay position on the
rate of only a single path {(s,T1),(r,T2)} reduces the problem
to its core by removing the clutter away. In other words, now
we only need to worry about the maximization of the flow
over this single path and the relay position that maximizes
the flow over this path also maximizes the multicast flow
RsT. This result of Theorem 1 motivates a pure geometric
interpretation of the problem. If we imagine the two hyperarcs
(s,T1) and (r,T2) to be two circles Csand Crcentered at s
and r with radii πsand πr, respectively, then the optimal relay
positioning problem could be stated as: For a given G(N,A),
find the point in C such that when r is positioned at this point,
max(α√γπs,πr) is minimized while r ∈ Csand the region of
union of two circles C∪= Cs∪ Crencompasses all T.
At first, it seems plausible to try a simple (preferably
convex) optimization framework to compute such a point,
but the condition that the two circles must encompass all N
brings in discreteness, which we avoid for obvious reasons. In
contrast, we propose a simple (polynomial time) algorithm to
compute such point in the next sections. Once the optimal relay
position is obtained, obtaining optimal power allocations (for
s and r) maximizing the multicast rate boils down to solving
a simple linear program involving only two paths. We divide
the development of this algorithm into two cases of γ = 1 and
γ ∈ (0,∞). The case of γ = 1 is easy to understand and holds
importance in its own right. In addition it develops the basic
intuition for the proposed algorithm and leaves the extension
to the case of all values of γ ∈ (0,∞), as straightforward.
IV. (Ps= Pr) - CASE AND ALGORITHM
In this section, we have γ = 1 and α ≥ 2 for a
given G(N,A) on the 2-D Euclidean plane. The optimal
relay positioning problem stated geometrically in the previous
section simply boils down to finding the point in C such that
max(πs,πr) is minimized while r ∈ Csand C∪encompasses
all T. We divide the problem in the following two cases based
on the topology of the given G(N,A).
A. s − tnmid-point case
Lemma 3: If r is placed at the mid-point of s − tn such
that the hyperarcs Cs and Cr each with radii
T, then it is the optimal relay position maximizing RsT.
The proof of Lemma 3 is a straightforward generalization
of Lemma 2 and therefore is omitted. Intuitively, Lemma 3
simply states that since the farthest node (from s) tn is also
the limiting node for maximizing RsT, if the rate is maximized
only to tnwhile guaranteeing it to all other nodes in T, then
this maximizes RsT as well. This means that if r is placed at
the mid-point of the segment s−tn(as this position maximizes
the rate to tn only) and if the two hyperarcs of the path
{(s,r),(r,tn)} ({Cs,Cr}) span all T, then clearly this is the
relay position that maximizes RsT.
Dstn
2
span all
B. General Case
In this case we tackle all topologies and case A becomes a
special case of it. Recall that, the entity LQ(C) represents the
coordinates of the point which is the argument of the objective
function of the output of program (A), and DQ is the value
of the objective function of the output of program (A).
Optimal relay positioning Algorithm (ORP)
Given: G(N,A).
1) Compute l0= L{N\r}(C) and build the set N0= {t ∈
T|Dst< Dl0t&Dl0t> Dsl0} = {t′
order of distance from s. If N0= {∅}, declare l0as the
optimal relay position and quit, else go to step 2.
2) Build the set N1 = {N\(r,N0)} and compute the
point l1 = LN1(C). Form the hyperarcs Cs and Cl1
of radii Dsl1and DN1, respectively. If C∪= Cs∪ Cl1
encompasses all T, output l1as the optimal relay position
and quit, else go to step 3.
3) Reform the hyperarc Csof radius Dst′
N2= {t ∈ T|Dst> Dst′
Declare l2as the optimal relay position and quit.
Algorithm ORP is a straightforward set of basic and intuitive
computational steps based on the properties of the point
l0 = LN\r(C). If there exist no node t′∈ T such that
t′/ ∈ Csand Dst′ < Dl0t′ (i.e. set N0is empty), that can be
directly reached by s rather than by a path through r, then l0is
certainly the optimal relay position. In contrast, if the set N0
is not empty, then there exist at least one receiver node in the
system that influences the computation of the optimal relay
position but can be served directly by Cs. Therefore, either
the nodes in N0 can be removed from the computation of
the optimal relay position (l1in Step 2) and max(πs,πr) can
be further reduced or we could reform the hyperarc Cswith
radius Dst′
then computing the point l2for the nodes that were not covered
by Csand thus reducing the value of max(πs,πr). Note that,
Algorithm ORP categorizes all possible topologies of the given
G(N,A) in three steps and there is no underlying iterative
process. This makes algorithm ORP behave like a numerical
formula, which we originally wanted from Theorem 1.
We leave the formal proof that ORP always outputs the
optimal relay position maximizing RsT to Appendix C and
extend this simple approach in a straightforward manner to
the case of all values of γ ∈ (0,∞) in the next section.
1,..,t′
m} in increasing
mand build the set
m} and compute l2= LN2(C).
m(where, t′
mis the farthest node in N0from s) and
V. Ps?= Pr- CASE AND ALGORITHM
In this section, we consider γ ∈ (0,∞) for a given
G(N,A) and α ≥ 2. Almost all the theory developed in
Section IV simply transcends to this section, with certain
notable differences. Mainly, that when γ ?= 1 it gives rise
to a bias in the positioning of r ( ref. Figure 2(e) and 2(f)).
Taking into account the bias while computing the optimal
relay position will be the main enhancement in this section.
Likewise previously, we first consider the s − tncase.

Page 5

A. s − tncase
Lemma 4: Given G(N,A), if r is placed on s − tn at a
distance of Dsr=
Cs∪ Cr spans all T, then it is optimal relay position that
maximizes RsT.
The line of argument for the proof of Lemma 3 (using
Lemma 2) could be simply generalized for Lemma 4.
Dstn
1+α√γfrom s, such that r ∈ Csand C∪=
B. General Case
In this case, like in Section IV, we generalize to all
topologies. As we know, that the values of γ (when not equal
to 1) and α inflict the bias on the relay position. The main
difference in case of Ps?= Pris the computation of the point
li= LQ(C) (i = {0,1}), given by,
?
and the computation of the set N0 = {t ∈ T|α√γDsl0>
Dl0t} = {t′
and the set N0 takes into account the bias induced by the
differences in the transmit power of the source and relay and
the value of α. The rest of the algorithm remains the same.
Now that we have an efficient algorithm for computing the
optimal relay position, we can be more ambitious to assess
the standing of our work in a more theoretical sense. One
of the important consequences of this work that signifies its
theoretical importance is shown in Figure 3. We computed the
difference between the optimal multicast rate R∗
position of r) and the cut set bound for |T| = 9 receiver nodes
network at 21 interesting positions, including the optimal relay
position computed by the Algorithm ORP. At the optimal relay
position (blue point), this difference is minimized, confirming
the fact that the optimal relay position not only results in gains
but the maximized multicast rate is theoretically closest to the
cut-set bound at the optimal relay position in our framework.
It is worth mentioning that the theory developed in this
paper well transcends to the low-SNR fading channels , which
we do not discuss here but can be easily generalized from the
results of [2] and [3].
li= LQ(C) ? argmin
i∈C
max
(j∈Q\s)(α√γDsi,Dij)
?
.
(B)
1,..,t′
m}, in the Algorithm ORP. Program (B)
sT(for a given
VI. CONCLUSION
We list the important deductions from our work in the
following points.
1) The problem of optimal relay positioning to maximize
the multicast rate for the achievable hypergraph model
of low-SNR networks using superposition coding and
FDMA, can be decoupled from flow optimization and
casted as a simpler geometric problem, as opposed to a
complex network optimization approach of [1].
2) The geometric properties of multicast are innately simple
and provide interesting insights for relay positioning
problem. This is largely due to the fact that all the
multicast flow is pushed over at most two paths which
is a direct consequence of the definition of the multicast
flow, and this results in simple geometric interpretation.
0
2
4
6
8
10
0
2
4
6
8
10
0
1
2
3
4
5
6
7
8
X Coordinate (meters)
Y Coordinate (meters)
R*
sT − Cut−set bound difference (nats/sec)
Fig. 3. |T| = 9 case with green receivers, red source and blue as the optimal
relay position. The optimal RsT and cut set bound difference (in nats/sec)
is calculated for 21 positions and is the lowest at the optimal relay position
(blue). We assume
N0=
Ps
Pr
N0= 1 (normalized) and α = 4.
3) Importantly, the benefits of determining the optimal relay
position are substantiated by the fact that the difference
between the maximized multicast rate and the cut-set
bound at the optimal position is minimized.
We now outline, what we think are certain important future
directions our work could take. The geometric properties of
multicast give great insights and are surprisingly easy to work
with. This motivates us to ask further, whether is it possible
to apply the simple techniques of our work for the optimal
relay positioning problem to moderate and high-SNR regimes
that are interference limited. Another natural and interesting
dimension is to look at the possibility of extending this work
to multicommodity flows.
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