Efficient Opportunistic Multicast via Tree Backbone for Wireless Mesh Networks

TL;DR: This paper proposes a new opportunistic multicast protocol to improve multicast throughput in Wireless Mesh Networks (WMN) by devising a Euclidean tree backbone as well as an efficient rate selection scheme to minimize the number of transmissions.

Abstract: In this paper, we propose a new opportunistic multicast protocol to improve multicast throughput in Wireless Mesh Networks (WMN). It builds upon opportunistic routing (OR) strategies that have been designed to improve unicast throughput in wireless networks. The key concept in our multicast protocol is a tree backbone. Our tree backbone protocol represents a tradeoff between traditional structured multicast protocols where a complete multicast tree is constructed and unstructured protocols where multicast is treated as a collection of unicasts. Tree backbone selects multiple nodes as intermediate nodes. Each pair of upstream and downstream nodes may be multiple hops away, and packet delivery between them takes advantage of OR. For single-rate WMNs, we show that constructing an efficient tree backbone that minimizes the number of transmissions is NP-hard, and we devise one effective heuristic algorithm for it. For multi-rate WMNs, we investigate the inherent rate-distance tradeoff and propose a Euclidean opportunistic multicast protocol by devising a Euclidean tree backbone as well as an efficient rate selection scheme to minimize the number of transmissions. In our simulations, our tree backbone multicast protocols outperform both the completely structured traditional multicast protocols and the completely unstructured unicast-based protocols augmented with OR in both throughput and delay.

Summary

Introduction

• For single-rate WMNs, the authors show that constructing an efficient tree backbone that minimizes the number of transmissions is NP-hard, and they devise one effective heuristic algorithm for it.
• The increase of the number of transmissions not only consumes more bandwidth resource, but also leads to more local interference among nearby transmissions, which may result in lower throughput.
• The authors have to reduce the number of transmissions when they exploit opportunistic routing for multicast, which helps to improve the system throughput.
• The “adjacent” backbone nodes may be multi-hop away in the network.
• Section III proposes the opportunistic multicast protocol in single-rate WMNs.

A. System Model

• WMNs consist of three types of nodes: gateways, mesh routers, and mesh clients.
• Mesh clients are the end users of a mesh network.
• The authors only study how to multicast packets to a group of mesh routers; then packets will be forwarded one more hop to the corresponding mesh clients that desire to receive the packets.
• Individual WMN links may have different qualities for a variety of reasons, thus the authors use the weight on each edge to denote the packet delivery ratio.
• The authors assume that each node has the same fixed communication range under the same rate.

B. Basic of Opportunistic Routing

• This set Fi j is carefully selected to minimize the number of forwarding nodes while maximizing the throughput improvement.
• Its forwarding candidates continue the forwarding based on their relay priority.
• That is, higher priority forwarding nodes are given the first chance to forward packets.
• When receiving packets, each forwarding node also determines if the newly received packet it receives should be forwarded, either by explicit coordination or by exploiting network coding properties.
• The above process repeats until the destination informs the source that enough packets have been received.

C. Motivation

• Traditional multicast protocols discover the least cost or highest throughput paths to the destinations.
• Building a shared tree is a common way in traditional multicast protocols, where transmissions to different destinations may share some hops in the tree to minimize the bandwidth cost.
• A natural multicast extension from OR is briefly introduced in [9], which requires the packet self route to all the destinations by OR.
• Fig. 1(a) shows a case of natural extension, which requires totally 10 hops.
• If the authors allow the transmissions to distinct destinations share some hops, it can decrease the total number of transmissions.

III. OPPORTUNISTIC MULTICAST IN SINGLE-RATE

• The authors first introduce the definition of tree backbone (TB), and explain the basic idea of their Opportunistic Multicast (OM) protocol.
• In order to save bandwidth and decrease interference, the tree backbone should minimize the number of transmissions along it.
• Afterwards, the authors present one heuristic algorithm to construct an efficient TB.

A. Basic Idea

• Opportunistic multicast builds the tree backbone instead of a multicast tree, which both allows the spatial opportunities given by OR and minimizes the number of transmissions by letting packets transmitted to different destinations share some hops.
• After the packet arrives at one tree backbone node, say t, it continues routing to t’s downstream backbone node, until it reaches the destination.
• Note that, since routing from the upstream node to the downstream node uses OR, different packets may travel along different paths.
• A good metric to evaluate the effectiveness of a TB is the expected number of transmissions for one packet to reach all the destinations through the TB.
• The authors prove that computing an MTB is NP-hard by proving its corresponding decision problem, TB-D defined below, is NP-hard.

B. NP-Hardness Proof

• Furthermore, the authors compute the total expected number of transmissions.
• Zuv, which denotes the total expected number of transmissions occurring in the network for one packet transmitted from u to v by OR.
• The decision problem TB-D is NP-hard, so the optimization problem of computing an MTB is also NP-hard.

IV. EUCLIDEAN OPPORTUNISTIC MULTICAST IN MULTI-RATE WMNS

• On one side, a higher data rate can be used to increase throughput, but it also has shorter transmission range and hence more hops to reach the destination.
• Besides, there are few spatial opportunities due to the low neighbor diversity in one hop.
• On the other side, lower data rate often has a longer transmission range and hence less hops in the selected path.
• The higher neighbor diversity brings more spatial opportunities, but the low rate disadvantage may counteract the above benefit.
• The inherent tradeoff between rate and distance is hereby worthy of a careful study.

A. Design Consideration

• Fi,d is the forwarding candidate set of ni, and the node order in Fi,d is based on the relay priority.
• When a node selects a rate, it should consider the expected bit advancement per second toward each destination, not just a specific one.
• This naive metric is not suitable for opportunistic multicast.
• On the other hand, the authors also need to propose a new tree backbone construction for multi-rate WMNs.

B. Euclidean Tree Backbone and Rate Selection

• The authors assume that a longer geometric distance requires more transmissions to reach the destination.
• This assumption is straightforward, but it does not apply to any case, since the number of transmissions depends on both the network topology and the link quality.
• At each iteration, the geometrically nearest destination is added to the partially constructed tree with a corresponding edge.
• Different packets received may aim to different next downstream backbone nodes, thus the forwarder needs to consider all those downstream backbone nodes, which are defined as the targets of the forwarder.
• For each rate R j, the authors calculate the largest I-MEAR based on Eq. 7, and then they select the best rate that yields the largest I-MEAR.

V. SIMULATIONS

• The authors evaluate their opportunistic multicast algorithms by comparing them with the natural multicast extension of OR [9] and a traditional multicast algorithm through the following metrics.
• The average number of packets each destination receives during a time unit, also known as Throughput.
• We can see that building an efficient tree backbone can improve the throughput, since it not only takes advantage of spatial diversity, but also minimizes the number of transmissions.the authors.the authors.
• This is because it does not take any spatial opportunities.
• The authors do not simulate the tree-based algorithm any more in the following subsections.

B. Delay Comparison

• The authors evaluate the delay of ST-B and the natural extension by comparing the average time each packet takes to reach the destination.
• The authors efficient tree backbone is able to minimize the number of transmissions as well as interference, which greatly speeds up the packet delivery.
• In addition, when the number of receivers increases, the multicast structure becomes bigger, thus the delay of all algorithms increases due to the increasing number of transmissions.
• It indicates that 18 Mbps has a short transmission range that greatly deceases the spatial opportunities.

A. Multicast Routing in Wireless Multi-hop Networks

• Multicast protocols in multi-hop wireless networks can be categorized into three groups: (i) tree-based, (ii) mesh-based, and (iii) stateless protocols.
• Tree-based protocols build an efficient multicast tree, and the packets are forwarded from the source to the destinations along tree paths [17]–[22].
• At the same time, mesh-based protocols build multiple trees among the group members [23]–[25], such that the packets are delivered to each destination along multiple paths.
• Compared with the first two groups of protocols, the authors do not build a complete tree or mesh.
• Compared with stateless protocols, the backbone the authors propose is able to minimize transmissions and decrease the overhead resulting from the destination list in the packet header.

B. Opportunistic Routing

• In recent years, opportunistic routing has become an interesting topic that improves the throughput and the transmission reliability in the face of unreliable wireless links [7].
• Some variants of opportunistic routing are proposed to improve throughput under different situations.
• The authors in [30] extend OR to enables devices with only one interface to operate on multiple channels, which reduces interference.
• The forwarding candidate set and relay priority are defined based on the nodes’ geographic information [31]–[33].
• In the ad hoc context, there are relevant works to consider, which takes advantage of diversity offered by multiple users [34].

C. Multi-rate Routing

• One of current wireless technical trends is that modern wireless devices are able to utilize multiple transmission rates to accommodate a wide range of conditions.
• In [10] [12], the authors study the impact of routing metrics on path capacity, investigate the impacts of several factors on the carrier sensing threshold in the multi-rate wireless networks, and propose the bandwidth distance product as a routing metric to improve throughput.
• In order to utilize multiple channels in multi-rate networks, the Data Rate Adaptive Channel Assignment algorithm is proposed in [37], which assigns links having the same data rates on the same channel to minimize the wastage of channel resources.
• None of the above work considers the multicast application in multi-rate networks.
• The authors work is different from previous approaches in two aspects: (i) the authors build a backbone structure for multicast routing, and (ii) they assign rates to nodes based on the location of the nearby backbone nodes.

VII. CONCLUSION

• The opportunistic multicast protocol in single-rate WMNs is first proposed, where a tree backbone is built to help the packets self route to destinations by OR.
• An efficient tree backbone should minimize transmissions to increase the throughput.
• The authors prove that computing a tree backbone with the minimum transmissions is NP-hard, and devise one heuristic algorithm for it.
• The authors also investigate the inherent ratedistance tradeoff in opportunistic multicast and propose the Euclidean opportunistic multicast protocol in multirate WMNs.
• Simulations show that their opportunistic multicast can achieve higher throughput and shorter delay than the natural multicast extension of OR and the traditional multicast.

Figures

Figure 3. EOM Example

Figure 4. Impact of Network Size

Figure 1. Motivation Example

Figure 5. Delay Comparison

Figure 2. ST-B Example

Figure 6. Impact of Multiple Rates

Full text

Efficient Opportunistic Multicast via Tree Backbone for Wireless Mesh
Networks
Guokai Zeng, Pei Huang, Matt Mutka, Li Xiao, Eric Torng
Department of Computer Science and Engineering
Michigan State University
East Lansing, Michigan, USA
Email: {zengguok, huangpe3, mutka, lxiao, torng}@cse.msu.edu
Abstract—In this paper, we propose a new opportunistic
multicast protocol to improve multicast throughput in
Wireless Mesh Networks (WMN). It builds upon oppor-
tunistic routing (OR) strategies that have been designed to
improve unicast throughput in wireless networks. The key
concept in our multicast protocol is a tree backbone. Our
tree backbone protocol represents a tradeoff between tra-
ditional structured multicast protocols where a complete
multicast tree is constructed and unstructured protocols
where multicast is treated as a collection of unicasts. Tree
backbone selects multiple nodes as intermediate nodes.
Each pair of upstream and downstream nodes may be
multiple hops away, and packet delivery between them
takes advantage of OR.
For single-rate WMNs, we s how that constructing an
efficient tree backbone that minimizes the number of
transmissions is NP-hard, and we devise one effective
heuristic algorithm for it. For multi-rate WMNs, we
investigate the inherent rate-distance tradeoff and propose
a Euclidean opportunistic multicast protocol by devising
a Euclidean tree backbone as well as an efficient rate
selection scheme to minimize the number of transmis-
sions. In our simulations, our tree backbone multicast
protocols outperform both the completely structured tra-
ditional multicast protocols and the completely unstruc-
tured unicast-based protocols augmented with OR in both
throughput and delay.
Keywords-Wireless Mesh Networks, Multicast, Oppor-
tunistic Routing, Multi-Rate
I. INTRODUCTION
Wireless mesh networks (WMN) have emerged as
an efficient means to expand the wireless reach of
metro-broadband deployments within a community or
within a company [1], [2]. For traditional wireless multi-
hop networks, such as MANETs and sensor networks,
route discovery and energy efficiency are major research
issues. However, these are relatively unimportant in
WMNs due to their static topology and rechargeable
mesh nodes. On the other hand, how to satisfy the
end users is paramount in WMNs, which usually aims
to maximize the system throughput under given band-
width [3]–[5].
Multicast communication is a critical component of
WMNs [6]. Many current or future services in WMNs
are bandwidth-sensitive and strongly based on many-
to-many interaction, such as Video on Demand. They
require efficient underlying multicast mechanisms when
throughput and delay are the critical concerns and
bandwidth may become a scarce resource.
Most traditional multicast protocols for wireless
multihop networks discover the least cost or highest
throughput paths to reach the destinations. Compared
with their wired counterparts, they build efficient mul-
ticast structures based on the wireless communication
facts: (i) the local broadcast enables multiple neighbors
to receive the packet, (ii) the packet loss ratios cannot be
ignored, and (iii) the packet delivery ratios are not the
same on different links. Thus, they usually choose one
or multiple next-hop destinations for each relay node,
and the links between selected one-hop neighbors have
good quality in the multicast structure.
However, these protocols do not fully take advantage
of the spatial characteristics of wireless communication.
It is unpredicted that which neighbors can receive the
packet in the current transmission. That is, when a
packet is transmitted, it is possible that some neighbor-
ing nodes receive the packet while the designated next-
hop destination does not. Therefore, some research work
[7], [8] has demonstrated that utilizing the cooperative
diversity to send packets through multiple relay nodes
concurrently can further improve the system throughput,
including the multicast throughput [9], which is known
as opportunistic routing (OR).
In opportunistic routing, any node that overhears the
packet transmission is encouraged to forward the packet
if it is closer to the destination. More concurrent relay
nodes give each transmission more chance to make
progress. The multicast extension of OR is discussed in
[9], but it does not build any efficient multicast structure
that is able to reduce the packet transmission. As we
know, OR brings an increase of the number of transmis-

sions because more neighbors participate in forwarding
the packets. The increase of the number of transmis-
sions not only consumes more bandwidth resource, but
also leads to more local interference among nearby
transmissions, which may result in lower throughput.
However, increasing throughput is paramount for mul-
ticast communication in wireless mesh networks. We
have to reduce the number of transmissions when we
exploit opportunistic routing for multicast, which helps
to improve the system throughput. On the contrary, tra-
ditional multicast protocols design the efficient multicast
structure that reduces packet transmissions, but they
lack of the spatial reuse of wireless communication that
also contributes to increasing throughput.
In order to achieve higher multicast throughput, we
propose the opportunistic multicast protocol by adopting
the OR strategy with an efficient multicast structure.
In this protocol, to utilize the spatial diversity, the key
step is to design a multicast tree backbone that is
different from a traditional multicast tree. Tree back-
bone specifies the packet transmission direction instead
of designating the exact next-hop destinations for the
relay nodes. The “adjacent” backbone nodes may be
multi-hop away in the network. An efficient backbone
structure must minimize the number of transmissions.
In this paper, we prove that computing a backbone
that minimizes the expected number of transmission is
NP-hard, and we devise one heuristic algorithm that
appears to work well for building an efficient backbone
in single-rate WMNs. Based on the tree backbone,
packets then self route from the upstream backbone
node to the downstream backbone nodes by OR until
they arrive at the destinations. Therefore, our protocol
not only takes advantage of spatial diversity of wireless
communication by utilizing OR, but also reduces the un-
necessary packet transmissions by building an efficient
tree backbone.
The recent trend in wireless communication is to en-
able devices with multiple transmission rates [10]–[12].
Generally speaking, low-rate communication provides
a long transmission range, while high-rate has to occur
at a short scope. The variance of transmission range
implies the variance of the neighboring node set, which
leads to different spatial opportunities. The inherent
rate-distance tradeoff for opportunistic unicast routing
has had impact on performance [11]. It is intuitive
to expect that this trade-off also affects opportunistic
multicast. In this paper, we investigate this t rade-off
and further propose a Euclidean opportunistic multicast
protocol by devising a Euclidean backbone structure
as well as an efficient rate selection scheme, which
minimizes the number of transmissions for multi-rate
WMNs .
The rest of this paper is organized as follows. Section
II describes the background and the motivation. Sec-
tion III proposes the opportunistic multicast protocol
in single-rate WMNs. We apply the basic idea of
opportunistic multicast to multi-rate WMNs in Section
IV. Section V presents simulation results. Section VI
surveys the related work, and the last section concludes
this paper.
II. B
ACKGROUND AND MOTIVATION
We start from the underlying network model by
introducing some terminology and the basic idea of op-
portunistic routing, which is followed by the motivation.
A. System Model
WMNs consist of three types of nodes: gateways,
mesh routers, and mesh clients. Gateways (access
points) connect mesh networks with the Internet. Mesh
routers form the backbone of a WMN. They typically
have minimal mobility, but they are often equipped
with a powerful ability to improve the flexibility and
capacity of WMNs. Mesh clients are the end users of a
mesh network. They are located within one-hop of mesh
routers so that they can access the Internet through the
mesh backbone. However, mesh clients usually do not
participate in transmitting data for other end users. In
this paper, we only study how to multicast packets to a
group of mesh routers; then packets will be forwarded
one more hop to the corresponding mesh clients that
desire to receive the packets.
To simplify the system model, we consider the net-
work as a weighted graph G =(V, E) with function
w, where V represents the set of gateways and mesh
routers, and E represents the physical links among
neighboring nodes (the node refers to the mesh router or
the gateway in this paper). Individual WMN links may
have different qualities for a variety of reasons, thus
we use the weight on each edge to denote the packet
delivery ratio. Since we only consider multicast among
the mesh routes that are usually fixed, the network
topology is considered as static.
Each node n
i
(1 ≤ i ≤∣V ∣) can transmit a packet at K
different rates R
1
, R
2
, ...R
K
. We assume that each node
has the same fixed communication range under the same
rate. That is, if nodes u and v use the same rate, and
if u can transmit packets directly to node v (and vice
versa), there is a link (u, v) in E. In this paper, we first
consider the multicast issues in single-rate WMNs, then
we apply our idea to multi-rate WMNs.
2

s
d
1
d
2
(a) Natural Extension
s
d
1
d
2
a
(b) Efficient Multicast Path
Figure 1. Motivation Example
B. Basic of Opportunistic Routing
There are different variations of OR. In the follow-
ing, we describe the basic details common to all OR
schemes.
A crucial component of OR is a forwarding set F
ij
for sending a message from node n
i
to node n
j
.This
set F
ij
is carefully selected to minimize the number
of forwarding nodes while maximizing the throughput
improvement. Furthermore, F
ij
is an ordered set where
typically nodes that are closer to destination node n
j
have higher priority. In this paper, we define closeness
to n
j
as the number of transmissions to move a packet
along the best traditional route to n
j
[7], [9], [13].
OR begins with the sender n
i
broadcasting a batch of
packets. Its forwarding candidates continue the f orward-
ing based on their relay priority. That is, higher priority
forwarding nodes are given the first chance to forward
packets. When receiving packets, each forwarding node
also determines if the newly received packet it receives
should be forwarded, either by explicit coordination
or by exploiting network coding properties. The above
process repeats until the destination informs the source
that enough packets have been received.
C. Motivation
Traditional multicast protocols discover the least cost
or highest throughput paths to the destinations. This
strategy is effective in wired networks, but not efficient
in wireless networks, since it does not exploit the
spatial characteristic of wireless communication. For
example, building a shared tree is a common way in
traditional multicast protocols, where transmissions to
different destinations may share some hops in the tree to
minimize the bandwidth cost. However, the shared tree
designates the next-hop destination for each relay node.
As a consequence, there are no spatial opportunities for
each transmission. We can safely conclude that the exact
shared tree is not suitable for opportunistic multicast.
However, if we utilize OR to realize the multicast ap-
plication without any structure, we have to face another
drawback. For instance, a natural multicast extension
from OR is briefly introduced in [9], which requires
the packet self route to all the destinations by OR.
Although utilizing the spatial diversities, this extension
also brings unnecessary transmissions and results in
more interference.
There is an example in Fig. 1, where s is the source,
and nodes d
1
and d
2
are destinations. Under the natural
extension, to deliver a packet, two copies may travel
along different paths to d
1
and d
2
byOR.Fig.1(a)
shows a case of natural extension, which requires totally
10 hops. However, if we allow the transmissions to
distinct destinations share some hops, it can decrease
the total number of transmissions. For example, the
packet is first desired to self route to node a, then
it is split into two copies that would be forwarded
to the two destinations by OR respectively. Fig. 1(b)
shows one case (solid line) of this improvement strategy,
which only needs 7 hops. Thus, we need to devise an
efficient opportunistic multicast protocol that achieves
high throughput by building an efficient backbone to
reduce transmissions.
III. O
PPORTUNISTIC MULTICAST IN SINGLE-RATE
WMNS
In this section, we first introduce the definition of
tree backbone (TB), and explain the basic idea of
our Opportunistic Multicast (OM) protocol. In order
to save bandwidth and decrease interference, the tree
backbone should minimize the number of transmissions
along it. We then prove that computing a TB with the
minimum transmissions in single-rate WMNs is NP-
hard. Afterwards, we present one heuristic algorithm to
construct an efficient TB.
3

A. Basic Idea
Opportunistic multicast builds the tree backbone in-
stead of a multicast tree, which both allows the spatial
opportunities given by OR and minimizes the number of
transmissions by letting packets transmitted to different
destinations share some hops.
Definition 1: Including a source s and a set of desti-
nations D ⊆ V (G), a set of nodes T ⊆ V (G) is selected
to be the backbone of multicast structure. The nodes in
T are called tree backbone nodes.
Definition 2: Among tree backbone nodes, if we des-
ignate that packets need to be delivered from backbone
node a to backbone node b by OR, we say that there
is a direction a → b. Node a is called the upstream
backbone node of b, and b is called the downstream
backbone node of a.
The tree backbone nodes and direction not only
illustrate the packet’s intermediate destinations, but also
indicate the packet forwarding direction. How to select
tree backbone nodes and decide direction is explained
in the following subsections, which helps to minimize
the number of total transmissions in the network.
Definition 3: In graph G, given a source node s,a
set of destinations D, a set of tree backbone nodes
T , and a set of directions R, the multicast backbone
TB(s, D, T, R) is called tree backbone.
The tree backbone indicates the multicast structure
for a specified multicast session (source s, and des-
tination set D). The packet forwarding is determined
by the tree backbone nodes and direction. That is,
after the tree backbone (TB) is built, starting from
the source node, the packets self route to the source’s
downstream backbone nodes by opportunistic routing.
After the packet arrives at one tree backbone node, say
t, it continues routing to t’s downstream backbone node,
until it reaches the destination. At each tree backbone
node, if the backbone node has multiple downstream
backbone nodes, the packet is split into multiple copies
and routed to the corresponding downstream backbone
nodes with different random paths by OR.
For example, in Fig. 1, T = {s
, a, d
1
, d
2
} and R =
{s → a, a → d
1
, a → d
2
}. So, the packets are desired
to be delivered from s to a, then from a to d
1
and d
2
respectively. Node a is the downstream backbone node
of s, while d
1
and d
2
are the downstream backbone
nodes of a. Note that, since routing from the upstream
node to the downstream node uses OR, different packets
may travel along different paths. For instance, from s to
a, the first packet may route along the solid line, while
the second packet may route along the dotted line.
A good metric to evaluate the effectiveness of a
TB is the expected number of transmissions for one
packet to reach all the destinations through the TB.
This is because the less transmissions means less band-
width cost and less interference, which results in higher
throughput and smaller delay. For short, we call this
metric the cost weight of TB. In order to improve mul-
ticast performance, we need to compute a TB with the
minimum cost weight. We refer to this as a Minimum
Tree Backbone or MTB. We prove that computing an
MTB is NP-hard by proving its corresponding decision
problem, TB-D defined below, is NP-hard.
INSTANCE: A weighted graph G =(V, E), a weight
function w on E, a source node s ∈ V , a subset D ⊆ V ,
and a positive number K.
QUESTION: Is there a TB(s, D, T, R)fors and D of
cost weight K?
B. NP-Hardness Proof
Lemma 1: TB-D is NP-hard
First, we explain how to calculate the expected num-
ber of transmissions Z
sd
for one packet to be transmitted
from source s to destination d in opportunistic routing.
For any two nodes i and j,leti < j represent that i is
“closer” to d than j, that is, i has smaller ETX [13] to
d than j.Letε
ij
denote the packet loss ratio from i to
j .Letz
sd
j
be the expected number of transmissions that
forwarder j must take to route one packet from s to d.
The expected number of packets that j must forward,
denoted by L
sd
j
,is[9]
L
sd
j
=
∑
i> j
(z
sd
i
(1 − ε
ij
)
∏
k< j
ε
ik
) (1)
Note that L
sd
s
is 1 since source s generates the packet.
From Eq. 1, the authors in [9] deduce that the expected
number of transmissions that j must make is:
z
sd
j
=
L
sd
j
1 −
∏
k< j
ε
jk
(2)
Suppose there are N nodes in the network. The calcu-
lation of z
sd
j
can be achieved in O(N
2
) [9]. Furthermore,
we compute the total expected number of transmissions
Z
sd
in the network by summing up all the nodes’
expected number of transmissions, that is,
Z
sd
=
∑
j∈V(G)
z
sd
j
(3)
Second, based on the above result, given a
TB(s, D, T, R), for any direction i → j ∈ R, the expected
4

a
s
d
1
d
2
Figure 2. ST-B Example
number of transmissions for delivering a packet from i
to j is Z
ij
. Hence the cost weight of the TB is:
λ
s
D
=
∑
∀i→ j∈R
Z
ij
(4)
Third, we construct a complete graph G
c
with a
weight function w, where V (G
c
)=V (G). For any edge
(u, v) on G
c
, the weight w(u, v)=Z
uv
, which denotes the
total expected number of transmissions occurring in the
network for one packet transmitted from u to v by OR.
It takes O(N
2
) operations to calculate z
sd
j
for each node
j , thus it takes O(N
3
) operations to compute Z
sd
, and
O(N
5
) to get graph G
c
. Graph G
c
is constructed at the
beginning of network, and it will not change as long as
the network topology does not change, thus computing
G
c
is a one-time activity regardless how many multicast
sessions are generated in the network.
For a given TB, we can find a corresponding Steiner
tree S on G
c
, where V (S)=T . An edge (u, v) appears
on S if and only if there is a direction u → v in R.The
cost weight of the TB is equal to the weight sum of
S. It i s well known that computing a Steiner tree with
weight sum K in a weighted graph is NP-hard [14], thus
TB-D is also NP-hard.
Corollary 1: Computing an MTB is NP-hard
Proof: The decision problem TB-D is NP-hard, so
the optimization problem of computing an MTB is also
NP-hard.
C. Heuristic Algorithm
We propose one Steiner tree-based heuristic algo-
rithms for MTB. For short, we call it ST-B. For this
algorithm, we suppose t hat the above complete graph
G
c
is constructed at the beginning of the network.
In this algorithm, under the well-known Takahashi-
Matsuyama (T-M) heuristic [15], a Steiner tree S is built
by an incremental approach to span over source s and
destination set D in G
c
. Initially, the tree contains only
s. At each iteration, the nearest unconnected destination
to the partially constructed tree S is found and the least-
cost path between them is added to the tree. Here, the
least-cost path P(u, v) refers to a path that connects u
and v, and the weight sum on P(u, v) is the smallest
among all paths between u and v.
During constructing S, we can get the backbone nodes
of the corresponding TB as follows:
1) Initially, T = {s}∪D.
2) At each iteration, when a least-cost path P(u, v)
is added to S, T = T ∪{u}∪{v}.
After the backbone nodes are determined, the set of
directions for TB is also discovered by this rule: for any
two nodes u, v ∈ T , if there is not any other backbone
node on path P(u, v) in S, and u is on path P(s, v) in S,
there is a direction u → v ∈ R.
We use a simple example to illustrate the process.
There is a complete graph G
c
in Fig. 2 (For ease of
reading, we do not draw the edges on G
c
), where s is the
source and d
1
, d
2
are destinations. Initially, s is included
in the t ree. Next, since d
1
is “closer” to s, the least-cost
path (s, a, d
1
) is added to the tree. Afterwards, since a
is the closest tree node to d
2
, the least-cost path (a, d
2
)
is added. Based on the Steiner tree, we can build the
tree backbone TB with T = {s, a, d
1
, d
2
} and R = {s →
a, a → d
1
, a → d
2
}
The time complexity of T-M heuristic is O(N
2
), and
searching for R also takes O(N
2
) operations. Therefore,
the time complexity of ST-B is O(N
2
).
IV. E
UCLIDEAN OPPORTUNISTIC MULTICAST IN
MULTI-RATE WMNS
Multi-rate capacity is a common feature of wireless
communication. On one side, a higher data rate can
be used to increase throughput, but it also has shorter
transmission range and hence more hops to reach the
destination. Besides, there are few spatial opportunities
due to the low neighbor diversity in one hop. On the
other side, lower data rate often has a longer transmis-
sion range and hence less hops in the selected path. The
higher neighbor diversity brings more spatial opportu-
nities, but the low rate disadvantage may counteract the
above benefit. The inherent tradeoff between rate and
distance is hereby worthy of a careful study.
A. Design Consideration
A local metric, called Expected Advancement Rate
(EAR) [11], has been proposed to find the best rate for
each node:
5

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Efficient opportunistic multicast via tree backbone for wireless mesh networks" ?

In this paper, the authors propose a new opportunistic multicast protocol to improve multicast throughput in Wireless Mesh Networks ( WMN ). For single-rate WMNs, the authors show that constructing an efficient tree backbone that minimizes the number of transmissions is NP-hard, and they devise one effective heuristic algorithm for it. For multi-rate WMNs, the authors investigate the inherent rate-distance tradeoff and propose a Euclidean opportunistic multicast protocol by devising a Euclidean tree backbone as well as an efficient rate selection scheme to minimize the number of transmissions. 

Q2. What is the effect of the tree backbone on the packet delivery?

Their efficient tree backbone is able to minimize the number of transmissions as well as interference, which greatly speeds up the packet delivery. 

Q3. What is the purpose of the routing strategies in [9], [29]?

The routing strategies in [9], [29] aim to combine network coding with opportunistic routing in a natural fashion, so that they can achieve the cooperative spatial diversity. 

Q4. What is the second step to determine the direction set R?

The second step is to determine the direction set R. For any edge (u,v) in the Euclidean Steiner tree, if u is added to the tree before v, there is a direction u→ v∈R. 

Q5. What is the metric to evaluate the effectiveness of a TB?

A good metric to evaluate the effectiveness of a TB is the expected number of transmissions for one packet to reach all the destinations through the TB. 

Q6. What is the heuristic method to build a multicast backbone?

At each tree backbone node, if the backbone node has multiple downstream backbone nodes, the packet is split into multiple copies and routed to the corresponding downstream backbone nodes with different random paths by OR. 

Q7. What is the metric used to calculate the number of transmissions?

Previous TB construction is based on the number of transmissions Zsd on each pair of source s and destination d. Calculating Zsd requires the information of link delivery ratios. 

Q8. What is the opportunistic routing of a multicast packet?

That is, after the tree backbone (TB) is built, starting from the source node, the packets self route to the source’s downstream backbone nodes by opportunistic routing. 

Q9. What is the heuristic algorithm to construct a multicast backbone?

Definition 1: Including a source s and a set of destinations D ⊆V (G), a set of nodes T ⊆V (G) is selected to be the backbone of multicast structure. 

Q10. What is the definition of closeness to n j?

In this paper, the authors define closeness to n j as the number of transmissions to move a packet along the best traditional route to n j [7], [9], [13].OR begins with the sender ni broadcasting a batch of packets. 

Q11. What is the heuristic for constructing a Steiner tree?

In this algorithm, under the well-known TakahashiMatsuyama (T-M) heuristic [15], a Steiner tree S is built by an incremental approach to span over source s and destination set D in Gc. Initially, the tree contains onlys. 

Q12. What is the heuristic algorithm to build a multicast backbone?

How to select tree backbone nodes and decide direction is explained in the following subsections, which helps to minimize the number of total transmissions in the network. 

Q13. What is the link between nodes u and v?

That is, if nodes u and v use the same rate, and if u can transmit packets directly to node v (and vice versa), there is a link (u,v) in E. 

Q14. What is the difference between the two papers?

Their work inherits the characteristic of opportunistic routing, but differs from the above papers, since it focuses on multicast and builds an efficient backbone.