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Multiple input multiple output (MIMO) wireless mesh networks (WMNs) aim to provide the last-mile broadband wireless access to the Internet. Along with the algorithmic development for WMNs, some fundamental mathematical problems also emerge in various aspects such as routing, scheduling, and channel assignment, all of which require an effective mathematical model and rigorous analysis of network properties. In this paper, we propose to employ Cartesian product of graphs (CPG) as a multichannel modeling approach and explore a set of unique properties of triangular WMNs. In each layer of CPG with a single channel, we design a node coordinate scheme that retains the symmetric property of triangular meshes and develop a function for the assignment of node identity numbers based on their coordinates. We also derive a necessary-sufficient condition for interference-free links and combinatorial formulas to determine the number of the shortest paths for channel realization in triangular WMNs.
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Research Article
On Modeling and Analysis of MIMO Wireless Mesh Networks
with Triangular Overlay Topology
Zhanmao Cao,1,2 Chase Q. Wu,2Yuanping Zhang,3Sajjan G. Shiva,2and Yi Gu4
1Department of Computer Science, South China Normal University, Guangzhou, Guangdong 510631, China
2Department of Computer Science, University of Memphis, Memphis, TN 38152, USA
3School of Computer Science & Education Soware, Guangzhou University, Guangzhou, Guangdong 510006, China
4Department of Computer Science, Middle Tennessee State University, Murfreesboro, TN 37132, USA
Correspondence should be addressed to Chase Q. Wu; chase.wu@memphis.edu
Received  September ; Revised  January ; Accepted  January 
Academic Editor: Hsuan-Ling Kao
Copyright ©  Zhanmao Cao et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Multiple input multiple output (MIMO) wireless mesh networks (WMNs) aim to provide the last-mile broadband wireless access
to the Internet. Along with the algorithmic development for WMNs, some fundamental mathematical problems also emerge in
various aspects such as routing, scheduling, and channel assignment, all of which require an eective mathematical model and
rigorous analysis of network properties. In this paper, we propose to employ Cartesian product of graphs (CPG) as a multichannel
modeling approach andexplore a set of unique properties of triangular WMNs. In each layer ofCPG with a single channel, we design
a node coordinate scheme that retains the symmetric property of triangular meshes and develop a function for the assignment of
node identity numbers based on their coordinates. We also derive a necessary-sucient condition for interference-free links and
combinatorial formulas to determine the number of the shortest paths for channel realization in triangular WMNs.
1. Introduction
e WiMax group advocated the last-mile broadband servi-
ces, IEEE . Standard [], which denes broadband back-
bones as wireless mesh networks (WMNs). Such networks
typically consist of two types of nodes, that is, base station
(BS) and subscriber station (SS). BS is a wireless gateway con-
nected to the Internet, while SS is a node that acts as a relay
station. In multiple input multiple output (MIMO) WMNs,
all nodes are equipped with multiple interfaces and support
both multicast and mesh modes. Particularly, in a mesh
mode, nodes can communicate with neighbors without the
help of BS and the relay strategy provides an economical way
to expand the mesh covering area. MIMO WMNs (in the rest
of the paper, we use the term WMNs for conciseness) are well
recognized as an ecient extension to the Internet backhaul
[].
WMNs possess some inherent characteristics that are
dierent from ad hoc or wireless sensor networks. Since the
nodes in WMNs are almost xed and typically powered by
electrical wires, the links or routing paths in WMNs generally
lastlongerthanthoseinmobileadhocnetworks.Also,every
node in WMNs typically has nonzero trac requests because
it needs to route aggregated trac from the terminal devices
in its region for either upload or download. e topology
of WMNs may be determined based on the predicted trac
requests or geographical environments. Since both BS and SS
canbeconsideredstatic,itisreasonabletoviewthemesh
topology as a xed graph.
e rapidly growing demand for ubiquitous Internet
access requires an eective mathematical model for WMNs as
it may simplify the tasks of routing, scheduling, and channel
assignment. To achieve a maximum fair usage of multiple
channels in WMNs, it is important to employ an ecient
channel allocation scheme and an appropriate overlay graph
topology for a given area []. In [], the virtual topology is
viewed as CPG to simplify the channel assignment problem
through a graph. In addition, the CPG model also brings
convenience for the analysis of routing and scheduling in
WMNs. As nodes are static in WMNs, they can be identied
Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2015, Article ID 185262, 11 pages
http://dx.doi.org/10.1155/2015/185262
Mathematical Problems in Engineering
geographically through their coordinates []. erefore, the
routing and scheduling problems can be analyzed using the
node coordinates. Interference is another fundamental issue
in either scheduling or routing, and the properties of interfer-
ence under a given coordinate scheme, if dened properly
and described rigorously, may bring benets to resource utili-
zation and interference avoidance.
In this work, we use CPG as a modeling approach and
explore a set of unique properties of WMNs with a triangu-
lar topology. In each layer of CPG with a single channel, the
network topology is a planar mesh. Our work makes several
theoretical contributions to the analysis of WMN properties:
(i) we design a coordinate scheme that retains the symmetric
property of triangular meshes and develop a function to
assign a unique identity number to a specic wireless node
based on its coordinates. (ii) We derive a necessary-sucient
condition for interference-free links under the proposed
node coordinate scheme. (iii) We derive combinatorial for-
mulasintermsofthenumberoftransceiversandchannels
to determine the number of the shortest paths for channel
realization in triangular WMNs.
e rest of the paper is organized as follows. Section 
surveys related work. Section  presents a channel-layered
CPG model with emphasis on interference detection. In
each layer of CPG, we propose a coordinate scheme, named
parallel cluster coordinate, derive a necessary-sucient con-
dition for interference avoidance, and develop a function for
node identity number assignment to support ecient WMN
maintenance and administration. Section  derives formulas
for determining the number of the shortest paths in WMNs.
2. Related Work
We conduct a brief survey of work directly related to mathe-
matical models for WMNs.
In the past decade, most eorts in WMN topology were
focused on interference and performance in planar meshes
[]. With geographical information from satellite or con-
trol channel communications, it is relatively easy to acquire a
planar topology since the nodes in WMNs are almost static.
For example, in IEEE Standard . for media-independent
handover, Media Independent Information Service (MIIS)
stores the geographical information of all access network
operators available in a particular region [].
Square and hexagonal meshes have been proposed to act
as wireless broadband backbones []. However, they are less
competitive than triangular meshes, as the latter outperforms
the former and other random meshes in terms of various
performance metrics such as coverage area, link quality,
per-user fair rate, and node density []. Hong and Hua
conducted a comparative evaluation of the throughput per-
formance between square, hexagonal, and triangular meshes.
eir experiments showed that triangular meshes achieve
higher throughput than others in several cases, and their
total throughput does not vary signicantly in response to
topology changes in large wireless networks with a constant
density []. erefore, we also adopt a triangular mesh
topology in our model.
A unied network model based on super graph may
further facilitate the analysis of various aspects of WMNs
such as interference, scheduling, routing, and channel assign-
ment. However, research eorts along this line are still quite
limited. Several researchers considered some of these aspects
simultaneously [,,], which motivates us to design a
unied model for WMNs.
In a given network topology, a properly designed coor-
dinate scheme may facilitate link interference detection and
path nding. In hexagonal meshes, Chin et al. proposed a
node coordinate scheme with three parallel line clusters [],
where a node is represented by a -tuple. In triangular meshes
consisting of BS nodes with a node degree of six, Cao et al. pro-
posed two BS-centered coordinate schemes [] and explored
interference and link groups in each of these schemes.
Furthermore, in addition to coordinates, a router should also
be assigned a unique identity number to support convenient
simulation, administration, and maintenance. In [,], Cao
et al. also represented the coordinates of a node by a -tuple
but did not tackle the identity number assignment problem.
e performance of WMNs is largely aected by link
interference. Most research eorts on this subject have been
made through generic methods or experimental studies,
instead of conclusive results in the form of necessary-
sucient conditions [,,]. In our work, we attempt to
design a suitable node coordinate scheme and then model the
interference in WMNs as a specic checking list based on set
theory.
Routing in WMNs is a -step procedure, that is, path
nding followed by channel assignment. One basic approach
to nd an alternative interference-free path is to count the
number of shortest paths and the number of all possible
channel assignment schemes. A tree-like path nding scheme
is proposed in []withoutanynodecoordinate.CaoandXiao
proposed path counting formulas for a source-destination
pair in square grids [], while the path counting problem in
triangular meshes is still le unexplored. In our work, based
on the proposed CPG model and coordinate scheme, we
tackle this problem in triangular meshes with multiple chan-
nels and interfaces. Channel assignment is another important
problem involving several network layers in WMNs, which is
essentially an NP-complete edge coloring problem [,].
3. A Channel-Layered Graph Model
AsmassiveMIMOisonitswayfromtheorytorealistic
deployment, one of the key problems is the interchannel
cooperation, which calls for the development of sophisticated
analytical channel models. Larsson et al. provide an overview
of massive MIMO and motivate researchers to develop
channel models capturing the essential channel behaviors
despite their limitations []. For example, the Kronecker
model, which is widely used to model channel correlation,
is not an exact representation of reality but provides a useful
model for certain types of analysis.
WMNs are conventionally modeled by a directed graph
where a directed edge between two neighbor nodes represents
a communication link over a specic channel. Since a node
Mathematical Problems in Engineering
(a) Mesh and virtual nodes (b) Layered mesh
(c) Links in a channel (d) Maximal links
A
A
A
A
B
B
B
B
C
C
C
C
D
D
D
D
E
E
E
E
c1
c2
F : e CPG-modeled virtual topology of a physical network with ve physical nodes and two channels.
equipped with transceivers may have (at most) simul-
taneous links over orthogonal channels (assuming that
more than channels are available), we can split a physical
node into fully connected virtual nodes, each of which is
equipped with a single transceiver. is way, we are able to
represent the original WMN as identical layers of networks,
each of which operates over a dierent channel.
3.1. Cartesian Product of Graphs. e topology-based model-
ing approach has been commonly used in wireless networks
for various purposes, but oen in a planar view [,,],
andmostofthediscussionsonscheduling,routing,and
channel assignment are also based on a planar topology [,
,,]. e recent development of MIMO WMNs calls
for a suitable model to describe MIMO-specic properties
and understand the cooperative activities across dierent
interfaces over multiple channels []. e planar topology
can be used to determine the internode interference []butis
insucient to provide a visual representation for analyzing
the cooperation between links or channels. On the other
hand, modeling MIMO WMNs as a super graph still remains
largely unexploited except the work in []. In this paper, our
goal is to develop an eective model to facilitate the analysis
of MIMO channel cooperation.
We propose to employ the CPG to model WMNs by
combining a triangular mesh of physical nodes and a graph of
fully connected virtual nodes. Together with the coordinates
of triangular overlay nodes, the CPG model provides a
convenient way to analyze the properties of interference
avoidance, channel assignment, and routing path counting.
is model retains the independence between orthogonal
channels while providing a general approach to analyzing link
behaviors over multiple channels.
Cartesian Product of Graphs. Given two graphs and ,the
Cartesian product ×is a graph such that
(i) the graph ×has a vertex set ()×();thatis,
avertexin×is denoted by a pair (V,V󸀠),V (),
and V󸀠 ();
(ii) any two nodes (,󸀠)and (V,V󸀠)and ,V∈ ()and
󸀠,V󸀠() are adjacent in ×, if and only if one
of the following holds: (a) =Vand 󸀠is adjacent to
V󸀠in ,or(b)󸀠=V󸀠and is adjacent to Vin .
For illustration, Figure (a) shows a mesh network of ve
physical nodes (le side) and a graph of two connected virtual
nodes (right side) corresponding to a physical node equipped
with two transceivers, each operating on a dierent channel
1or 2.Figure (b) shows a channel-layered virtual topology
of the original mesh network modeled by CPG.
In this example, the CPG of two graphs in Figure (a)
results in a two-layered graph in Figure (b). A solid directed
Mathematical Problems in Engineering
edge in the top or bottom planar meshes in Figures (b),
(c), and (d) represents a communication link =(,)
𝑐,
transmitting data from node to node over a channel ,
while a dashed edge has a conict with some active links.
emaximumpossiblenumberofconcurrentactive
links on a given channel in a mesh is largely aected by
the selection of the senders. As shown in Figure (b), when
node in the top layer is sending data on channel 1,the
three neighbors of , that is, nodes ,,and,inthe
same layer (over channel 1)cannotsenddata.Furthermore,
node canbeselectedasasender,butneitherofits
neighbors, that is, nodes and ,canreceivedatafrom
without interference, illustrated as the dashed directed edges
in Figure (c). In this case, there is no other active link except
(,)𝑐1. However, if we choose the initial senders properly,
as shown in Figure (d), there could exist two concurrent
interference-free links on channel 𝑖,=1,2,thatis,(,)𝑐𝑖
and (,)𝑐𝑖.
Since all active links must be interference-free on the
same channel at the same time, links are generally sparsely
distributed in a planar mesh with respect to a certain
channel. Note that more links mean better service to trac
requests. e CPG model allows us to consider concurrent
paths in multiple layers over dierent channels through
radio cooperation among channel layers for a given trac
task, hence providing more capacity and higher throughput
comparedtothesituationwithasinglechannelandradio.
is is consistent with the experimental results presented
by Draves et al. []. e proposed CPG model is aligned
well with existing research in terms of interference relation,
link activity, and network throughput, as well as routing and
scheduling [,,,,] and enables us to conduct deeper
theoretical analysis of WMNs.
One advantage that CPG brings is to simplify the expres-
sion of multichannel links. Since dierent layers (operating
on dierent channels) are of the same topology, the schedul-
ing strategy derived in one layer is readily applicable to
another layer. For example, there exists a certain link distri-
bution pattern among concurrent links. In CPG, it is obvious
that such an interference-free link distribution pattern on one
channel also exists on others.
3.2. Coordinate Scheme. A well-designed coordinate scheme
may facilitate the analysis of WMNs. Since all channel
layers are of the same topology, we only need to design the
coordinate scheme for one layer or channel. e channel
information can be added to the node coordinates to uniquely
identify a specic layer.
Chin et al. proposed a coordinate scheme in hexagonal
cellular networks where each node has a degree of three
[]. Inspired by their work, we propose a parallel cluster
coordinate scheme in a local triangular mesh with one BS
node and a number of SS nodes with a degree of six. is
scheme can be readily extended to larger networks with
multiple BSs by inserting the BS information to the node
coordinates.
3.2.1. Parallel Cluster Coordinate Scheme. In a triangular
mesh, we rst dene three clusters of parallel lines along three
A
B
C
D
E
j
F
BS
i−2 i−1 i+1
j+2
j+1
j−1
j−2
k−2 k−1 k+1
k
i
F : e parallel cluster coordinate scheme.
dierent directions, that is, north-east, east, and south-east.
Sinceeachnodeisapointintersectedbythreelines,each
fromoneofthethreeclusters,weproposeaparallelcluster
coordinate scheme (PCCS), which uses a -tuple (,,) to
represent the coordinates of a node intersected by the th, th,
and th line in the corresponding clusters []. As illustrated in
Figure , the BS node with the coordinates (0,0,0)is located
at the center, and the SS nodes ,,,,,and,which
are one hop away from the BS node, have the coordinates
 = (1,0,1), = (0,1,1), = (−1,−1,0), = (−1,0,−1),
= (0,−1,−1),and = (1,1,0).
Note that not every combination of three integers can
represent a node in a triangular mesh because some lines
dened by combinatorial -tuples do not intersect at a
common point. We summarize such lines as follows:
=  1,1,,+1,+1,,, 1,+1,
,+1,1,+ 1,, 1, 1,,+ 1. ()
Under the proposed PCCS scheme, for a given sender
(,,), there are six possible receivers, which form its
neighbor set 𝑏(,,):
𝑏,,=  1,+1,,+1,1,,
,+1,+1,,1,1,
+1,,+1,−1,,−1.
()
PCCS retains the symmetric nature of a triangular mesh
and facilitates the calculation of the distance between a pair
of nodes.
3.2.2. Symmetric Property of PCCS. In PCCS, the coordinates
of any two nodes that are symmetric with respect to the
central BS node located at (0,0,0) are negated. For example,
the pairs of nodes and ,and ,andand in Figure 
are symmetric, and their coordinates are negated from their
counterparts.
e symmetric properties of CPG and Cayley graph
have been well studied []. According to the vertex/edge
Mathematical Problems in Engineering
transitive properties, a link group can be transited to generate
another one in any layer.
Similarly, due to the symmetric property, we are able to
transform a link group to another one through rotation. For
example, link group of ((0,2,2),(0,1,1))𝑐and ((2,−2,0),
(1,−1,0))𝑐is interference-free. Aer a clockwise rotation of
/3, a new group of interference-free links is of ((2,0,2),
(1,0,1))𝑐and ((0,−2,−2),(0,−1,−1))𝑐,respectively.With
another clockwise rotation of /3, the derived interference-
free links are ((2,−2,0),(1,−1,0))𝑐and ((−2,0,−2),(−1,0,
−1))𝑐, respectively. Note that the rotation operation is edge
transitive, and it generates a new link group because of the
symmetric property.
3.2.3. Distance between Two Nodes. Asmosttracisupload/
download (to/from BS), we need to count the number of hops
fromarouternodetoBSinPCCS.Wehavethefollowing
properties.
Property 1. In triangular WMNs with PCCS, the minimum
number of hops from = (,,) to BS = (0,0,0)is
=||++||
2=max ||,,||()
which is consistent with the one in [].
Property 2. Suppose that BS is positioned at an arbitrary
location (0,0,0),insteadof(0,0,0).Node(,,) can be
translated to (,,)(0,0,0)by a translation function:
:,,→ ,,0,0,0.()
In general, suppose that (0,0,0)is the destination of a
trac path; we may virtually view (0,0,0)as the BS node
aer applying the translation of ().en,theformulain()
may facilitate further analysis.
Given two nodes =(
1,1,1)and =(
2,2,2)in
PCCS, the distance between and , denoted by (,),
is the minimum number of hops between them, which is
calculated as (,) = (|1−
2|+|
1−
2|+|
1−
2|)/2.
3.2.4. Mapping to 2D Points. In order to draw a planar mesh,
we need to map -tuple coordinates (,,)to -dimensional
(D) points (,).Todothis,werstoverlaparectangular
plane coordinate system to the PCCS triangular mesh. Let
the -axis overlap the axis =0while keeping the positive
rightward direction. Meanwhile, the -axispassestheBS
node and is vertical to the line =0with a positive upward
direction. en, we can determine (,)by projecting (,,)
onto axes and .ePCCSsupportsafunctionmapping
(,,)to (,)as follows.
Property 3. A one-to-one mapping function maps node
coordinates (,,)in PCCS to D coordinates (,):
,= ,, =
=+
2
=. ()
For example, node (0,1,1) is mapped to (1/2,1),and
(−1,3,2) is mapped to (1/2,3).Ifanodeisonthe-axis, =
−always holds. For example, (−1,2,1)is mapped to (0,2).If
node is located on =0in PCCS, =/2. For example,
(−2,2,0) is mapped to (−1,2).ispropertyfacilitatesthe
plotting of a triangular mesh in PCCS.
3.3. Interference-Free Conditions. To analyze the interference
between links, we need to consider node interference rela-
tions, which are critical to scheduling links in WMNs. As
interference is an inherent nature for radio media, the wire-
less communication performance may be severely degraded
if radios operate without a proper scheduling scheme [].
Minimizing interference has been extensively investi-
gated in the literature [,]. Subramanian et al. discussed
channel assignment in a multiradio situation []. Tan et al.
designed algorithms to set up a skeleton of minimum interfer-
ence for a single channel []. Scheduling links in a coopera-
tive way will improve the energy eciency and reduce colli-
sion. ese discussions assume variable transmitting power
or interface channel switching. However, the variation of
transmitting power may lead to the variation of network
topology, which may cause changes in the interference
relation.Xuetal.useasensingschemetoachievepowere-
ciency for convergence communication []. eir method
canhelpsettheinitialpowerinanalmost-staticWMN
topology while promising interference-free cognitive access
with link status as busy or idle.
When sender is sending over channel 0, neither can
receive data over channel 0, nor can its neighbors in its
eective radio coverage send data. If a valid neighbor node
receives data over channel 0,wehavealink(,)𝑐0.
Furthermore, the neighbors of receiver cannot send data
over channel 0at the same time. erefore, to be link
interference-free in multiradio multichannel environments
using PCCS, we need to consider three classes of node inter-
ferences: sender-to-sender, receiver-to-receiver, and sender-
to-receiver.
3.3.1. Sender-to-Sender. Givenacertainchannel,anytwo
sendersmustbeatleasttwohopsawayinatriangularmesh
to avoid mutual interference, which could help construct an
interference-free candidate set for possible senders.
Denition 1. Arouterset𝐶is called a sender candidate set if
every two nodes in 𝐶are at least two hops away.
Given two nodes 𝑖=(,,)and 𝑗=(,,)in 𝐶,the
distance between them must satisfy (𝑖,𝑗)≥2.
For a given node 𝑗∉
𝐶,wecanadd𝑗to 𝐶only if it
is at least two hops away to any node 𝑖in 𝐶.Wehavethe
following necessary condition on interference-free links:
𝑖,𝑗=|−|+|−|+−
2≥2. ()
3.3.2. Receiver-to-Receiver. Two nodes 𝑖and 𝑗can each act
as a receiver simultaneously if they have a distance larger than
. Similar to the sender candidate set, we use 𝐶to denote
Mathematical Problems in Engineering
T : Conditions for simultaneous links over one channel.
ree neighbor pairs Link interference-free conditions
Nodes Nodes Necessary Sucient
in 𝐶in 𝐶condition condition
𝑖,𝑗∀ =,(𝑖,𝑗)≥2()
𝑖,𝑗∀ =,(𝑖,𝑗)≥1() () () ()
𝑖𝑗∀ =,(𝑖,𝑗)>1()
a receiver candidate set, in which any two nodes 𝑖=(,,)
and 𝑗=(,,)satisfythefollowingcondition:
𝑖,𝑗=|−|+−+−
2≥1. ()
Note that both 𝐶and 𝐶are candidate sets, and the
actual sender and receiver sets are a subset of 𝐶and 𝐶,
respectively. e relations between nodes largely depend on
the previous selected links.
Property 4. If links (1,1)𝑐and (2,2)𝑐can be scheduled at
the same time, then (1,2)must satisfy condition () and
(1,2)must satisfy condition ().
Note that Property  is only a necessary condition for
interference-free links.
3.3.3. Sender-to-Receiver. Given a certain channel, a node
cannot receive data if it is in the eective radio range of
the sender of any other active link; otherwise, interference
occurs. If two links 𝑖and 𝑗( =) coexist, the nodes involved
in these two links must satisfy the following condition:
𝑖,𝑗=|−|+−+−
2>1, =, ()
where 𝑖is the sender of link 𝑖and 𝑗is the receiver of link
𝑗.
For example, in Figure (d), since ’s neig hb o r does not
conict with link (,)𝑐1,can be a receiver of another link
(,)𝑐1.
We summarize three necessary conditions for link coex-
istence in the aforementioned three classes in Tab l e  .
We have the following theorem.
eorem 2. Two lin k s 𝑖and 𝑗can be simultaneously sched-
uled (coexist) on the same channel, if and only if they satisfy all
of the three necessary conditions in Table 1.
eorem  is based on the PCCS scheme and the set
theory. As long as the PCCS node coordinates are given,
we are able to determine the interference between links.
ese known conditions are helpful to nd as many links as
possible, while contributing to concurrent central scheduling.
3.4. Transformations of Link Groups. We at t e m p t t o  n d
as many coexisting links as possible in a given local area.
Since a link can be established only between a valid sender
candidate and a valid receiver candidate, the actual (nal)
k
j
(i−2,j+3,k+1) (i,j +3,k+3)
(i − 1, j + 3, k + 2)
(i−1,j+2,k+1) (i, j + 2, k + 2)
(i−1,j+1,k) (i+1,j +1,k+2)
(i,j+1,k+1)
(i, j, k) (i + 1, j, k + 1)
i
F : Coexisting links in a local area.
scheduled link set is a subset of 𝐶×𝐶. Generally, coexisting
links are sparsely distributed in the network. Based on a
known group of coexisting links around one triangle, we wish
to obtain a new group of coexisting links through certain
transformations.
Starting from a link 1=(
1,1)𝑐1,where1= (,,)and
1= (, +1,+ 1), we want to set up a dense link group in
alocalarea.esetof1s neighbor nodes is 𝑏(1)={V|
(V,(,,)) ≤ 1,V∈}.esetof1s neighbor nodes is
𝑏(1)={V|(V,(, + 1,  + 1)) 1,V∈}.Asany
neighbor node of the sender or the receiver of an active link
should remain silent if it does not receive data from the sender
1, to avoid interference with 1, nodes in 𝑏(,,)𝑏(1)
cannot send data on 1when 1is active.
To expand the active link group containing 1=(
1,1)𝑐1,
two nodes 2= (, + 3, + 3) and 2=(,+2,+2),
which are not in 𝑏(1)and 𝑏(1),maynegotiateforanew
link. If successful, (2,2)𝑐1is added to the active link group.
According to the conditions in eorem ,link2=(
2,2)𝑐1
can coexist with 1.
Any node in 𝑏(1)∪
𝑏(2)cannot be a new sender,
except 1and 2. However, it is completely dierent on the
receiverside,whereonenodemaybethereceiverofanew
link,evenifitisin𝑏(1)∩𝑏(2). For example, in Figure ,
3=(2,+3,+1)and 3= (1,+2,+1)form a new
link 3=(
3,3)𝑐1, which can coexist with 1and 2.Around
the central triangle in Figure ,wendthreeinterference-free
links coexisting over one channel.
Although the three nodes of the central triangle are
involved in three active links as shown in Figure , neither of
these three nodes can act as a sender while the other two are
receiving data from their respective senders. To obtain the
same number of new coexisting links, we need to keep the
receivers unchanged, while considering a certain switching
to the three senders of active links 1,2,and3.
Mathematical Problems in Engineering
By passing the sending token from the current sender
toitsneighbornodeclockwisewhilekeepingtheoriginal
receiver, we are able to establish three possible new links:
󸀠
1= 1,+1,,,+1,+1𝑐1,
󸀠
2= +1,+2,+3,,+ 2, +2𝑐1,
󸀠
3=−1,+3,+2,−1,+2,+1𝑐1.
()
Since the sender (−1,+3,+2)of 󸀠
3is one hop
away from the receiver (,+2,+2)of 󸀠
2,whichviolates
the third condition in eorem ,󸀠
2and 󸀠
3cannot coexist on
channel 1,whichmeansthat󸀠
1,󸀠
2,and󸀠
3cannot coexist. In
other words, switching the senders of interference-free links
may incur new interferences. erefore, we must perform
interference check aer switching the sender of an active link.
Sender switching may generate a new dense link group in
a local mesh. For example, we obtain a new link
1= (( +
1,,+1),(,+1,+1))𝑐1, by switching the sender of link 1
to its neighbor node anticlockwise. e three links
1,2,and
3are interference-free. Similarly, link
3=((−2,+2,),(−
1,+2,+1))𝑐1is also interference-free with links 1and 2.
Since triangular meshes possess symmetric properties,
transformations such as rotation and translation can retain
the interference-free features, which may save computing
time in nding new link groups [,].
3.5. Node Identity Number Assignment. e nodes in tri-
angular meshes can be viewed not only as wireless router
nodes, but also as resources or data sets. Assigning a unique
identication (ID) number to each node brings several
benets. For example, such IDs can help to locate or identify
nodes quickly for various administration or maintenance
purposes.
Given the coordinates (,,)of a node in PCCS, there
exists a general function that maps (,,) to a unique
integer. For example, in Figure ,wecanmapBS = (0,0,0)to
0, while =(1,0,−1)to 1,and = (−1,1,0)to 2.
Denition 3. In PCCS, -circle in a triangular mesh is a set
of nodes that have exact distance of hopstothenodeBS.
Suppose that (,,) is in -circle, where =max{||,
||,||} according to (). To construct a mapping function
(,,), we classify nodes according to their coordinates. For
any node  = (,,), we consider the following seven cases
in Figure , which is a logical route extracted from Figure .
e rst and special case =0,(0,0,0) = 0.enodeIDs
in -circle increase along an anticlockwise direction. ese
cases are applicable to any -circle. e parallel solid line
segments are corresponding to identity counting piecewise
functions; for example, the nd case includes three parallel
segments of dierent -circle in Figure , in which nodes on
these segments get their ID number following the nd one in
().edashedlinemeansthattheinnernodeiscounted
already, and the outer node is the new start of the next circle.
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
F : e cases for node identity number assignment.
To assign an identity to each node on -circle, we need
to know the total number of nodes inside this circle:
=1+6(1+2+⋅⋅⋅+−1)=1+3(−1).()
Let = 3(1);wehave=+1. For example, there
is one node inside circle =1,thereare+1=7nodes
inside circle =2,andthereare+1=18+1=19nodes
inside circle =3.
e number of nodes on -circle depends on the value
of . According to (),forany-circle, with the six solid
line segments in the order as shown by arrows in Figure ,
we assign each node a unique integer on -circle with an
ID number in {+1,+2,...,++1,...,+6}.
Aer overlapping Figure  on Figure ,anodeisassigned
with its identity number in one of the seven cases in ()
corresponding to the line segment case in Figure :
,, =
0=0st
++1 =0∧= 2nd
+−+1 =0∧= rd
+2−+1 =0=− 4th
+3−+1 =0∧=− 5th
+4++1 =0∧=− 6th
+5++1 =∧ =,0 7th.
()
For example, in the th case, node (1,−4,−3)is assigned
with an ID number of 3×4×(4−1)+4×4+1+1=54.
Some nodes may satisfy two cases in () while getting the
same ID. For example, node (0,3,3),whichisonthe3-circle
and the line segment in parallel with -axis, is assigned with
an ID number of 3××(−1)++1=3×3×2+3+1=
22 in the nd case in (). Meanwhile, node (0,3,3),asitis
on one line segment of Case  in Figure ,cangetitsIDof
3××(−1)+−+1=3×3×2+3+1=22by
followingtherdcasein().
We shall provide more explanations for the nd and
th cases to facilitate a better understanding of ().Inthe
nd case where  =0=,thecoordinateof
node (,,)must be positive. Node (,,) on this -circle
(=) segment should be assigned with an ID number in
Mathematical Problems in Engineering
{+1,+2,...,++1,...,+}.Inthiscase,node(,,)
is located above the lines =0and >0,asshowninCase 
in Figure . Hence, with the increasing , each node (,,)is
assigned with an ID number (,, ) = +  +1. Generally,
on each -circle (>0), from the nd case to the th case,
each with nodes, the total number of nodes is 6.
In the th case, we need to avoid repeatedly counting the
node on line =0,asshowninFigure .enumberofhops
to the BS is =.elastnodeonthiscircleisonaline
segment of the -axis parallel cluster with =0according to
Figure . e condition ==excludes the nodes
already counted in the previous Case  in this circle, while
==0prevents the st case from being reconsidered.
Meanwhile, the condition  =ensures that we count nodes
on the next circle following the nd case again where =.
In the th case, a node can be assigned with an ID number in
aniteset{+5+1,...,+5++1,...,+6}.With
the increasing ,where=0,1,...,1,node(,,)on
thissegmentisassignedwithanIDnumberof+5++1.
Note that the mapping function (,,) is a segmented
linear function, which is invertible for any nite set.
4. Path Counting
Routing is one fundamental problem in WMNs. To develop
a good routing scheme, one needs to know the number of
alternative paths and the number of channel assignments
foragivenpairofsourceanddestination.Wediscusspath
ndingandrealizationbasedontheproposedCPGmodel
and coordinate scheme. e total number of shortest paths in
grid meshes was discussed in []. In this section, we tackle
the path counting problem in triangular meshes.
We use (,)to denote the distance between source =
(1,1,1)and destination =(
2,2,2).Firstly,(,) =
(|1−
2|+|
1−
2|+|
1−2|)/2. In order to transmit data
from to , we need to select one path from (,),which
is the set of all shortest paths from to .
4.1. Path Alternatives. Every step along the shortest path from
to is one hop forward in one of the directions ,,and.
e two smaller numbers of {|1−2|,|1−2|,|1−2|}indicate
the lines of parallel clusters that form a grid mesh for path
selection, as illustrated in the grid of dashed lines in Figure .
We refer to the two corresponding directions from to in
the grid mesh as the correct directions.
e correct directions ensure that the data is transmitted
through one of the shortest path, where every hop selection
makes one hop closer to the destination. rough the use
of correct directions, we are able to reduce the problem of
counting all shortest paths from to in triangular meshes
to a problem in grid meshes.
We provide an example in Figure  to count the number
of paths, where |1−2|=2,|1−2|=3,and|1−2|=5.e
grid with dashed lines contains all the shortest paths from
to .
Inasimplesituationwhereoneof{|1−2|,|1−2|,|1
2|}is , the dashed grid degrades to a line. erefore, there
is only one shortest path available.
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F : e shortest path alternatives in triangular meshes.
For convenience, let ℘ = {|1−
2|,|1−
2|,|1
2|}. e correct directions are consistent with min(℘) (the
minimum of three elements in )andmid(℘) (the middle
of three elements in ). To select a shortest path, it is
necessary to remove the direction corresponding to max(℘)
(the maximum of three elements in ); otherwise, it would
lead to a longer path.
If the smallest in {|1−
2|,|1−
2|,|1−
2|} is 1or 2,
in the corresponding direction, the sender and receiver must
be on two neighbor parallel lines or two parallel lines with
one line between them. We can use the corresponding grid to
calculate the number (𝑆,𝐷) of paths from to as follows:
(𝑆,𝐷) =
1if min ℘ = 0,
mid ℘+1 if min ℘ = 1,
mid ℘+1mid ℘+2
2if min ℘ = 2.
()
For example, in Figure ,thenumberofpathsfromto
is determined by two directions and ,as|1−2|=2and |1
2|=3are smaller than |1−2|=5.etotalnumberofpath
alternatives from to is.However,onestepalongthe
direction of (it becomes either +1or −1) obviously leads
to a longer path. We present two more lemmas on direction-
related properties as follows.
Lemma 4. On a shortest path, the coordinate displacements
between and along the three directions satisfy
min +mid =max .()
Proof. e number of hops on a path dened by the correct
direction of min(℘)and mid(℘)is min(℘) +mid(℘),andthe
path traverses exactly min(℘)+mid(℘)+1dierent points. As
each node is intersected by three lines, each from one cluster,
one of the three lines must belong to the max(℘) cluster. It
follows that the path traverses min(℘) + mid(℘) + 1 lines
in the max(℘) cluster. Hence, the displacement of the two
line numbers (i.e., max(℘)) traversing and is equal to
theheightofthetree(min(℘) + mid(℘) + 1) − 1.Wehave
max(℘) =(min(℘)+mid(℘)+1)1 = min(℘)+mid(℘).
Lemma 5. e distance from source to destination is
(,)=min ℘+mid ℘. ()
Mathematical Problems in Engineering
e proof of Lemma  simply follows the denition of
(,)and Lemma .
In triangular WMNs, given a source-destination pair
and , the number of shortest paths satises the following
theorem.
eorem 6. When min(℘) > 0,thenumber(𝑆,𝐷) of paths is
determined in a grid of the two correct directions corresponding
to min(℘)and mid(℘)as follows:
(𝑆,𝐷) =(,)
min ℘, ()
which is the number of min(℘) combinations chosen from
(,)objects.
Proof. Without loss of generality, let and be the correct
directions. Note that is determined by min(℘), while is
determined by mid(℘). e coordinates of the next receiver
node would lead to one hop closer to the destination along
the direction or .
e number (𝑆,𝐷) of paths is equal to the number of
strings of ’s a n d ’s : is repeated min(℘) times, and is
repeated mid(℘)times in a permutation of min(℘)+ mid(℘)
elements, that is, (,) in Lemma .etotalnumberof
permutations is (,)! = (min(℘) + mid(℘))!,butwith
repetitions. Note that the same permutated strings can only
be counted once. e number of duplicated permutations is
min(℘)!and mid(℘)! in the direction of and ,respectively.
en, the total number of dierent paths is obtained by
dividing the total number of permutations by the number of
duplications in both directions:
(,)!
min ℘!×mid ℘! =(,)
min ℘=(,)
mid ℘. ()
For example, in Figure ,={2,3,5}with and being
the correct directions, where appears min(℘) =2 times and
appears mid(℘) = 3 times in every permutation. A valid
routing path is determined by the number of ’s and ’s as
well as their relative positions in the string. For example, ,
,andare all valid paths with two ’s and t h re e ’s .  e
path 1=,whichisinthe
set (,)of all shortest paths from to , can be expressed
as a constrained permutation of the two correct directions,
that is, .
Given the node coordinates, data packets are delivered
hop by hop along the correct directions, taking the receiver of
the current hop as the sender of the next hop, until max(℘)=
0.iscanbedonerecursivelyandmayhelpavoidthe
overhead of maintaining a routing table.
4.2.PathCountingwithChannelAssignment. For a given
path from to withaconstantnumberofavailableorthogo-
nal channels, we need to decide the number of feasible
channel assignment schemes for implementing this path by
using three channels.
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c1
c2
c3
c4
F : Channel assignment to realize a path from to .
To illustrate this problem, we show a path with a channel
assigned to each hop in Figure , which is derived from
Figure :→
𝑐2→
𝑐1→
𝑐3→
𝑐2→
𝑐1,wherethe
label 𝑖between two neighbor nodes is the channel assigned
to the corresponding link.
e directed edges distributed in dierent layers form a
feasible orthogonal channel assignment to a path from to .
A valid path realization allows simultaneous transmission of
all the component links on the path. For example, in Figure ,
links (,)𝑐2,(,)𝑐2,(,)𝑐1,(,)𝑐1,and(,)𝑐3can
coexist at the same time.
Let the correct directions be and .Givenorthogonal
channels, the rst hop has 𝜛
1channel choices, the second
hop has 𝜛−1
1channel choices, and the third hop has 𝜛−2
1
channel choices. en, the three selected channels can be
used repeatedly without interference; that is, the fourth hop
selects the same channel as the rst hop, and the h hop
selects the same channel as the second hop and so on.
Channel 𝑖canbeselectedatmost(,)/3times.
In realizing a path using three channels at time ,therst
three hops determine the channel assignment order in every
three downstream hops. For example, if the rst three hops
are arranged in a channel order of 2,1,and3, then the
second three hops should be assigned channels in the same
order. Otherwise, a realization of the path from to would
requiremorechannels.Forexample,with(,) = 4,ifthe
rst three hops use channels in the order of 2,1,and3,the
fourth hop cannot use channels 1and 3,butonly2or a new
channel.
Note that using  channels, the channel assignment
permutations of the rst three hops (i.e., 3!)areallpossible
schemes for downstream three-hop groups.
Lemma 7. When min(℘) =0,withorthogonal channels, the
number of valid channel assignment schemes for a given path
from source to destination using three channels at the same
time is
=
3×3!. ()
 Mathematical Problems in Engineering
Based on eorem  and Lemma ,wehavethefollowing
theorem.
eorem 8. When using three channels to realize a path, the
number of channel assignments for all shortest paths from to
is
(,)
min ℘⋅. ()
Proof. When min(℘) = 0, there is only one shortest path, and
the number of channel assignments for this path is obtained
by Lemma .
We focus on a gen e r a l c a s e wh er e m i n (℘) > 0.e
number of channel assignments can be counted in two
independent steps.
e rst step is to count all shortest paths from to in a
plane mesh. is step does not assign channels. For example,
Figure  shows one such path: →→→→
→.
e second step assigns channels to the selected path
using three channels without interference. For example,
Figure  shows the channel assignment for 1from to
in Figure :→
𝑐2→
𝑐1→
𝑐3→
𝑐2→
𝑐1.
Since the above two steps are independent, by the mul-
tiplication principle, the total number of path realization
schemes satises (). is result could be extended to
dierent numbers of channels assigned to one path.
5. Conclusion
We conducted a theoretical exploration on mathematical
models and combinatorial characteristics of MIMO WMNs.
For a single-channeled mesh, we designed a coordinate
scheme and a node identity assignment scheme and derived
the interference-free conditions. For multiradio multichannel
WMNs, we derived rigorous formulas to count the number of
shortest paths from source to destination.
It is of our future interest to nd some transformations
to generate new link groups from the known ones. Along
this direction, we plan to investigate the CPG vertex/edge
transitive properties for performance improvement.
Conflict of Interests
e authors declare that there is no conict of interests
regarding the publication of this paper.
Acknowledgments
e research is funded in part by Nature Science project of
Guangdong Province under Grant no. S and
China Scholarship Council no. . is research is
also partly sponsored by U.S. Department of Energy’s Oce
of Science under Grant no. DE-SC with University of
Memphis.
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... To simplify CA, Cao et al. suggested a virtual model with Cartesian product of graphs (CPG), which decomposes the complex layered structure [7]. Cao et al. developed a Cartesian product of graphs (CPG) model to simplify channel assignment. ...
... In the CPG model, the links of a path are distributed in various channel layers as shown in [7]. The links in one channel layer are collected in a greedy way to have sufficient interference-free links for an initial path and other interference-free links to support other paths. ...
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