Mapping Hex-Cells into Two-Dimension Mesh Network
Abdulelah Saif
*
Department of Computer Science, King Khalid University, Dhahran Al-Janoob, K.S.A.
Keywords: Parallel Computing, Interconnection Networks, 2D Mesh, Hex-Cells, Tree-Hypercube, Mapping.
Abstract: Graph mapping is an important aspect for interconnection networks used for communication between
processors in parallel systems. Some parallel algorithms use communication structures which can be
represented by Hex-Cells HC. In order to run these algorithms on Two-Dimension 2D Mesh multiprocessor
system, without changing the current topology and the running application, their communication graphs
need to be embedded into 2D mesh. In this paper, we have developed an algorithm for embedding Hex-
Cells HC(i) into 2D mesh M(2i,4i-1), where i = 1,2,3,…..i.e. To measure the efficiency of the algorithm, a
comparison is done between 2D Mesh and Tree-hypercube in terms of dilation, congestion and expansion.
As a result, the embedding of Hex-cells into 2D Mesh has dilation 1, congestion 1, expansion (4i-1)/3i,
where i is the level of HC which is better than Tree-hypercube. Moreover, 2D Mesh embeds hex-cells for
any level whereas Tree-hypercube embeds hex-cells for two levels.
*
https://mysite.kku.edu.sa/site/absaif/home
1 INTRODUCTION
Chang and Chen (1997) state that one of the most
important characteristics of any interconnection
network is that it should be able to simulate any
other interconnection networks without any costs.
This feature puts such interconnection network a
good candidate for working in general purpose
parallel machine. Chang and Chen(1997) and Xinbo
at al (2018),and Peng at al (2018) state that graph
embedding is an important feature of parallel
computing in order to convert from one network to
another so that every element and edge in the guest
graph is transformed into a node and edge in the host
graph. This embedding has two benefits, the first
reduces the time that processes spend by exchanging
information among them, and the second reduces the
total time because some processes are busy while
others are not. This enhances the communication
between processes as well as overcoming the
congestion problem while the processes are
communicating with each other. Additionally, there
is also an important advantage of embedding, which
is: If the guest graph is transferred to the host graph,
the host graph can simulate the guest graph work
(execute tasks in parallel) without any losses. If the
guest graph is included in the host graph and there is
less delay, less congestion, and less expansion, then
the inclusion is optima. Thus, the goal of the
embedding is to minimize the dilation, the
congestion and the expansion cost. Reducing delay
leads to reduced processes time and reduced
congestion leads to reduced pressure in the host
graph and reduced expansion reduces the hardware
required in the host graph.
2D mesh is the popular general purpose networks
with fixed node degree Emad (2008)) symmetry
Grama at al (2003), and has the ability to embed
other regularly networks. These characteristics of the
2D Mesh network make it able to map other
networks efficiency. The problem of mapping Hex-
Cells HC(d) network into 2D-Mesh has not attracted
the attention of researchers towards it.
In this paper, We have developed an algorithm
for embedding hex-cells HC(i) into 2D mesh M (2i,
4i-1), where i = 1,2,3,…..i.e. To measure the
efficiency of the algorithm, a comparison is done
between 2D Mesh and Tree-hypercube in terms of
dilation, congestion and expansion. As a result, the
embedding of Hex-cells into 2D Mesh has dilation 1,
and congestion 1, expansion (4i-1)/3i, where i is the
level of HC which is better than Tree-hypercube.

Moreover, 2D Mesh embeds hex-cells for any level
whereas Tree-hypercube embeds hex-cells only for
two levels.
Section 2 presents related works, section 3
presents meshes, section 4 presents hex-cell
network, section 5 presents embedding hex-cells into
2d mesh, section 6 presents embedding hex-cells
into 2d mesh with wraparound link(torus), and
section 7 summarizes and concludes the paper.
2 RELATED WORKS
Mesh is one of the most commonly used
interconnection networks and, therefore, embedding
between different meshes becomes a basic
embedding problem. Not only does an efficient
embedding between meshes allow one mesh-
connected computing system to efficiently simulate
another, but it also provides a useful tool for solving
other embedding problems. This work shows an
embedding of an s1* t1 mesh into an s2 * t2 mesh,
where si <= ti (i = 1, 2), s1t1 = s2t2, such that the
minimum dilation and congestion can be achieved
and presents a lower bound on the dilations and
congestions of such embeddings for different cases.
Also, the work presents an embedding with dilation
└s1/s2┘ + 2 and congestion └s1/s2┘ + 4 for the
case s1 =>s2, both of which almost match the lower
bound ┌s1/s2┐. Finally, for the case s1 < s2, the
work presents an embedding which has a dilation
less than or equal to 2* sqrt(s1), Shen (1997).
Sang and Hyeong (1996) considered the problem of
embedding complete binary trees into meshes using
the row-column routing and obtained the following
results: a complete binary tree with 2p-1 nodes can
be embedded (1) with link congestion one into a
9/8(√2p)×9/ 8(√2p) mesh when p is even and a
√(9/8*2p)×√(9/8*2p) mesh when p is odd, and (2)
with link congestion two into a √(2p)×√(2p) mesh
when p is even, and a √(2p-1)×√(2p-1) mesh when p
is odd . Yang at al (2008), state that embedding torus
in hexagonal honeycomb torus, states that a number
of parallel algorithms admit a static torus-structured
task graph. Hexagonal honeycomb torus (HHT)
networks are considered as good candidates for
interconnection networks. To execute a torus-
structured parallel algorithm efficiently on an HHT,
it is necessary to include the tasks to processors such
that the communication overhead is minimum. This
paper showed that a (3n, 2n) torus can be included
into an nth-order HHT with congestion 4, dilation 3,
expansion 1 and load factor 1. Consequently, a (3n,
2n) torus task graph can be executed on an nth-order
HHT efficiently using a parallel algorithm. In
Michael (2008), states that an undirected source
graph G was included in a host graph EM. This
paper presented an algorithm which was showed
how to map G into EM with time and space O(|V |2)
using the new ideas of islands and bridges. An island
is a subgraph in the host graph which was mapped
from one node in the guest graph while a bridge is
an edge connecting two islands which was mapped
from one edge in the guest graph. This work was
motivated in real applications related to quantum
computing and there was a need to map source
graphs efficiently in the extended grid. CAHIT
(1998) state that, a cubic tree is a tree in which all its
internal vertices are of degree three except pendent
vertices. This paper explores embedding cubic trees
into rectangular grid of minimum size such that the
edges are either horizontal or vertical segments. The
method is based on the minimum area embedding of
the three complete binary trees. The author gives
necessary and sufficient conditions for cubic trees
embeddable into a rectangular grid.
3 MESHES
In a Mesh network, the nodes are arranged in a k
dimensional lattice of width w, giving a total of wk
nodes or w*w in the case of 2D Mesh. Usually k=1
(linear array) or k=2 (2D array or 2D Mesh).
Communication is allowed only between
neighboring nodes. All interior nodes are connected
to 2k other nodes, Mehdipourm (2016). A two-
dimensional Mesh illustrated in Figure 1(a) is an
extension of the linear array to two-dimensions.
Each dimension has p nodes with a node identified
by a two-tuple (i,j). Every node (except those on the
periphery) is connected to four other nodes whose
indices differ in any dimension by one. A variety of
regularly structured computations map very
naturally to a 2D Mesh. For this reason, 2D Meshes
were often used as in parallel machines Grama at al
(2003). Some data transfers in 2D Mesh may require
2((w*w)½-1) links to be traversed. This can be
reduced by using wraparound connections between
nodes on same row or column as in Figure 1(b) or
when k=3 (three dimensions) as in Figure 1(c)
Mehdipourm (2016).

a)
b)
c)
Figure 1: Meshes :(a) k=2 , w=4 without wraparound,
(b) k=2 , w=3 with wraparound, and (c) k=3 ,w=3 with
wraparound.
4 HEX-CELL NETWORK
A hex-cell network which has a depth d is defined
by HC (d) and is constructed recursively using
hexagonal cells, each hexagon has six nodes. HC (d)
has d levels numbered from 1 to d, where, level 1 is
the level with one hexagon cell. Level 2 is the six
hexagon cells surrounding the hexagon at level 1.
Level 3 is the 12 hexagon cells surrounding the six
hexagons at level 2, as shown in Figure 2 the HC(d)
network levels are labeled from 1 to d. Each level i
has Ni nodes, which are the processing elements
interconnected in a ring structure. Addressing nodes
in HC is shown in Figure 3, Sharieh, et al, (2008).
Figure 2: (a) HC (one level) (b) HC (two levels) (c) HC
(three levels).
Figure 3: Addressing nodes in HC.
In HC(d), the number of nodes at level i is : Ni =
6(2i -1). The total number of nodes in HC(d) is : N =
6d2. The number of links in HC(d) is L = 9d2 -3d.
The diameter of HC(d) is 4d-1.
5 EMBEDDING HEX-CELLS
INTO 2D MESH WITHOUT
WRAPAROUND LINK
s * t 2D Mesh, M, is a network in which the nodes
are ordered in a Mesh of s rows numbered from 0 to
s - 1 from top to bottom, and t columns numbered
from 0 to t – 1 from left to right. The node at row i
and column j is defined as M(i,j). The 2D Mesh is
one of the most important networks and has received
extensive studies due to numbers of reasons, among
them many data structures especially arrays and
matrices, fit into a Mesh-connected system.
Moreover, some real multiprocessor computer
systems have been produced based on Meshes. The
study of mapping between meshes is used in many
applications. One application is to allow a mesh to
simulate other meshes of various ratios, which mean
matrices of various shapes can be efficiently mapped
to a mesh-connected system Shen (1997). Hex-cell
can be included into 2D Mesh by including hex-cell
nodes into 2D mesh nodes, and hex-cell edges into 2D
Mesh edges; without adding edges (i.e. dilation 1),

Map_Hex-Cell_Into_2 D_Mesh( int d) // d is depth of Hex_cell network
{
int node(2*d,4*d-1)
max_columns_in_row =( 4 * d – 1) //determine the maximum columns in each row
First_row_element=d-1 //determine the first element in each row
For j=1 to d-1
max_columns_in_row = max_columns_in_row -2 // end of loop
For i=1 to d-1
{
For j=1 to max_columns_in_row
{
node (i,j+ First_row_element) in 2D Mesh = node (i,j) in HC
if(i>1 and (link is found between node (i,j) and node (i-1,j-1) in HC) and d>1)
Connect Node (i, j) in 2D Mesh with node (i-1, j) in 2D Mesh through a link
// end of if statement
}// end of internal loop
First_row_element = First_row_element-1
max_columns_in_row = max_columns_in_row +2
}// end of external loop
For i=d to d+1
{
For j=1 to max_columns_in_row
{
node (i,j) in 2D Mesh = node (i,j) in HC
if(i>1 and (link is found between node (i,j) and node (i-1,j-1) in HC) and d>1)
Connect Node (i, j) in 2D Mesh with node (i-1, j) in 2D Mesh through a link
// end of if statement
}// end of internal loop
}// end of external loop
First_row_element = 1
For i=d+2 to 2*d
{
max_columns_in_row = max_columns_in_row -2
For j=1 to max_columns_in_row
Figure 4: The mapping algorithm of Hex-Cells HC(d) into 2D Mesh M(2d,4d-1).

{
node (i,j+ First_row_element) in 2D Mesh = node (i,j) in HC
if(i>1 and (link is found between node (i,j) and node (i-1,j-1) in HC) and d>1)
Connect Node (i, j) in 2D Mesh with node (i-1, j) in 2D Mesh through a link
// end of if statement
}// end of internal loop
First_row_element = First_row_element +1
}// end of external loop
}// end of Mapping function
Figure 4: The mapping algorithm of Hex-Cells HC(d) into 2D Mesh M(2d,4d-1) (cont.).
congestion 1 and with lower expansion. Smaller
dilation leads to shorter communication delay where
the host graph (2D Mesh) emulate the guest graph
(hex-cell), and smaller expansion leads to more
efficient utilization of the processor where the host
graph (2D Mesh) emulates the guest graph (Hex-
Cell). In the next section, the algorithm for mapping
hex-cell into 2D mesh, M (i,j) is presented.
5.1 The Proposed Algorithm for
Embedding Hex-Cells into 2D Mesh
We have designed an algorithm for embedding hex-
cell HC(d) into 2D Mesh M(i,j) , where i=2d, j=4d-1
where d=1,2,3,…; without having additional number
of edges when mapping edges of hex-cell (i.e.
dilation one and congestion one), with expansion 1
when d =1, expansion 1.16 when d =2, expansion
1.22 when d=3 and expansion 1.3 for any value of d
where d>10. Mapping nodes of hex-cell into nodes
of 2D Mesh is done by mapping the addresses and
edges of the hex-cell into the addresses and edges of
2D Mesh. Algorithm for embedding hex-cells into
2D Mesh networks is in Figure 4.
5.2 Examples on the Embedding
Algorithm, and Discussion
Example 5.2.1; mapping hex-cell HC(1) (Figure 5)
into 2D Mesh M(2,3). Figure 6 illustrates this
example. In figures of 2D Mesh, the addresses
within the nodes are for 2D Mesh and the addresses
above the nodes are for HC.
Example 5.2.2; mapping hex-cell HC(2)
(Figure 7) into 2D Mesh M(4,7), Figure 8 illustrates
this example.
Figure 5: HC(1).
Figure 6: Mapping of HC(1) into M(2,3).
Example 5.2.3; mapping hex-cell HC(3) (Figure 9)
into 2D Mesh M(6,11), Figure 10 illustrates this
example.
This process continues in the same manner for
mapping any level of hex-cells into 2D Mesh.
Figure 7: HC(2).

Figure 8: Mapping of HC(2) into M( 4,7).
Figure 9: HC(3).
Figure 10: Mapping of HC(3) into M( 6, 11).
5.3 A Note about Mapping Hex-Cells
into 2D Mesh
In the following table, we list the total number of
nodes in HC(d), M(2d,4d-1), and the number of
extra nodes in the embedding of hex-cell into 2D
Mesh.

Table 1: Number of extra nodes in embedding hex-cells into 2D Mesh.
D
Total number of nodes in HC(d) Total number of nodes in M(2d,4d-1)
Number of extra nodes in
embedding
1
6 6 0
2 24 28 4
3
54 66 12
4
96 120 24
5
150 190 40
Table 2: Comparison among TH and 2D Mesh after embedding HC into them.
d Total number
of nodes in
HC(d)
Total number
of nodes in
TH(2,2d)
Total number
of nodes in
M(2d,4d-1)
Number of extra nodes in
embedding HC(d) into
TH (2,2d)
Number of extra nodes in
embedding HC(d) into
M(2d,4d-1)
1 6 7 6 1 0
2 24 31 28 7 4
3 54 127 66 73 12
4 96 511 120 415 24
5 150 2047 190 1897 40
Figure 11: Comparison among TH and 2D Mesh after embedding HC into them.
5.4 Comparison between
Tree-Hypercube TH Qatawneh
(2008) and 2D Mesh
From the above table, and Figure 1, we notice that
when mapping hex-cells into 2D Mesh, the number
of extra nodes is less than the number of extra nodes
when mapping hex-cells into tree-hypercube and
therefore the expansion of embedding HC(d) into
M(2d,4d-1) is less than expansion of embedding
HC(d) into TH (2,2d).
6 EMBEDDING HEX-CELLS
INTO 2D MESH WITH
WRAPAROUND LINK
The most common topology attainable with nodes of
four links is the mesh or square grid. By connecting
the ends of the mesh around, a Torus is produced,
Wilson.
Hex-Cells can be embedded into 2D mesh with
wraparound link in the same way of embedding hex-
cells into 2D mesh without wraparound link by the
same algorithm mentioned above in section 5. But
the benefit of wraparound link in the 2D mesh lies in
the routing where the short path from the source to
0
500
1000
1500
2000
12345
Number of extra nodes in
embedding HC(d) into TH(2,d)
Number of extra nodes in
embedding HC(d) into
M(2d,4d-1)

the destination is less than that of 2D mesh without
wraparound link because the diameter of torus is less
than that of 2D mesh without wraparound link.
Example 6.1.1; mapping hex-cell HC(1) (Figure
12) into torus, M(2,3). Figure 13 illustrates this
example. In figures of torus, the addresses within the
nodes are for torus and the addresses above the
nodes are for HC.
Figure 12: HC(1).
Figure 13: Mapping of HC(1) into torus(2,3).
Example 6.1.2; mapping hex-cell HC(2)
(Figure 14) into 2D mesh M(4,7), Figure 15
illustrates this example.
Figure 14: HC(2).
Figure 15: Mapping of HC(2) into torus(4,7).
As before, this process continues in the same
manner for mapping any level of hex-cells into 2D
mesh with wraparound link (torus).
7 CONCLUSIONS AND FUTURE
WORK
One of the most important characteristics of any
interconnection network is that it should be able to
simulate any other interconnection networks without
any costs. This feature puts such interconnection
network a good candidate for working in general
purpose parallel machine. Graph embedding is an
important feature of parallel computing in order to
convert from one network to another so that every
element and edge in the guest graph is transformed
into a node and edge in the host graph.
In this paper, We have developed an algorithm
for including hex-cells HC(i) into 2D mesh M (2i,
4i-1), where i = 1,2,3,…..i.e. To measure the
efficiency of the algorithm, a comparison is done
between 2D Mesh and Tree-hypercube in terms of
dilation, congestion and expansion. As a result,
including Hex-cells into 2D Mesh has dilation 1,
congestion 1, and expansion (4i-1)/3i, where i is the
level of HC which is better than Tree-hypercube.
Moreover, 2D Mesh embeds hex-cells for any level
whereas Tree-hypercube embeds hex-cells only for
two levels.
For future work, we suggest mapping hex-cells
into other topologies that have diameter less than
that of hex-cells, such as X-Torus and An nth-order
HHT, and with lower dilation, congestion and
expansion as possible.
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