European Journal of Scientific Research

ISSN 1450-216X Vol.22 No.2 (2008), pp.457-468

© EuroJournals Publishing, Inc. 2008

http://www.eurojournals.com/ejsr.htm

Hex-Cell: Modeling, Topological Properties and

Routing Algorithm

Ahmad Sharieh

Computer Science Department

King Abdulla II School for Information Technology

University of Jordan

E-mail: sharieh@ju.edu.jo

Mohammad Qatawneh

Computer Science Department

King Abdulla II School for Information Technology

University of Jordan

E-mail: drqat@yahoo.com

Wesam Almobaideen

Computer Science Department

King Abdulla II School for Information Technology

University of Jordan

E-mail: wesmoba@ju.edu.jo

Azzam Sleit

Computer Science Department

King Abdulla II School for Information Technology

University of Jordan

E-mail: azzam.sleit@ju.edu.jo

Abstract

This paper introduces a new class of interconnection networks referred to as Hex-

Cell. Amongst the attractive features of the proposed topology is the embedding capability

of topological structures such as linear array, ring, tree and mesh. Hex-cell employs an

efficient routing algorithm which requires less knowledge of the network interconnections

and brings about less communication cost. These features make it a suitable candidate for

general purpose applications. Hex-Cell is expandable in an incremental fashion with

minimal cost due to its recursive structure. The comparison of Hex-Cell with other

networks of interests in terms of degree, diameter, and network-cost establishes merits

when compared with others.

Keywords: Hex-Cell, Interconnection Topology, Parallel Computing, Routing

Hex-Cell: Modeling, Topological Properties and Routing Algorithm 458

1. Introduction

Parallel processing elements provide high processing power for solving problems in major application

areas such as image processing and scientific computing [1, 2, 9,10, 11]. The underlying

communication network and the matching of the algorithm with the network structure strongly affect

the performance of parallel machines. Consequently, some features in the network of parallel machine

are highly desirable such as minimal communication cost, efficient routing, and the capability of

embedding topological structures such as ring, linear array, tree, and mesh.

Among the wide variety of interconnection networks structures proposed for parallel computing

systems is hypercube which received much attention due to the attractive properties inherited in their

topology [3]. The embeddability, symmetry, regularity, strong resilience, and simple routing have

made hypercube superior to many other multicomputer networks. However, in a hypercube structure,

the number of communication ports and channels per processor increases with the increase of the

network size which is a disadvantage of the structure. Ideally, we would like to construct a new

network that combines features such as less communication, efficient routing and capability of

embedding other static topologies. This motivates us to develop a new interconnection network to meet

the demands of highly parallel systems.

Although, the hexagon topology was addressed as a 3D hexagonal network [4], the node degree

and number of links are still considered high. In this paper, we introduce a new network called Hex-

Cell which will have better node degree and number of links than the existing topologies. We compare

the proposed structure with other know topologies such as 3D hexagonal network, linear array, ring,

binary tree, and hypercube as shown in Table I.

Section 2 presents the features of Hex-Cell and compares them with those of other topologies.

Section 3 presents the routing algorithm of Hex-Cell and section 4 summarizes and concludes the

paper.

2. System Architecture

Throughout this paper, an interconnection network will be viewed as an undirected graph, in which the

vertices correspond to processors and edges correspond to the bi-directional communication links

between processing elements. We propose a hex-cell network based on the hexagon topology.

2.1. Definition of Hex-Cell Network

A Hex-Cell network with depth d is denoted by HC(d) and can be constructed by using units of

hexagon cells, each of six nodes. A Hex-Cell network with depth d has d levels numbered from 1 to d,

where, level 1 represents the innermost level corresponding to one hexagon cell. Level 2 corresponds

to the six hexagon cells surrounding the hexagon at level 1. Level 3 corresponds to the 12 hexagon

cells surrounding the six hexagons at level 2 as shown in figure 1. The levels of the HC(d) network are

labeled from 1 to d. Each level i has N

i

nodes, representing processing elements and interconnected in

a ring structure.

459 Ahmad Sharieh, Mohammad Qatawneh, Wesam Almobaideen and Azzam Sleit

Figure 1: (a) HC (one level) (b) HC (two levels) (c) HC (three levels)

Lemma 1:

In a Hex-Cell network, the number of nodes at level i is:

N

i

= 6(2i – 1)

Proof:

Using the previous definition of HC(d), it can be easily seen that at level 1, N

1

= 6 since there is a

single hexagon cell with six vertices. Level 2 introduces six hexagon cells (one at each side of the

hexagon at level 1). It can be seen that level (i+1) has 12 nodes in addition to corresponding nodes to

those at level i. Therefore,

N

i

= 6+ (i – 1)* 12 = 6(2i – 1)

■

Lemma 2:

The total number of nodes in a Hex-Cell network HC(d) is:

N = 6d

2

Proof:

Using Lemma 1, the total number of nodes in a Hex_Cell network with d levels can be expresses as

follows:

(1)

Therefore,

N = 12 ∑ i – 6 ∑ 1 = 6d(d+1) – 6d = 6d

2

■

Equivalently, we can conclude that the number of levels of HC needed to accommodate N

nodes is estimated to be √(N/6).

Hex-Cell: Modeling, Topological Properties and Routing Algorithm 460

Table 1: Comparison of parameters of several topologies.

Topology Name

Maximum

Node Degree

Diameter

Number of

Links

Bisection

Bandwidth

Remarks on network size

Hex-Cell 3 4√(N/6)-1 3N/2 -3√(N/6) 2√(N/6) N is the number of nodes

Hypercube log

2

N log

2

N log

2

N (N/2) N/2 N is the number of nodes

Binary Tree 3 2(log

2

N -1) N-1 1 N is the number of nodes,

Linear Array 2 N-1 N-1 1 N nodes

Ring 2 N/2 N 2 N nodes

2D-Torus 4 2(r/2) 2N N/2 r × r torus where r = √N

Cube-Connected Cycles 3 2k-1+[k/2] 3N/2 N/(2k) N= k × 2

k

nodes with a cycle length k ≥ 3

3D Hexagonal 6 1.16 √N 3N -8.66√N 2.32 √N N is the number of nodes

Node Degree: The node degree on an interconnection network is defined as the maximum number of

edges (links or channels) that a node can have in the network. It is a desirable feature in an

interconnection that the number of ports does not grow at the same rate as a function of the number of

nodes in the network. In other words, a network topology which secures constant node degree is highly

desirable. Constant node degree facilitates modularity in building blocks for scalable systems [5,7,8]. It

can be easily seen that the node degree of a Hex-Cell with depth 1 (i.e. HC(1)) is 2 and that for HC(d)

is 3, for d > 1. Figure 2 compares node degree against network size of the HC with those of hypercube,

linear array, binary tree and 2D-torus. It is worth noting that the degree of the HC is constant when d >

1 and less than the degree of hypercube and falls in between of linear array and 2d-torus.

Figure 2: Degree against the network size

Diameter: The diameter (D) of a network is the maximum shortest path between any two nodes

[5,7,8]. The path length is measured by the number of links traversed. The network diameter indicates

the maximum number of distinct hops between any two nodes, thus providing figure of communication

merit for the network. Therefore, the network diameter should be as small as possible. It will not only

reduce the traveling time for messages, but also minimize message density in the links of the network.

The diameter of HC(d) is 4d-1. Figure 3 compares the diameter against network size of the HC with

that of hypercube, linear array, binary tree and 2D-torus respectively.

461 Ahmad Sharieh, Mohammad Qatawneh, Wesam Almobaideen and Azzam Sleit

Figure 3: Diameter against network size

Number of Links: The number of links in HC(d) is quantified in the following lemma/

Lemma 3:

The number of links in HC(d) is L = 9d

2

– 3d

Proof:

The number of nodes of HC at level i is N

i

= 6(2i -1). Each node represents a processing element and

interconnected in a ring topology. The number of links at each level is 6(2i – 1) + 6(i-1), or

equivalently,

L

i

= 18i – 12 (2)

Consequently, the total number of links in HC(d), L is equal to:

(4)

Figure 4 compares the number of links against network size of the HC with that of hypercube,

and binary tree. As shown in Figure 5, we can see that the total number of links for HC is less than

hypercube but greater than binary tree.

Hex-Cell: Modeling, Topological Properties and Routing Algorithm 462

Figure 4: The number of links against network size

Bisection width: when a given network is cut into two equal halves, the minimum number of edges

(channels) along the cut is called the channel bisection width b [5,7,8]. In case of a communication

network, each edge corresponds to a channel with w bit wires. Therefore, the wire bisection width is B

= b*w. This parameter B reflects the wiring density of a network. When B is fixed, the channel width

(in bits) w=B/b. Thus, the bisection width provides a good indicator of the maximum communication

bandwidth along the bisection of a network.

Lemma 4:

The bisection of HC(d) is 2d.

Proof:

It follows from the previous discussion. ■

3. Routing in Hex-Cell Network

Communication or data routing is the most fundamental function of interconnection networks. Data

routing is the act of moving information across an interconnection network from a source to a

destination [6]. Various routing modes have been designed. These modes include point-to-point

routing, one-to-many broadcasting, many-to-many broadcasting, personalized communication and data

gathering. This paper focuses on developing optimal point-to-point routing algorithm for Hex-Cell

networks. Routing for this mode will be shown to be optimal in time and space.

3.1. Point-to-Point Routing for HC.

An optimal routing algorithm in any network would route information or message from a source to a

destination along a shortest path. The routing algorithm can be centralized or distributed. In centralized

routing, a single processing element determines the shortest path form a given source node to a given

destination node. Then, the message is sent along that path. Finding the shortest path by a single central

processing element is very slow and leads to a major bottleneck. On the other hand, if every source

node takes the responsibility of computing the shortest path to the destination and sends that path along

with the message to guide it through the intermediate nodes, the bottleneck problem is avoided, but

traffic is increased and routing remains relatively slow.

In distributed, routing, however, all intermediate nodes on the shortest path cooperate to find

the shortest path using the destination address. Therefore, each intermediate node needs only the

463 Ahmad Sharieh, Mohammad Qatawneh, Wesam Almobaideen and Azzam Sleit

destination address to determine which neighbor falls on the shortest path to a given destination. In the

routing algorithm, each intermediate node determines, in constant time, which of its neighbors will

receive the message.

The routing of a message can be viewed as a sequence of changes made on the source address

label to become the destination address label. These changes are done at every intermediate node on

the path. When the message is received by an intermediate node, it will consider itself as a new source.

Due to the recursive structure of the Hex-Cell, routing can be done easily in constant time and space.

3.2. Proposed Routing- Algorithm

In order to explain the following algorithm, we use the level numbering scheme as follows. Each node

in the HC is identified by a pair (X.Y), where X denotes the line number in which the node exists, and

Y denotes the location of the node in the line as in figure 5. A node with the address 1.1 is the first

node that exists at line number 1. 1.2 refers to the second node that exists at line number 1, and so on.

Figure 5: Addressing nodes in HC.

In the proposed routing algorithm for HC as shown in Figure 6, the movement from one node to

another node can be done by using one of the following three cases:

CASE 1:

If (X

s

> X

d

)

moveUp(X

s

,Y

s

,X

d

,Y

d

).

Here we have two directions, moveUp/Left-to-Right, and moveUp/ Right-to-Left.

CASE 2:

If (

X

s

<

X

d

)

moveDown

(X

s

,Y

s,

X

d

,Y

d

)

Here we have two directions, moveDown/ Left-to-Right and moveDown/Right-to-Left.

CASE 3:

If (X

s

= X

d

)

moveHorizontal(X

s

,Y

s

, X

d

,Y

d

).

Here also we have two directions, moveHorizontal/Left-to-Right and moveHorizontal/Right-

to-Left.

Where, X

s

is the line number of source node, Y

s

is the location of source node in line the

number, X

d

is the line number of destination node, and Y

d

is the location of destination node in the line

number. One of the previous cases will be called recursively until the destination has been reached.

Hex-Cell: Modeling, Topological Properties and Routing Algorithm 464

Figure 6: Routing algorithm for HC(d)

3.3. Discussions via examples

The following examples explain the above routing cases. For this purposed, we consider a network of a

Hex-cell topology with 4 levels; i.e. HC(4) as shown in Figure 7.

465 Ahmad Sharieh, Mohammad Qatawneh, Wesam Almobaideen and Azzam Sleit

Figure 7: HC(4) topology with labeled nodes and levels

moveUp / Left-to-Right

Let (X

s

,Y

s

) = (5,6) be the source node and (X

d

,Yd) = (3, 9) be the destination node. When executing

the routing algorithm, CASE 1 will be applied (moveUp / Left-to-Right). To reach the destination, the

algorithm goes through six steps; i.e., (5,6)→ (5,7) →(4,7) → (4,8) → (3,7) → (3,8) → (3,9), as shown

in Figure 8.

Figure 8: MoveUp/ Left-to-right

moveUp / Right-to-Left

Let (8,9) and (4,8) be the source and destination nodes respectively. CASE 1 of the routing algorithm

will be applied (i.e., moveUp / Right-to-Left) going through the following steps: (8,9)→ (7,10)

→(7,9) → (6,10) → (6,9) → (5,10) → (5,9) → (4,9) → (4,8), as shown in Figure 9.

Hex-Cell: Modeling, Topological Properties and Routing Algorithm 466

Figure 9: MoveUp / Right-to-Left

moveDown/ Left-to-Right

Let (1,7) be the address of the source node and assume that it is trying to send a message to the

destination node (4,12). Applying the algorithm in Figure 7 will trigger CASE 2 (moveDown / Left-to-

Right). To reach the destination the algorithm passes through five steps; i.e., (1,7)→ (2,8) →(2,9) →

(3,10) → (3,11) → (4,12), as shown in Figure 10.

Figure 10: moveDown / Left-to-right.

moveDown / Right-to-Left

Let (4,9) and (7,4) be the address of the source and destination nodes respectively. CASE 2 of the

routing algorithm (i.e., moveDown / Right-to-Left) will bring about the following path: (4,9)→ (5,9)

→(5,8) → (6,7) → (6,6) → (7,5) → (7,4), as shown in Figure 11.

Figure 11: movedown / Right-to-Left.

467 Ahmad Sharieh, Mohammad Qatawneh, Wesam Almobaideen and Azzam Sleit

moveHorizontal / Left-to-Right

Let (2,4) and (2,7) be the source node and destination nodes. Case 3 (i.e., moveHorizontal/Left-to-

Right) will be triggered taking us through the following path: (2,4)→ (2,5) →(2,6) → (2,7), as shown

in Figure 12.

Figure 12: Horizontal / Left-to-Right

moveHorizontal / Right-to-Left

For the source (X

s

,Y

s

) = (8,4) and destination (X

d

,Y

d

) = (8,1), CASE 3 will be used. Applying the

algorithm, the case will be 3 (moveHorizontal/Right-to-Left). To reach the destination we applied the

algorithm and passed through 3 steps, (8,4)→ (8,3) →(8,2) → (8,1), as shown in figure 13.

Figure 13: moveHorizontal / Right-to-Left

4. Conclusion

This paper introduces a new interconnection topology called Hex-Cell. The proposed architecture has

some of the interesting characteristics of hypercube, binary tree, linear array, star and 2D-torus. The

degree and total number of links of the HC is less than those of hypercube. In hypercube, the node

degree of each node is a logarithmic function of the total number of nodes, which is a drawback of the

topology. Consequently, the hypercube topology is not a good candidate for interconnection network

for a very large parallel computer that might contain hundreds of thousands of nodes due to limitations

concerning integrated circuit technology and port number. We proposed a new interconnection network

suitable for large parallel computers, that can connect hundreds of thousands nodes with 3 links per

node.

The proposed topology has the ability to efficiently simulate programs written for architectures

such as linear arrays, rings and meshes. The Hex-Cell topology can easily embed these structures into

it. Also, the paper proposed an efficient routing algorithm for HC which requires no detailed

knowledge of the network, and can be utilized in wide range network applications such as mobile

systems, and Ad-Hoc mobile networks.

Hex-Cell: Modeling, Topological Properties and Routing Algorithm 468

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