Routing in Hexagonal Networks under a Corner-Based Addressing Scheme

Abstract

In this letter, a new addressing scheme for hexagonal networks is proposed. Using the new addressing scheme, many routing algorithms designed for networks using square-based topologies such as mesh and torus can also be applied to hexagonal networks. Methods of applying the turn model to hexagonal networks are derived, with some new minimal and partial adaptive routing algorithms obtained. Simulations of the new routing algorithms under different working conditions are carried on hexagonal networks of various sizes. The results show that the proposed algorithms can offer lower packet delay and loss rate than the popular dimension order routing algorithm.

Full text

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IEICE TRANS. INF. & SYST., VOL.E89–D, NO.5 MAY 2006
1755
LETTER
Routing in Hexagonal Networks under a Corner-Based Addressing
Scheme∗
Huaxi GU†a), Jie ZHANG††b), Zengji LIU†, and Xiaoxing TU†, Nonmembers
SUMMARY
works is proposed. Using the new addressing scheme, many routing algo-
rithms designed for networks using square-based topologies such as mesh
and torus can also be applied to hexagonal networks. Methods of applying
the turn model to hexagonal networks are derived, with some new mini-
mal and partial adaptive routing algorithms obtained. Simulations of the
new routing algorithms under different working conditions are carried on
hexagonal networks of various sizes. The results show that the proposed
algorithms can offer lower packet delay and loss rate than the popular di-
mension order routing algorithm.
key words: routing algorithm, hexagonal network, addressing, turn model
In this letter, a new addressing scheme for hexagonal net-
1. Introduction
The Direct Interconnection Network (DIN) has been widely
usedinvarious areas,suchas parallelmulti-computer/multi-
processor systems and networks of workstations. Recently,
some companies use DIN in terabit routers. For example,
Avici Systems uses 3D torus as switching fabrics in their
terabit router AVICI TSR[1], while Pluris makes use of hy-
percube in TeraPlex20[2]. DIN can be represented as a
graph, where a processor (or a cluster of a few closely lo-
cated processors) is represented as a node and the commu-
nication channel between each two nodes is represented as
an edge. A survey of such networks can be found in [3],[4].
There are three possible tessellations of a plane with
regular polygons of the same kind: square, triangular and
hexagonal, corresponding to dividing a plane into regu-
lar squares, triangles and hexagons, respectively[5]. The
hexagonal network is based on triangular tessellation. It has
found applications in various fields such as chemistry[15],
computer graphics, image processing[9] and also wireless
ad hoc networks[16] and interconnection networks.
the context of interconnection networks, hexagonal network
was used in HARTS project[6] at the University of Michi-
gan. Unlike k-ary n-cube topology, which has been used in
many projects, the research on hexagonal networks is rather
limited[3]–[6]. The reason is that the square-based topol-
ogy facilitates the design of simple routing algorithms. For
In
Manuscript received April 12, 2005.
Manuscript revised September 23, 2005.
†The authors are with the State key lab of ISN, Xidian Univer-
sity, Xi’an, 710071 China.
††The author is with Dept. of Computing and Information Sys-
tems, Univ. of Luton, LU1 3JU, UK.
∗This work was supported by NSFC (No.60402040).
a)E-mail: hxgu@xidian.edu.cn
b)E-mail: jie.zhang@luton.ac.uk
DOI: 10.1093/ietisy/e89–d.5.1755
example, three partial adaptive routing algorithms derived
from the turn model are widely used in multi-computer sys-
tems[3],[4],[10]. It is veryeasy to implement them without
adding extra virtual channels. However, it is very difficult to
apply them to hexagonal networks based on the existing ad-
dressingschemesin[7]–[9]. Inthisletter, wepresentanovel
property of hexagonal networks and propose a new corner-
based addressing scheme. It is very convenient to describe
nodes and channels of hexagonal networks with the new ad-
dressing scheme. Hence, routing algorithms designed for
networks of square-based topology can be easily applied to
hexagonal networks.
This letter will stimulate further research on hexago-
nal networks. The rest of this letter is organized as fol-
lows. In Sect.2, we briefly introduce hexagonal networks
and present the new corner-based addressing scheme. In
Sect.3, how to use the turn model to derive new routing
algorithms for hexagonal networks is presented. Section 4
describes the simulation results and the performance of the
new routing algorithms obtained in Sect.3. Concluding re-
marks are given in Sect.5.
2.The Corner-Based Addressing Scheme
Hexagonal networks can be classified into hexagonal mesh
(H-Mesh) and hexagonal torus (H-Torus). An H-Mesh of
size 3 is illustrated in Fig.1(a) and an H-Torus can be ob-
tained from an H-Mesh by using the C-type wrapping pro-
posed in [7]. If we rotate an H-Mesh by 30◦clockwise,
we can find the novel property of a hexagonal network,
i.e., a hexagonal network of size t can be looked as three
Fig.1
An H-Mesh of size 3.
Copyright c ? 2006 The Institute of Electronics, Information and Communication Engineers

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IEICE TRANS. INF. & SYST., VOL.E89–D, NO.5 MAY 2006
(t − 1) × (t − 1) squared surfaces. For example, as shown in
Fig.1(b), an H-Mesh of size 3 can be looked as three sur-
faces of a 2 × 2 × 2 3D mesh (like a corner of the cubic
box). Using this property, a new corner-based addressing
scheme can be proposed, and the routing algorithms used in
square-based topology networks can be adapted to hexago-
nal networks.
Corner-Based Addressing Scheme: The H-Mesh is placed
in X–Y–Z frame and each node is labeled as (x,y,z). There
is at least one zero element in (x,y,z) because all the nodes
are onthe surfaces of the 3Dmesh. If two elements are zero,
the node must be on an axis.
The addressing scheme is named based on the fact that
the hexagonal network can be taken as a corner of the cubic
box. We can obtain the following remarks with the corner-
based addressing scheme.
Remark 1. A node (x,y,z) belongs to a hexagonal mesh or
a hexagonal torus of size t if and only if x ∗ y ∗ z = 0 and
0 ≤ x,y,z ≤ t.
Remark 2. For a hexagonal mesh of size t, the node N1=
(x,y,z) and the node N2 = (x?,y?,z?) are connected if and
only if |x−x?| = |y−y?| = z+z?+1 = 1 (on the X–Y surface)
OR |y − y?| = |z − z?| = x + x?+ 1 = 1 (on the Y–Z surface)
OR |z − z?| = |x − x?| = y + y?+ 1 = 1 (on the X–Z surface)
OR |x − x?| + |y − y?| + |z − z?| = 1 (other cases).
Remark 3. For a hexagonal torus of size t, the node N1=
(x,y,z) and the node N2 = (x?,y?,z?) are connected if and
only if the following equations can be satisfied.

z?= z +
t

axis, (x?,y?,z?) must satisfy Eq.(1); If N2is the neighbor of
N1in the negative direction of y axis, (x?,y?,z?) must satisfy
Eq.(2).
The details of the derivation can be found in [10],
which is omitted here due to space limit. The equations of
the four other neighbors in positive and negative directions
of X or Z axes can be derived in a similar manner.

z?= z +
t
x?= x+
z+
?y+1−ε(x·z)
t
?
× (t−1)
t

× (t−1)−ε(x·z)
y?= (y + 1 − ε(x · z)) mod t
?y + 1 − ε(x · z)
z+
?
?
× (t − 1)
−
?y+1−ε(x·z)
t
× (t−1)
t

× t−ε(x·z)(1)
If N2is the neighbor of N1in the positive direction of y
x?= x +
z +
?y − 1 + Υ
t
?
× (t − 1)
t

× (t − 1) + Υ
y?= (y − 1 + Υ) mod t
?y − 1 + Υ
?
× (t − 1)
−

z +
?y − 1 + Υ
t
?
× (t − 1)
t

?0
× t + Υ
(2)
where Υ = δ(y)ε((t − 1 − x)(t − 1 − z))
?0
ε(∆) =
∆ ≤ 0
∆ > 01
and δ(∆) =
∆ ? 0
∆ = 01
3.Routing in Hexagonal Networks
Turn model is an elegant technique for designing par-
tially adaptive routing algorithms without virtual channels.
It prohibits minimum number of turns to prevent cycles
in the channel dependency graph so that deadlock can
be avoided. Three partially adaptive routing algorithms,
namely negative-first, west-first and north-last, were de-
rived from the turn model for n-dimensional meshes. In the
following, methods of applying negative-first routing algo-
rithms to hexagonal network are shown, with two new algo-
rithms obtained, namelyNFHM (Negative First forHexago-
nal Mesh) and NFHT (Negative First for Hexagonal Torus).
North-last and west-first routing algorithms can be applied
to hexagonal networks in a similar way.
Figure 2(a) shows the turns allowed and prohibited
by negative-first routing algorithm in 2D mesh. The cor-
responding turns prohibited by NFHM and NFHT are de-
fined in Fig.2(b). Thus, to route in a negative direction, the
packets must start in the negative direction. We can define
NFHM routing function as follows:
Definition 1
Let GH-Mesh= (NH-Mesh, CH-Mesh) be an H-
Mesh, where NH-Mesh and CH-Mesh represent node set and
channel set respectively, let P be the packet set, and p ∈ P,
represents a single packet. ndst, ncur, nformer, nnext∈ NH-Mesh
where ndst, ncur, nformerand nnextare the destination node,
the current node, the former node and the next node respec-
tively, and ndst = dst(p). The NFHM is a partial adaptive
routing function, defined as:
NFHM(ncur, nformer, p)={nnext|(if<nformer,ncur> = (+)
then <ncur,nnext> = (+)) AND (|xdst− xnext| ≤ |xdst− xcur|)
AND (|ydst− ynext| ≤ |ydst− ycur|) AND (|zdst− znext| ≤ |zdst−
zcur|)}
where xdstis the x-coordinate of the destination node, and
xnext, xcur, ydst, ynext, ycur, zdst and znext are of the similar
meaning. <nformer,ncur> = (+)meansthe channel from node
nformerto ncuris in positive direction. The routing function
Fig.2
Turns prohibited by different algorithms.

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LETTER
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NFHM maps the current node, the previous node and the
destination node to the next node, nnext, on the route from
ncurto ndst. NFHM only chooses channels in the positive
direction for the packets that arrives at the current node in
positive direction. Hence, turns shown as dashed lines in
Fig.2 can be prohibited. At the same time, according to def-
inition 1 the packet would be closer to the destination after
each hop.
As for H-Torus, NFHM is not deadlock free because of
the existence of the wraparound channels. With wraparound
channels all the nodes form a cycle. Deadlock may oc-
cur at heavy traffic load especially for small size H-Torus.
This issue has been ignored in the previous research[7],[9].
We use two virtual channels[11] (VC0 and VC1) to solve
this problem. The virtual channel assignment is defined as
Rule 1.
Definition 2
Let GH-Torus= (NH-Torus,CH-Torus) be an H-
Torus and wraparound channel set Cwraparoundis defined as
follows:
Cwraparoud = {<(0,0,1),(0,0,0)>,<(1,0,0),(0,0,0)>,
<(0,1,0),(0,0,0)>}
Rule 1
Use VC 0, if the wraparound channel has not been
used; use VC 1, if the wraparound channel has been used.
By adding Rule 1 to Definition 1, we can obtain the
routing algorithm NFHT as follows.
Algorithm NFHT (ncur, nformer, p)
{
If ncur= ndestSend p to the local node and Exit;
else
{
nnext= NFHM(ncur, nformer, p);
/*Obtainsuitablenode*/
if Vtag= 0 then VC num = 0
else if Vtag= 1 then VC num =1;
send p to the node nnexton the virtual chan-
nel VC num;
}
EXIT.
/* Rule 1*/
}
VC num is the virtual channel number 0 or 1. Vtagis
a field in the packet and initially set to 0 when the packet
is generated. If the wraparound channel has been used, the
Vtagwill be set to 1.
Function NFHM(ncur, nformer, p) helps choosing suit-
able nodes to forward packet p. As Definition 1 shows, the
nodes should not form the prohibited turns shown in Fig.2.
Hence, deadlock can be avoided. As NFHM(ncur, nformer, p)
selects nodes that are closer to the destination, livelock will
not occur.
The Dimension Order (DO) routing algorithm[3] is
widely used in many real products. However, it allows use
of only one fixed physical path from the source to desti-
nation, even there exist many alternative paths. This may
lead to contention for the same network channel by multi-
ple packets, while other channels on alternate paths remain
idle. The NFHT algorithm can solve this problem by pro-
viding enough adaptability for the packets. When NFHT is
employed, there can be multiple outgoing channels for the
incoming packets to be routed. The choice can be made ran-
domly or based on the current levels of contention. Another
advantage of NFHT is that it can tolerate faults in the net-
works. The DO routing algorithm cannot handle faults be-
cause of its non-adaptive nature. In networks implementing
theDO algorithm, a singlefaultmaydisablecommunication
channels between many pairs of nodes. However, NFHT is
able to continue to work with faults present, as it has multi-
ple routing choices for the packets.
4.Simulation Results
OPNET[12] is used to evaluate the proposed routing algo-
rithms. Simulations are carried out on H-Torus and H-Mesh
hexagonal networks. IP traffic is used for simulation as the
future telecommunications and computer networks (LAN)
will be dominated by IP networks.
Only store-and-forward switching mechanism[3] is
used in the simulations, but the algorithms are also suitable
for wormhole switching[4] and virtual-cut-through switch-
ing[13]. The nodes generate packets at time interval that
follows the negative exponential distribution. Two virtual
channels are used for each algorithm.
The traffic patterns used in simulations include the uni-
form and the hotspot traffic[3],[9]. In uniform traffic pat-
tern, each node sends packets to all other nodes with the
same probability. In hotspot traffic pattern, one or more
nodes are designated as the hotspot nodes, which receive
hotspot traffic in addition to the regular uniform traffic.
Without loss of generality, node (0, 0, 0) has been chosen
as hotpot with 5% hotspot traffic.
Packet length distribution is a specific distribution SP
(Size and Percent) that is based on the IP (Internet Protocol)
packet size and percentages sampled over a two-week pe-
riod[14]: 40bytes (56 percent of all traffic), 1500bytes (23
percent), 576bytes(16.5percent)and52bytes(4.5percent).
Packets arriving at a destination node are consumed
immediately. If the node buffer is full, the packet will be
dropped. The performance of the routing algorithms is eval-
uated in terms of two main QoS (Quality of Service) met-
rics: ETE (End To End) delay and packet loss rate. The
ETE delay is defined as the average time from the packet
generation to the time when it reaches the destination. The
simulationmeasures thenumberofpacketsinjected(Pi)into
the network and the number of packets dropped (Pd) within
the network. The loss rate is defined as the ratio of Pdto Pi.
We only show the results on H-Torus of size 3 below.
Larger size of hexagonal torus and H-Mesh were also tested
with similar results obtained and omitted due to space limit.
We compare NFHT with the popular DO algorithm[3].
DO routes the packet first along the lowest dimension and
then along higher dimension until the packet arrives at the
destination. But one point that needs to be paid attention
to is that the routing path must be on the surface of the 3D
mesh, i.e. there is at least one zero element in the node co-

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Fig.3
patterns.
ETE delay against offered load under uniform and hotspot traffic
Fig.4
traffic patterns.
Packet loss rate against offered load under uniform and hotspot
ordinates (x,y,z). For example, if the packet is transferring
from (1, 0, 1) to (1, 1, 0), the path it will take is (1, 0, 1) →
(1, 0, 0) → (1, 1, 0) but not (1, 0, 1) → (1, 1, 1) → (1, 1, 0)
because H-mesh is only a hollow part of a 3D mesh.
Figures 3 and 4 illustrate the ETE delay and through-
put performance of the two routing algorithms under differ-
ent traffic patterns. Under both traffic patterns, the adap-
tive NFHT performs better than DO. For example, under
uniform traffic mode, DO saturates when the offered traffic
is about 5.5Mbps/node whereas NFHT saturates at a traf-
fic load of 9Mbps/node. It is obvious that similar conclu-
sions can be reached when hotspot traffic presented. DO
saturates earlier and yields higher ETE latency because it
cannot distribute traffic evenly among the links around the
hotspots. NFHT algorithm performs better on balancing the
hotspot traffic. The reason is because NFHT makes the traf-
fic choose alternative routes to bypass the hotspot, thus low-
ering the burden of the hotspot. For the full range of the
traffic, NFHT offers lower latency and smaller loss rate than
DO. Therefore, the simulations show that NFHT can in-
crease the network throughput and decrease the average la-
tency under various conditions.
5.Conclusion
In this letter, a novel property of hexagonal network has
been found. By using such property, many routing algo-
rithms designed for mesh or torus can be applied to hexag-
onal networks. This facilitates the design of routing algo-
rithms for hexagonal networks and will stimulate further
research on hexagonal networks. We only show the meth-
ods to apply unicast routing algorithms to hexagonal net-
works. Under the corner-based addressing scheme, broad-
castingis veryeasy toimplement. It canbecarriedoutin the
three surfaces of the 3D mesh. Therefore, applying popular
broadcasting algorithms designed for mesh-like topology to
hexagonal networks is our future research topic.
Acknowledgments
The authors are extremely grateful to the associated editor
and the anonymous reviewers, who offered the authors very
helpful suggestions to improve this paper. The authors also
thankDr. KunWang and Dr. Jun Yangfor their helpful com-
ments.
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