Hamiltonian Cycle Within Extended Otis Arrangement Network

Published Date: 02 Nov 2017

Awwad Ahmad

University of Petra

Faculty of IT, CS Dept

Amman - Jordan

[email protected]

Abstract—In this paper we propose the construction of a Hamiltonian cycle in the Extended OTIS-Arrangement network. The Extended-OTIS-Arrangement network has many attractive topological properties Including regular degree, semantic structure, low diameter, and ability to embed graphs and cycles. As constructing a Hamiltonian cycle is one of the important advantages for any topology due to the importance of broadcast messages between different nodes.

The main contribution of this paper is to propose the Hamiltonian Cycle algorithm which will make the paper more realistic as realistic candidate for HSPC topology. Throughout the paper we present an algorithm which constructs a Hamiltonian cycle in the extended OTIS-Arrangement interconnection network. Besides highlighting on the topological properties of the extended OTIS-Arrangement we have introduced detailed examples to show how a Hamiltonian cycle is constructed.

Keywords—Parallel and Distributed System; Interconnection

Networks; Extended OTIS-Arrangement; Topological

Propertie;, Routing Algorithm; Hamiltonian Circuit.

I. INTRODUCTION AND MOTIVATION

Recently, there has been an increasing interest in a class of interconnection networks called Optical Transpose Interconnection Systems "OTIS-networks" [4, 5, 10, 24]. This type of networks was proposed by Marsden et al [11]. More attractive results for the OTIS have been reported in [1, 2, 8, 9, 15]. Since the achievable terabit throughput at a reasonable cost makes the OTIS a strong competitor to the electronic alternatives [7, 23, 27]. These encouraging findings prompt the need for further testing of the suitability of the OTIS for real-world parallel applications.

A number of computer architectures have been proposed in which the OTIS was used to connect different processors [13, 15, 18, 27]. Krishnamoorthy et al [7] have shown that the power consumption is minimized and the bandwidth rate is maximized when the OTIS computer is partitioned into N groups of N processors each. [1, 2, 3, 6]. Furthermore, the advantage of using the OTIS as optoelectronic architecture lies in its ability to maneuver the fact that free space optical communication is superior in terms of speed and power consumption when the connection distance is more than few millimeters [7, 11]. In the OTIS, shorter (intra-chip) communication is realized by

electronic interconnects while longer (inter-chip) communication is realized by free space interconnects [11, 12, 14, 27]. Using Arrangement as a factor network will yield the Extended OTIS-Arrangement in symbolizing this network.

There are some studies in literature that address more improvements on the topological properties of the OTIS interconnection network [21, 22, 23, 27]. In [23], an Extended OTIS-Arrangement topology was introduced; the new proposed network has many good topological features such as regular degree, semantic structure, low diameter, and ability to embed graphs and cycles.

Utilizing the attractive properties of Arrangement network beside the OTIS technology in proposing the Extended-OTIS -An,k. This paper introduces the construction of Hamiltonian cycle for the Extended OTIS-Arrangement which is constructed by "multiplying" an arrangement topology by itself [17, 19]. The set of vertices is equal to the Cartesian product on the set of vertices in the factor arrangement network [3, 6, 16, 19]. The set of edges E in the OTIS-Arrangement consists of two subsets, one is from the factor Arrangement, called arrangement-type edges, and the other subset contains the transpose edges. Throughout this paper the terms "electronic move" and the "OTIS move" (or "optical move") will be used to refer to data transmission based on electronic and optical technologies, respectively.

The Extended OTIS-Arrangement network outperforms the OTIS-Arrangement in many feature including semantic structure, regularity, smaller diameter, and other exceptional properties [23].

Embedding of topologies with regular structure and also irregular structure has been broadly investigated in the literature, e.g [25]. Embedding structures and other topologies is one of the key features of interest in interconnection networks. The load of an embedding is the maximum number of nodes in a graph assigned to any node in the embedded graph. We are interested in this research only in one -to-one mappings to embed a Hamiltonian cycle, so the load of any embedding is one [25, 26].

In the mathematical field of graph theory, a Hamiltonian path is a path in an undirected graph which visits each node exactly once. A Hamiltonian cycle is a cycle in an undirected graph which visits each node exactly

once and also returns to the starting node. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem [21, 27].

The Hamiltonian path seeks whether there is a route in a directed network from a starting node to an ending node, visiting each node exactly once. This challenge has inspired researchers to broaden the definition of computer computations. The Hamiltonian problem arises in many real world applications [27].

This paper proposes a theoretical study on the routing properties in general and embedding Hamiltonian cycle in specific for the Extended OTIS -Arrangement due to its attractive properties. Section 2 presents notations and preliminary definitions.

Details of constructing a Hamiltonian cycle in the Extended OTIS-Arrangement topology will be discussed in section 3. Section 4 concludes the paper.

II. PRELIMINARY AND DEFINITIONS

The arrangement graph which was proposed by Day [3] has been shown as an attractive alternative to star network [20].

Definition 1: The (n,k)-arrangement graph An,k = (V, E) is an undirected graph given by:

V= { p1 p2… pk pi in<n> and pi≠pj for i≠j} = Pkn ,

And

E = {(p,q) p and q in V and for some i in 〈k〉, pi≠qi and pj = qj for j≠i}.

That is the nodes of An,k are the arrangements of k

elements out of n elements of 〈n〉, and the edges of An,k connect arrangements which differ exactly in one of their k

positions. For example in A5,2 the node p = 23 is connected

to the nodes 21, 24, 25, 13, 43, and 53. An edge of An,k connecting two arrangements p and q which differ only in

one position i, it is called i-edge . In this case, p and q is called the (i,q)-neigbour of p.

The OTIS-Arrangement network is one of the known networks which have been proposed to be used in real life systems [20]. In the OTIS-Arrangement the notation 〈g, p〉 is used to refer to the group and processor addresses, respectively. Two nodes 〈g1, p1〉 and 〈g2, p2〉 are connected

if, and only if, g1 = g2 and (p1, p2)∈E0 (such that E0 is the set of edges in arrangement network) or g1 = p2 and p1 = g2, in this case the two nodes are connected by transpose edge.

The following are the basic topological properties for the OTIS-Arrangement. For Instance if the factor arrangement network is of size |n !/(n-k)!|, degree is k(n-k) and diameter is 1.5 k [3]. Then the size, the degree, the diameter, number of links, and the shortest distance of the OTIS-Arrangement network are as follows [20]:

• Size of |OTIS-An,k | = |n!/(n-k)!|2.

Degree of OTIS-An,k = Deg.(An,k), if g = p.

Deg. (An,k) + 1, if g ≠ p.

Diameter of OTIS-An,k = 2 1.5 k +1.

Number of Links: Let N0 be the number of links in the An,k and let M be the number of nodes in the An,k. The number of links in the OTIS- An,k =

(M 2 − M ) / 2 + N02

Dist. of OTIS- An,k

min( d(p1, g2 ) + d(g1, p2 )

+ 1, d(p1, p2 ) + d(g1, g2 ) + 2 )

if g1 ≠ g2

Dist. (p1,p2 )

if g1 = g2

where d(p1, p2) is the length of the shortest path between any two processors 〈g1, p1〉 and 〈g1, p2〉.

Definition 2: The cross product G=G 1⊗G2 of two undirected connected graphs G1=(V1, E1) and G2=( V2, E2) is the undirected Graph G=(V, E), where V and E are given by:

V={〈x1, y〉 | x1 V1 and y V2} and

E={(〈 x1,y〉, 〈y1,y〉) | (x1,y1) ∈E1} ∪ {(〈x,x2〉,〈x,y2〉) | (x2,y2) ∈E2}.

So for any u =〈x1, x2〉 and v =〈 y1, y2〉 in V, (u, v) is an edge in E if, and only, if either (x1, y1) is an edge in E 1 and

x2 = y2, or ( x2, y2) is an edge in E2 and x1 = y1. The edge (u, v) is called a G1-edge if (x1, y1) is an edge in E1, and it is

called G2-edge if (x2, y2) is an edge in E2 [6].

The Extended OTIS-Arrangement is obtained by "multiplying" an arrangement topology by itself and connects different groups by extra optical links when processor index is equal to group index. The vertex set is equal to the Cartesian product on the vertex set in the factor arrangement network. The edge set consists of edges from the factor network and new edges called the transpose edges. The formal definition of the Extended OTIS-arrangement is given below.

Definition 3: Let x, y be either group’s addresses or processor’s addresses of two nodes in an Extended OTIS-Arrangement labeled as k symbols taken out of n symbols

〈x1, x2, .. xk 〉 , 〈y 1, y2, .. yk 〉 ; x is called an opposite of y if and only if they differ in the kth symbol of the group and

processor addresses.

The following example show the opposite of x and y in case of Extended-OTIS-A3,2.

3131 3232; 1212 1313; 32323 2121

Definition 4: Let An,k = (V0, E0) be an undirected graph representing an arrangement network where. The Extended

OTIS-Arrangement = (V, E) network is represented by an undirected graph obtained from An,k as follows V = { 〈x, y〉 | x, y ∈ V0} and E = {(〈x, y〉, 〈x, z〉) | if (y, z) ∈E0} ∪ {(〈x,

y〉, 〈y, x〉 ) | x, y ∈ V0} ∪ {(〈x, x〉,〈y, y〉) | x, y ∈ V0 | x is an opposite of y}.

Two nodes 〈g1, p1〉 and 〈g2, p2〉 are connected if one of the following cases occurs:

1- If g1 = g2 and (p1, p2) ∈ E0 (such that E0 is the set of edges in arrangement network), in this case the two nodes are connected by electronic edge.

2- If g1 = p2 and p1 = g2, in this case the two nodes are connected by transpose edge.

3- If g1 = p1 and g2 = p2 and g1 is an opposite of g2, in this case the two nodes are connected by transpose edge too.

Definition 5: The Topological properties of the Extended OTIS- Arrangement are defined as follows:

1- Size: If the arrangement factor network of size N, then the size of the Extended OTIS-Arrangement is N2.

Figure 1 shows Extended OTIS-A3,2 Interconnection Network, where the dashed bold lines represents the opposite OTIS link between two groups, on the other hand the solid line between different groups represents normal OTIS link.

Fig. 1 Extended OTIS-A3,2 Interconnection Network

- Distance: The distance in the Extended OTIS-Arrangement is defined as the shortest path between any two processors, 〈g1, p1〉 and 〈g2, p2〉. To transmit data originated in the source node 〈g 1, p1〉 to the destination node 〈 g2, p2〉 we follow one of the five possible paths shown above i, ii, …, v . The distance length of the shortest path between the nodes 〈g1, p 1〉 and 〈g 2, p2〉 is:

i- When g1 = g2 then the path involves only

electronic moves from source node to the destination node.

ii- When g1 is opposite of g2, and if the number of optical moves is an odd number of moves, then

the paths can be compressed into a shorter path of the form:

E

O

E

•

〈g1, p1〉 →〈g1, g1〉 →〈g2, g2〉 →〈g2,

p2〉

E

O

E

or 〈g1, p1〉 →〈g1, g2〉 →〈g2, g1〉 →〈g2,

p2〉; whichever is shorter, where the symbols O

and E stand for optical and electronic moves

respectively.

iii- When p2op = g1or p1op = g2, and the path involves

an even number of optical moves. In this case the paths can be compressed into a shorter path

of d( p1, g2 ) + d( p2 , g1) +1 or

one of

the

following two cases:

E

O

E

• 〈g1, p1〉 →〈g1, g1〉 →〈 p2, p2〉 →〈

O

p2, g2〉 →〈g2, p2〉 if p2op= g1.

O

E

O

•

〈g1, p1〉 →〈p1, g1〉 →〈p1, p1〉 →〈g2,

E

g2 〉 →〈g2, p2〉

if p1op= g2,

where

op

means opposite.

iv- When g1 ≠ g2 and if the number of optical moves

is an even number of moves, then the paths can be compressed into a shorter path of the form:

E

O

E

•

〈g1, p1〉 →〈g1, p2〉 →〈p2, g1〉 →〈p2,

O

g2〉 →〈g2, p2〉

v- When g1 ≠ g2, and the path involves an odd

number of optical moves. In this case the paths can be compressed into a shorter path of the form:

E O E

• 〈g1, p1〉 →〈 g1, g2〉 →〈 g2, g1〉 →〈 g2, p2〉

III. HAMILTONIAN CYCLE STRUCTURE IN THE EXTENDED OTIS-ARRANGEMENT

This section presents the Hamiltonian cycle structure of the recently proposed interconnection topology Extended OTIS-An,k [20]. The following notations and theorems will be needed in constructing the Hamiltonian cycle for this topology.

Theorem 1. If the arrangement factor degree is n, then any node in the Extended OTIS-An,k is regular and the node degree of this network is k(n-k)+1.

Proof.

Every node has k(n-k) electronic edges based on the

properties of the An,k factor network. Also every node ; 〈g, p〉; has an additional optical edge based on the Extended

OTIS-An,k topology rule: {(〈 g, p〉,〈p, g〉) | g, p∈V0} ∪{(〈g, g〉,〈p, p〉 | g, p ∈V0 ∩ g is an opposite of p}.

So if g = p then 〈p, p〉

O

〈g, g

〉 else 〈g, p〉

O

〈p,

g〉.

→

op

op

→

Since every node has an n! /(n-k)! electronic edges , in addition to one optical edge, then by definition the topology is regular.

So it follows that degG of the extended OTIS-Arrangement is:

DegOTIS- An,k(g,p)= DegG0(p)+1= k(n-k) +1

Regularity in degree is considered one of the attractive topological properties of any potential network, this property allows the network to be semantic and we can easily apply any mathematical tasks on such networks where it is needed to equally divide such operations across the network processors, e.g. load balancing [26].

Theorem 2. Let 〈g1, p1〉 and 〈g 2, p2〉 be two different nodes in the Extended OTIS-An,k. The length of shortest path from

the source node 〈 g1, p1〉 to the destination node 〈g2, p2〉 is defined mutually exclusive as in the following order:

d(p1, p2) if g1 = g2

d(p1, g1) + d(g1opposite, p2)+1if g1 = g2opposite

Length = d(p1, g1) + d(p1opposite, g2)+1if p1 = p2opposite d(p1, p2) + d(g1, g2)

if g1 = p1 and g2 = p2 and d(g1 = g2)

min d(p1, g2)+ d(p2, g1)+1, d(p1, p2)+ d(g1, g2) +2 Otherwise

Where d (p1, p2) is the number of electronic moves between p1and p2 labels.

Proof:

By following one of the five possible paths; i, ii, iii, iv, and v shown above. The length of the shortest path between the

nodes 〈g1, p1〉 and 〈g1, p1〉 can be as follows:

- If both nodes are in the same group then the shortest path is guaranteed by generating electronic moves toward the

destination; d(p1, p2).

- If g 1= g2op it means that one optical move is needed to move toward the destination group via a group opposite

edge. To reach the destination, some electronic moves

might be needed first at one source group to reach 〈g1, p1〉, then one optical move to reach the intermediate group; finally other electronic moves at the destination group might be needed to reach the destination node.

- If g1= p2op or g2= p2op it means that two optical moves are needed to reach the destination group through an

intermediate group equal to p1op. This requires some electronic moves to perform the two optical moves, and finally to reach the destination node at minimal distance.

- If g1≠ g 2, and d(p1,p2) = 3/2 k it means that two optical moves in addition to some electronic moves are needed to reach the destination group through an intermediate group g1op. First an opposite move is required to reach, then n1/(n-

k)!-1electronic moves to reach 〈p2, g2〉, then an optical move to reach 〈g2, p2〉.

- Otherwise we choose the shortest path based on the factor OTIS moves [23].

Since Hamiltonian is a cycle in an undirected graph which visits each node exactly once and finally returns to the starting node.

The following summarize the necessary steps for constructing the algorithm of Hamiltonian circuit for the Extended OTIS-An,k topology:

1. Some assumptions that must be taken in consideration in constructing this algorithm:

a.The start and the destination nodes are 〈g1, p1〉 and 〈g2, p2〉 where g and p are group and processor

addresses respectively, as shown in example 1 below choose 〈13, 13〉 and 〈12, 12〉 as start and destination nodes.

b. Let n and k are two integers satisfying 1<= k <= n-1 and denote 〈n〉 = {1,2,….,n} and 〈k〉 = {1,2,…,k}.

c.Let Pkn taken k at a time, the set of arrangements

of k elements out of the n elements of 〈n〉.

d.Any node in the proposed topology are labeled

such that: V= { p1 p2… pk pi in 〈n〉 and pi≠pj for i≠j}.

Do n!/(n-k)!-1 factor moves towards a potential target node in the current group according to the following rules:

a.In the 1st group, fix the first k-1 symbols of the permutation of the processor address of the factor network.

b.Change the symbol kth of the processor address by decrease the last symbol of the label by one for all possible of last symbol label.

c.If the result of the subtraction = 0, choose the largest symbol of the processor label.

Repeat step 2 for all the different permutations of k-1 symbols. i.e. Pkn−1 .

If the target factor address matches an already visited node in first group do an optical move from 〈g, p〉 to 〈p, g〉 in the last node, such that {( 〈x, y〉, 〈y, x〉) | x, y ∈ V0} ∪ {(〈x, x〉,〈y, y〉) | x, y

∈ V0 | x is an opposite of y}.

Repeat steps 2, 3 and 4 until all the n!/(n-k)! groups are visited.

Finally an OTIS move back toward the start node.

The following examples show how a Hamiltonian Circuit is constructed for 2 different factor arrangement networks of different sizes.

Example 1: Hamiltonian cycle within an Extended OTIS-A3,2 graph, Figure 2 shows a representation of such a Hamiltonian cycle, where the size of the factor network is 6 and the size of the extended is 36 nodes.

Example 2: Hamiltonian cycle within an Extended OTIS- A4,2 graph, Figure 3 shows a representation of such a Hamiltonian cycle.

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