A Novel Interconnection Network with Improved Network Cost Through Shuffle-Exchange Permutation Graph

The interconnection network represents an interconnected structure of processors that strongly determines the performance quality of a parallel processing system. The shuffle-exchange permutation (SEP) network with three degrees has high fault tolerance and can be efficiently simulated through star, bubble-sort, and pancake graphs. This study proposes a new interconnection network: the new SEP (NSEP), which improves the diameter and reduces network cost by adding one edge to the SEP network, and presents its graph properties and routing algorithms. The NSEP network, with a degree of connectivity of four, demonstrated maximum fault tolerance and Hamiltonian cycle. Furthermore, the diameter was seen to be improved by 40% or more and the network cost by 20% or more.


Introduction
With the explosive increase in data size owing to the recent advancements in information technology, the demand for high-performance computers with large computational power is increasing, particularly for big data and artificial intelligence applications. In response to these demands, high-performance computers with various computational processing units, such as graphics processing units (GPUs) and multicore processors, in addition to the conventional central processing units (CPUs), have been developed. Such computers are constantly evolving in response to new demands and requirements [1].
A parallel computer is a computer system that divides a given task and processes tasks among processing units operating in parallel. Parallel computers are classified into shared memory multiprocessors and message-passing multicomputers [2]. In the former, the memory system affects the overall system performance [3]. The interconnection network refers to the location and connection structure between processors and is one of the factors that determine the performance of a parallel processing system [4]. Hence, continuous research on interconnection networks is required to improve the performance of parallel processing computers.
Network cost is one of the measures of interconnection networks and is represented by the product of the number of degrees and the diameter. The number of degrees is related to the hardware cost and the diameter to the software cost. The network cost may be reduced by reducing the number of degrees or the diameter. The number of degrees is inversely correlated to the diameter. It is difficult to reduce the network cost because reducing the number of degrees increases the diameter, whereas reducing the diameter increases the number of degrees [4].
The interconnection networks mesh the hypercube and star graph classes depending on the number of nodes. The SEP [5] network is a star graph class with n! nodes, with node and edge symmetry, has excellent scalability through recursive structures, and has a very small number of degrees and diameters over hypercube [6][7][8]. The existing SEP network has a maximum fault tolerance with a degree of connectivity of three, and efficient simulation can be performed for star, bubble sort, and pancake graphs. Thus, one can still get the advantage of the fixed degree of the network (independent of the size) [5]. In addition, NSEP networks with increased degree one also predict that simulation will be efficient for star, bubble sort, and pancake graphs.
For n-dimensional NSEP proposed in this study, when n = 2k, the distance between two nodes, n 2 , was reduced to one by adding an edge to the SEP. The proposed NSEP has a fixed number of degrees of four and has the properties of the existing SEP. The NSEP network has a maximum fault tolerance with a degree of connectivity of four and has a Hamilton cycle. Compared to the SEP network, the diameter was improved by more than 40% and the network cost by more than 20%.
In Section 2, we examine the network measure of the interconnection network, a constant-degree graph. In Section 3, we define the new interconnection network, NSEP n , present the theoretical properties of graph and routing algorithm, and analyze the diameter. Finally, Section 4 concludes this study.

Related Works
In this chapter, we first consider the importance of the network measure and the advantages of the fixed number of degrees. Next, we examine the Hamiltonian cycle and SEP with an improved network cost. In a multiprocessor system, a connected network for supporting the communication between each processor is called a multiprocessor interconnection network [9]. The interconnection network can be represented as an undirected graph representing each processor as a node and a communication link between processors as an edge. An edge is placed between any two processors with a link between them. This edge is an undirected edge that can bidirectionally transmit data. The interconnection network of parallel computers is represented as an undirected graph as follows: where V(G) is the set of nodes of graph G; that is, V = {0, 1, 2, · · · , N − 1} and E(G) is the set of edges of graph G. The edge of graph G is a pair of arbitrary two nodes, v and w of V(G). The necessary and sufficient condition for the existence of an edge (v, w) is the presence of a communication link between nodes v and w [10][11][12][13][14][15].
Network measures for evaluating interconnection networks include number of degrees, diameter, network cost, connectivity, fault tolerance, and symmetry [10,15]. The number of degrees for a node v refers to the number of edges adjacent to the node v, and the number of degrees for graph G refers to the maximum value among the number of degrees of the nodes belonging to V(G). A network that has an equal number of degrees for all nodes in graph G is called a regular network. The diameter is the maximum value of the shortest path between any two nodes in the network and is the lower limit of the delay time required to transmit information to the entire network. A network having a relatively small diameter compared to the number of nodes, despite the short distance between nodes, has a disadvantage that it is difficult to design the network in terms of hardware as the number of nodes increases [16][17][18]. The interconnection networks that have been proposed until now can be classified into the following three types according to the number of nodes: mesh class with k × n nodes, hypercube class with 2 n nodes, and star graph class with n! nodes [6].
The mesh structure has been widely used as a planar graph to date, and commercialized in various systems [19,20]. An m-dimensional mesh M m (N) consists of N m nodes and mN m − mN m−1 edges. Each node's address is represented by an m-dimensional vector, and when the addresses of any two nodes differ by one in one dimension, there is an edge between them. Because low-dimensional meshes are easy to design and are useful from the algorithmic viewpoint, they are widely used as a network of parallel processing computers. The higher the dimension of a mesh, the smaller its diameter and the larger the bisection width, and various parallel algorithms can be rapidly executed; however, it is costly [6]. Structures that improve the diameter of a mesh with a typical lattice structure, hexagonal mesh, toroidal mesh, diagonal mesh, honeycomb mesh, and torus have been proposed [19,21].
The Hamiltonian path of the interconnection network is a path that passes through all nodes of G only once. The Hamiltonian cycle of the graph G refers to a path with the same starting and destination nodes as the path that passes through all nodes only once. If the network has a Hamiltonian path or Hamiltonian cycle, a ring or a linear array can be easily implemented, which can be utilized as a useful pipeline for parallel processing [22]. If the graph v contains a Hamiltonian cycle, it is appropriate to include the Hamiltonian path.
The n-dimensional SEP graph SEP n is a regular network that represents nodes by permutation of each symbol and has three degrees [5]. In this respect, this study interchangeably uses nodes and permutation. There are three edges of SEP n − {g 12 , g L , g R } according to the conditions. If an arbitrary node of SEP n is S = s 1 s 2 s 3 · · · s n−1 s n , adjacent nodes are as follows.

1.
Edge g 12 : Connects the nodes in which the leftmost first and the second symbols are exchanged in permutation. For example, it corresponds to a node g 12 (S) = s 2 s 1 s 3 · · · s n−1 s n that is adjacent by an edge g 12 in a node S.

2.
Edge g L : All symbols in the permutation are moved one digit to the left, and the leftmost symbol is moved to the rightmost position. For example, it corresponds to a node g L (S) = s 2 s 3 · · · s n−1 s n s 1 that is adjacent by the edge g L in node S.

3.
Edge g R : All symbols of the node permutation are moved to the right by one digit, and the rightmost symbol is moved to the leftmost position. For example, it corresponds to a node g R (S) = s n s 1 s 2 s 3 · · · s n−1 that is adjacent by the edge g R in node S. In node S, any node that is adjacent by edge operation g 12 is represented by g 12 (S), and the same method is applied to edge {g L , g R }. If the order of the edge sequence is g L , g 12 , g R when the edge operation is applied in the node S, the permutation change of this node is represented as S → g L (S) → g 12 (g L (S)) → g R (g 12 (g L (S))). The permutations of the node to which the edge sequence order g L , g 12 , g R is applied in the node S are S = s 1 s 2 s 3 · · · s n−1 s n , g L (S) = s 2 s 3 · · · s n−1 s n s 1 , g 12( g L (S) = s 3 s 2 · · · s n−1 s n s 1 , and g R( g 12( g L (S))) = s 1 s 3 s 2 · · · s n−1 s n . Thus, when edge sequence g L , g 12 , g R is applied in node S, the last node is g R( g 12( g L (S) = s 1 s 3 s 2 · · · s n−1 s n . Figure 1 shows a 4D SEP 4 graph.

Definition and Properties of Graph
The ( − ) graph is a regular graph with four degrees obtained by adding one edge to the existing graph ( = 2 , ≥ 2). One edge added to = ⋯ ⋯ , a node of , is an edge that connects the permutations in which the symbols , ~ , and ~ have been exchanged. Let be an added edge of the graph. The node is ( ) = ⋯ ⋯ ad- The SEP graph can be easily simulated on graphs based on permutation groups, such as a Cayley graph, and its algorithms can be efficiently executed in new graphs with minimal changes. The diameter of the SEP graph is 1 8 9n 2 − 22n + 24 , and its degree of connectivity is three, having a maximum fault tolerance [5]. Because SEP n is a Cayley graph, it has a node symmetric property [10]. The cycle, whose path length is n and composed of edges g L (or g R ) in SEP n , is called s − cycle [5]. In the SEP graph, the positions of symbols are exchanged using edge operation g 12 , and the symbol to be exchanged is moved to the leftmost position using edge operation < g L , g R >. This study improved the diameter value by adding one edge in which the symbol of the position n 2 + 1 can be quickly moved to the leftmost position in the node permutation of the graph SEP n .

Definition and Properties of NSEP n Graph
The NSEP n (New − SEP n ) graph is a regular graph with four degrees obtained by adding one edge to the existing SEP n graph (n = 2k, k ≥ 2). One edge added to S = s 1 s 2 s 3 · · · s n 2 s n + 1 2 · · · s n−1 s n , a node of NSEP n , is an edge that connects the permutations in which the symbols n 2 , s 1 ∼ s n 2 , and s n + 1 2 ∼ s n have been exchanged. Let g n 2 be an added edge of the NSEP n graph. The node is g n 2 (s) = s n + 1 2 · · · s n−1 s n s 1 s 2 s 3 · · · s n 2 adjacent by the edge g n 2 in the node S = s 1 s 2 s 3 · · · s n 2 s n + 1 2 · · · s n−1 s n . Therefore, the NSEP n graph has four edges g 12 , g L , g R , g n 2 for each node. The four nodes g 12 (S), g L (S), g R (S), g n 2 (S) adjacent to the node S = s 1 s 2 s 3 · · · s n 2 s n + 1 2 · · · s n−1 s n of the NSEP n graph are shown below. g 12 (S) = s 2 s 1 s 3 · · · s n 2 s n+1 2 · · · s n−1 s n g R (S) = s n s 1 s 2 s 3 · · · s n 2 s n + 1 2 · · · s n−1 g L (S) = s 2 s 3 · · · s n 2 s n + 1 2 · · · s n−1 s n s 1 g n 2 (S) = s n+1 2 · · · s n−1 s n s 1 s 2 s 3 · · · s n 2 Figure 2 shows an example of the NSEP 4 graph. In Figure 2, the thick line represents the edge g 12 , the solid line represents the edge g L (or g R ), and the dotted line represents  Because the graph has one extra edge over the graph, the latter is a subgraph of the former. Cycles whose path length is and which comprise the edges (or ) of the are called s-cycles. For example, an s-cycle with the path length of four at the node S (= 1234) of is = (1234) 2341 3412 4123 1234( = ) A cluster in has several important properties. These properties can be used to confirm that the graph has a Hamiltonian cycle. The following definitions define the cluster and show its properties in Attributes 1, 2, and 3.

Definition 1.
In the graph , a partial graph consisting of nodes constituting s-cycles and the edge connecting the nodes in the s-cycles is called a graph . Because the NSEP n graph has one extra edge over the SEP n graph, the latter is a subgraph of the former. Cycles whose path length is n and which comprise the edges g L (or g R ) of the NSEP n are called s-cycles. For example, an s-cycle with the path length of four at the node S (= 1234) of NSEP 4 is A cluster in NSEP n has several important properties. These properties can be used to confirm that the NSEP n graph has a Hamiltonian cycle. The following definitions define the cluster and show its properties in Attributes 1, 2, and 3.

Definition 1.
In the graph NSEP n , a partial graph consisting of nodes constituting s-cycles and the edge g n 2 connecting the nodes in the s-cycles is called a graph C n .
In NSEP 4 , one cluster C 4 containing the node S (=1234) is a partial graph consisting of four edges g L (or g R ), and two edges g n 2 . Figure 3 shows C 4 , the cluster of NSEP 4 .
subgraph of the former. Cycles whose path length is and which comprise the ed (or ) of the are called s-cycles. For example, an s-cycle with the path le four at the node S (= 1234) of is = (1234) 2341 3412 4123 1234( = ) A cluster in has several important properties. These properties can b to confirm that the graph has a Hamiltonian cycle. The following definiti fine the cluster and show its properties in Attributes 1, 2, and 3.

Definition 1. In the graph , a partial graph consisting of nodes constituting s-cy the edge
connecting the nodes in the s-cycles is called a graph .
In , one cluster containing the node S (=1234) is a partial graph con of four edges (or ), and two edges . Figure 3 shows , the cluster of

Property 1.
There are (n − 1)! C n clusters in the NSEP n graph.
Proof. The total number of nodes in NSEP n is n!. A cluster C n is s-cycles with n different nodes and consists of n 2 g n 2 edges that connect nodes along the path constituting s-cycles. Moreover, the number of nodes in each cluster C n is n by s-cycles. Therefore, the number of C n clusters is n! n = (n − 1)!.

Property 2.
The cluster C n of NSEP n has n 2 g n 2 edges.
Proof. There are n nodes in each cluster in the NSEP n graph, and they are adjacent to each other by g n 2 edges that exchange n 2 symbols. Because there is only one node with such an adjacent relationship for one node, two nodes form a pair. Therefore, there are n 2 g n 2 edges connecting n nodes that constitute the cluster.

Property 3.
A node U constituting one cluster C n of NSEP n is adjacent to the node g 12 (U) of another cluster C n by the edge g 12 . The n nodes constituting a cluster C n are adjacent to nodes of n different clusters C n by the edge g 12 .
Proof. By the definition of NSEP n it can be seen that n nodes of a cluster C n are adjacent to n nodes of different clusters C n by the edge g 12 .
Due to the added edge g n 2 , a new cycle with ( n 2 + 1) nodes exists. Definition 2 defines the cycles of NSEP n and the associated theorem is shown in Lemma 1, 2, and 3. In the Lemma 4, 5, and 6, we show that there is a Hamiltonian cycle between two adjacent nodes in cluster C n of NSEPn.

Definition 2.
When there is an arbitrary node U in the cluster C n , let V = g n 2 (U) be the node U and the node adjacent to the edge g n 2 (however, U = V). Let the ( n 2 + 1)-cycle be the path from node U to the node V constituting the edge g L (or g R ) at the path distance of n 2 , and the path constituting the edge g n 2 at the node U. Assume that there are nodes U( = 1234) and V(= 3412) at NSEP 4 . The three-cycle path containing node U(= 1234) and V(= 3412) is given as U(= 1234)

Lemma 1.
When there is a node U in one cluster C n , let V = g n 2 (U) be the node U and the node adjacent to the edge g n 2 (however, U = V). There are two ( n 2 + 1)-cycles that share the edge g n 2 connecting nodes U and V.
Proof. By Definition 2, ( n 2 + 1)-cycle is a path consisting of g n 2 edge g R (or g L ), and 1 g n 2 edge. It can be seen that these ( n 2 + 1)-cycles can create cycles using the edge g R and the edge g L , respectively. Therefore, there are two ( n 2 + 1)-cycles that share the edge g n 2 .
Proof. By Property 2, each cluster has n 2 g n 2 edges, and by Lemma 1, there are two ( n 2 + 1)-cycles that share an edge g n 2 . Therefore, because n 2 × 2 = n, there are n ( n 2 + 1)-cycles.
Lemma 3. The number of ( n 2 + 1)-cycles in the network NSEP n is n!.
Proof. By Lemma 2, there are n ( n 2 + 1)-cycles in each cluster, and by Property 1, the number of C n clusters is (n − 1)!. Therefore, the number of ( n 2 + 1)-cycles in the NSEP n network is (n − 1)! × n = n!.

Lemma 4.
There is a Hamiltonian path, whose path length is n, including an arbitrary node U in the cluster C n , and a node g R (U) (or g L (U)) adjacent to the edge g R (or g L ) from the node U.
Proof. Let U be an arbitrary node of the cluster C n . Let V1(= g L (U)) be the adjacent node by node U and edge g L , and the node adjacent by node and edge. Because each cluster C n has s-cycles of NSEP n as a partial graph, the path connected by node U and edge g R (or g L ) has cycles including nodes V1 and V2. Therefore, there is a Hamiltonian cycle with the path length of n from node U to an adjacent node V1(= g L (U)) by an edge g L , and an adjacent node V2(= g R (U)) by the node U and the edge g R .

Lemma 5.
There is a Hamiltonian cycle between an arbitrary node U that constitutes a cluster C n , and nodes V1 = g n 2 (U) connected from the node Uto the edge g n 2 .
Proof. Let U be the starting node of the cluster C n , and the target node V1 = g n 2 (U) be the node that is connected to node U and the edge g n 2 . By the definition of NSEP n graph, the distance between the nodes U and V in s-cycles consisting of edges g L (or g R ) is n 2 . Let S1 be a node at a n 2 − 1 distance along the s-cycle from the starting node U. The node S2 = g n 2 (S1) connected by the node S2, and the edge g n 2 has a distance of n 2 in s-cycles. Therefore, node S2 is a node adjacent to U located at a distance of n 2 − 1 from the node S1. A node at a n 2 − 1 distance along the s-cycle from a node S2 becomes a target node V. Because nodes U and V1 = g n 2 (U) are adjacent to the edge g n 2 , a Hamiltonian cycle is formed. Therefore, there is a Hamiltonian cycle with a length n, connecting two adjacent nodes in the cluster C n .

Lemma 6.
There exists a Hamiltonian cycle that includes two adjacent nodes U, V in the cluster C n .
Proof. By Lemmas 4 and 5, there is a Hamiltonian cycle that includes two adjacent nodes U, V in the cluster C n . The reduced graph RS n−1 of NSEP n represents the reduced s-cycles in NSEP n to one node. The node whose leftmost symbol is one in the permutation of n nodes constituting the s-cycles of NSEP n is called the leader node. The node address of the reduced graph RS n−1 is represented by the remaining permutation addresses except one in the permutation of the leader node. In s-cycles 1234-4123-3412-2341, shown in Figure 4, the leader node is 1234, and s-cycle is represented by the super node 234 in the graph RS 3 . Definition 3 defines a subgraph of RS n−1 as RS k n−2 relative to the rightmost symbol. The theorem about it is shown in auxiliary Lemmas 7-9, and NSEP n shows that in Theorem 1 has a Hamilton cycle.

Definition 3.
A bubble-sort graph, which is a partial graph that includes all nodes of the reduced graph , is known as . Furthermore, is a ( − 2) normal Cayley graph [5]. Therefore, when is ≥ 5, it includes − 1 subgraph , and all are adjacent to each other. Because the nodes belonging to have the same rightmost symbol, when the rightmost symbol is k, the of is defined as (2 ≤ ≤ ).

The
network has an even number of = 2 symbols representing node addresses. In a network having an even number of symbols in , does not exist. If has a Hamiltonian cycle, it is natural that there is a Hamiltonian cycle when is even. After showing that there is a Hamiltonian cycle in , we show that there is also a Hamiltonian cycle in .

Lemma 7.
There is a Hamiltonian path between any two arbitrary nodes of , and it has a Hamiltonian cycle.
Proof. Let and be the starting and destination nodes, respectively. can be divided into two areas A and B with the same number of nodes. The thick lines correspond to the edges within each area. There are three nodes constituting one area, and all nodes are adjacent. In addition, the nodes constituting the area have cycles in a complete graph, and there is always a Hamiltonian path between two nodes. In each node, there are two edges connecting to nodes in other areas. There are two cases of the relationship with nodes and , as shown in Figure 5. The edges can be present in one area as shown in Figure 5-1, or in different areas as shown in Figure 5-2. Let ′(≠ ) and ′ be a node adjacent to in area A and a node adjacent to in area B, respectively. There is a node adjacent to ′ in area B, and there is a Hamiltonian path between this node and ′. Therefore, in Case 1, there is a Hamiltonian path between and . Now we move on to Case 2. Let ′(≠ ) be the node connected through the Hamiltonian path from in area A. Because both ′ and are present in area B, there is a Hamiltonian path. That is, there is a Hamiltonian path between and in Case 2 as well. Therefore, because there is a

Definition 3.
A bubble-sort graph, which is a partial graph that includes all nodes of the reduced graph RS n−1 , is known as RSB n−1 . Furthermore, RSB n−1 is a (n − 2) normal Cayley graph [5]. Therefore, when RS n−1 is n ≥ 5, it includes n − 1 subgraph RS n−2 , and all RS n−2 are adjacent to each other. Because the nodes belonging to RS n−2 have the same rightmost symbol, when the rightmost symbol is k, the RS n−2 of RS n−1 is defined as RS k n−2 (2 ≤ k ≤ n).
The NSEP n network has an even number of n = 2k symbols representing node addresses. In a network having an even number of symbols in NSEP n , RS 4 does not exist. If RS n−1 has a Hamiltonian cycle, it is natural that there is a Hamiltonian cycle when n is even. After showing that there is a Hamiltonian cycle in RS n−1 , we show that there is also a Hamiltonian cycle in NSEP n .

Lemma 7.
There is a Hamiltonian path between any two arbitrary nodes of RS 3 , and it has a Hamiltonian cycle.
Proof. Let U and V be the starting and destination nodes, respectively. RS 3 can be divided into two areas A and B with the same number of nodes. The thick lines correspond to the edges within each area. There are three nodes constituting one area, and all nodes are adjacent. In addition, the nodes constituting the area have cycles in a complete graph, and there is always a Hamiltonian path between two nodes. In each node, there are two edges connecting to nodes in other areas. There are two cases of the relationship with nodes U and V, as shown in Figure 5. The edges can be present in one area as shown in Figure 5-1, or in different areas as shown in Figure 5-2. Let U ( = V) and V be a node adjacent to U in area A and a node adjacent to V in area B, respectively. There is a node adjacent to U in area B, and there is a Hamiltonian path between this node and V . Therefore, in Case 1, there is a Hamiltonian path between U and V. Now we move on to Case 2. Let U ( = V ) Electronics 2021, 10, 943 8 of 16 be the node connected through the Hamiltonian path from U in area A. Because both U and V are present in area B, there is a Hamiltonian path. That is, there is a Hamiltonian path between U and V in Case 2 as well. Therefore, because there is a Hamiltonian path between any two nodes of RS 3 , and another Hamiltonian path between adjacent nodes, this has a Hamiltonian cycle.
Therefore, the number of in satisfies the following equation. in . Therefore, it can be seen that is always adjacent h Proof. By Definition 3, RS n−1 has n − 1 RS n−2 subgraphs as a cluster. Figure 6 shows a subgraph on the RS n-1 . For example, if n = 5, then CN 4 , which is the number of RS 3 in RS 4 becomes CN 4 = CN 3 × 4. Because RS n−1 is hierarchical, let us assume CN n−1 = CN n−2 × (n − 1). We prove that the formula CN n−1 = (n−1)!
10, x FOR PEER REVIEW 10 of 18 Proof. By Lemma 9, has a Hamiltonian cycle, which regards the cluster of as a super node. A node in an adjacent cluster of is adjacent to another cluster through an adjacent node [5]. By Lemma 6, there is a Hamiltonian path between adjacent nodes of the cluster . Thus, the network has a Hamiltonian cycle. Figure 7 shows a Hamiltonian cycle of RS4. □ Therefore, the number of RS 3 in RS n−1 satisfies the following equation.
Proof. By Definition 3, all n − 1 RS k n−2 subgraphs are adjacent to each other in RS n − 1 . Let R i (1 ≤ i ≤ (n − 1)!) and R j (1 ≤ j ≤ (n − 1)!, i = j) be any two nodes adjacent to each other in RS n−1 . Adjacent relationships are indicated by dotted lines. All nodes in RS n−1 are adjacent to RS k n−2 through adjacent nodes. There is always adjacent RS k n−3 in RS n−2 . Therefore, it can be seen that RS k 3 is always adjacent hierarchically in the same manner up to RS 4 . As we have shown in Lemma 7, that there exists a Hamiltonian path between any two nodes, which implies that there exists a Hamiltonian cycle in RS n−1 . Theorem 1. The NSEP n network has a Hamiltonian cycle.
Proof. By Lemma 9, RS n−1 has a Hamiltonian cycle, which regards the cluster C n of NSEP n as a super node. A node in an adjacent cluster of NSEP n is adjacent to another cluster through an adjacent node [5]. By Lemma 6, there is a Hamiltonian path between adjacent nodes of the cluster C n . Thus, the NSEP n network has a Hamiltonian cycle. Figure 7 shows a Hamiltonian cycle of RS 4 .
Electronics 2021, 10, x FOR PEER REVIEW Proof. By Lemma 9, has a Hamiltonian cycle, which regards the cluste as a super node. A node in an adjacent cluster of is adjacent to a cluster through an adjacent node [5]. By Lemma 6, there is a Hamiltonian path b adjacent nodes of the cluster . Thus, the network has a Hamiltonian cycl ure 7 shows a Hamiltonian cycle of RS4. □

Routing Algorithm and Diameter Analysis
Routing refers to the path from one node to another. Because a partial g is a Cayley graph, it is node symmetric [10]. Therefore, the path of the s node and the destination node D can be regarded as the path of the starting and the ID node. Let the ID node be 123 ⋯ . The algorithm proposed in this stu method of placing the symbols in sequence up to n by iteratively applying the me checking the positions of symbols 1 and 2, placing symbol 2 on the right side of sy and symbol 3 on the right side of symbol 2. The position of the symbol is represe definition 4, and the formulas used by the algorithm are represented in Lemmas 1

Routing Algorithm and Diameter Analysis
Routing refers to the path from one node to another. Because SEP n a partial graph of NSEP n is a Cayley graph, it is node symmetric [10]. Therefore, the path of the starting node S and the destination node D can be regarded as the path of the starting node S and the ID node. Let the ID node be 123 · · · n. The algorithm proposed in this study is a method of placing the symbols in sequence up to n by iteratively applying the method of checking the positions of symbols 1 and 2, placing symbol 2 on the right side of symbol 1, and symbol 3 on the right side of symbol 2. The position of the symbol is represented in Definition 4, and the formulas used by the algorithm are represented in Lemmas 10-13.

Definition 4.
The position of the symbol s 1 in the current node S(= s 1 s 2 s 3 · · · s i · · · s n−1 s n ) is represented by p(s i ) (1 ≤ i ≤ n).

Lemma 10.
In node S(= s 1 s 2 s 3 · · · s i · · · s n−1 s n ), the path of the node adjacent to the node by the edge sequence g L , g 12 , g R is as follows. The last node permutation is g R (g 12 (g L (S))) = s 1 s 3 s 2 · · · s n−1 s n in the path to which the edge sequence g L , g 12 , g R is applied in node S. The path in node S is as given below.
S(= s 1 s 2 s 3 · · · s i · · · s n−1 s n ) → g L (S) = s 2 s 3 · · · s i · · · s n−1 s n s 1 → g 12 (g L (S)) = s 3 s 2 · · · s i · · · s n−1 s n s 1 → g R (g 12 (g L (S))) = s 1 s 3 s 2 · · · s i · · · s n−1 s n Lemma 11. The number of iterations for an edge or edge sequence is denoted by × [i] . For example, when i = 3, g A , g B × [3] = g A , g B , g A , g B , g A , g B . The number of iterations of the edge sequence is incorporated as given below.
When p(i) = a, p(i + 1) = b, Lemma 12. The value of the number of iterations of the edge sequence, which is less than 0, is subject to the reverse operation.
Lemma 13. The distance between the symbols s i and s j at the node address S = s 1 s 2 s 3 · · · s i · · · s j · · · s n n = 2 k , 1 ≤ k ≤ log 2 n is denoted by p(s i ) − p s j .
The routing algorithm is outlined as follows.
[STEP 1] Symbol 2 is placed to the right of symbol 1. When the node address is divided by half, that is, n 2 , the positions of the two symbols, p(1) and p(2) are checked, and the algorithm is executed according to the following cases. The cases are divided into the cases of p(1), p(2) ≤ n 2 ; p(1), p(2) ≥ n 2 ; p(1) ≤ n 2 and p(2) > n 2 , or p(2) ≤ n 2 and p(1) > n 2 . [STEP 2] i + 1 is placed to the right of symbol i. When the node address is divided by half, the positions of the two symbols, p(i) and p(i + 1) are checked, and the algorithm is executed according to the following cases. The cases are divided into the cases of p(i), p(i + 1) ≤ n 2 , p(i), p(i + 1) > n 2 , and p(i) ≤ n 2 AND p(i + 1) > n 2 OR p(i + 1) ≤ n 2 AND p(i) > n 2 .
[STEP 3] In this algorithm, n is placed at the rightmost position while the relative positions from 1 to n are arranged in an ascending order, and this is the step of matching with the target node ID.
The routing shown in Algorithm 1.
Proof. In the worst case in [STEP 1], the diameter is n.
Because the worst case of p(i + 1) is n 2 + 3, p(i + 1) − p(i) = 1 3 n. The result is obtained as follows.
Because 1 < i < n − 1 in STEP 2, it is iterated by n − 3 times as follows.
In the worst case in STEP 3, the diameter is n 2 − 2. The worst case is p(n) < n 2 − 1, and the algorithm is described as follows.
Therefore, in STEP 3, the worst case is n 2 − 2. As a result, it can be seen that the diameter in the worst case of [STEP 1, 2, 3] is 2 3 n 2 − 3 2 n + 1 or less.
For example, when n = 6, in the worst case, the length is 16 as follows. Proof. The network cost is represented by the degree number X diameter. The network cost of NSEP n is as follows.
In Table 1, the network cost of NSEP n was compared with the constant branching class connections. NSEP n increases the number of nodes rapidly as n increases. Thus, some of the network costs for each network were rearranged in Table 2 when the number of nodes was equal, and the results were shown in Table 3 and Figure 8 as a graph. Here, the network cost of NSEP n is always less than that of SEP n , and we can see that when it is n > 10, the network cost of NSEP n is the smallest.   In Figure 8, the five circles at the right end of the chart represent network costs of the mesh, honeycomb, SEP, torus, and NSEP, in that order from the top, when the number of nodes is 4 × 10 8 .

Conclusions
The SEP interconnection network has three degrees and a diameter of (9 − 22 + 24). This study proposed a new interconnection network NSEP by adding a new edge to the SEP network. In the NSEP network, the diameter and network cost were improved by reducing the distance between two nodes in a distance to one by adding one edge to the existing SEP network.
The interconnection network proposed in this study has the same number of nodes as SEP, having four degrees, a diameter of − + 1, and a network cost of ( ). The interconnection network shows excellent results by reducing the diameter by 40% or more and the network cost by 20% or more, while increasing the number of degrees by one in comparison to SEP. The interconnection network NSEP is a network with a Hamiltonian cycle and SEP as a subgraph. Because the NSEP network is defined to only have an even number of nodes (n = 2k), a generalized graph definition is additionally required. The algorithm designed in this paper is an algorithm that sorts symbols 1 through n. In some cases, the opposite arrangement of n through 1 may be effective. Further research will be required under conditions that allow us to select efficient algorithms between the two algorithms. It is hoped that this will lead to research on interconnected networks to improve the performance of parallel processing computers.  In Figure 8, the five circles at the right end of the chart represent network costs of the mesh, honeycomb, SEP, torus, and NSEP, in that order from the top, when the number of nodes is 4 × 10 8 .

Conclusions
The SEP interconnection network has three degrees and a diameter of 1 8 9n 2 − 22n + 24 . This study proposed a new interconnection network NSEP by adding a new edge to the SEP network. In the NSEP network, the diameter and network cost were improved by reducing the distance between two nodes in a n 2 distance to one by adding one edge to the existing SEP network.
The NSEP n interconnection network proposed in this study has the same number of nodes as SEP, having four degrees, a diameter of 2 3 n 2 − 3 2 n + 1, and a network cost of O n 2 . The interconnection network NSEP n shows excellent results by reducing the diameter by 40% or more and the network cost by 20% or more, while increasing the number of degrees by one in comparison to SEP. The interconnection network NSEP is a network with a Hamiltonian cycle and SEP as a subgraph. Because the NSEP network is defined to only have an even number of nodes (n = 2k), a generalized graph definition is additionally required. The algorithm designed in this paper is an algorithm that sorts symbols 1 through n. In some cases, the opposite arrangement of n through 1 may be effective. Further research will be required under conditions that allow us to select efficient algorithms between the two algorithms. It is hoped that this will lead to research on interconnected networks to improve the performance of parallel processing computers.