Abstract
An embedding of a guest network into a host network is to find a suitable bijective function between the vertices of the guest and the host such that each link of is stretched to a path in . The layout measure is attained by counting the length of paths in corresponding to the links in and with a complexity of finding the best possible function overall graph embedding. This measure can be computed by summing the minimum congestions on each link of , called the congestion lemma. In the current study, we discuss and characterize the congestion lemma by considering the regularity and optimality of the guest network. The exact values of the layout are generally hard to find and were known for very restricted combinations of guest and host networks. In this series, we derive the correct layout measures of circulant networks by embedding them into the path- and cycle-of-complete graphs.
1. Introduction
Nowadays, there is an emerging demand for high-performance concurrent functional in different fields which can be successfully achieved through parallel processing techniques. The core of a parallel processing system is the interconnected network by which the system processors are connected. One of the important challenges in parallel processing techniques is how to allocate the subprocesses to the processors within the system in such a way that the total communication cost is minimized. This issue in parallel processing can be reduced to a graph embedding problem [1, 2]. For this purpose, the network topology is formulated as a simple graph, in which the vertex set denotes the system processors and the edge set denotes the links connecting them.
In this paper, the collection of vertices and edges of a simple graph network are, respectively, represented by and . A graph embedding of a guest network GN into a host network HN is a kind of vertex and edge labeling denoted by a and onto mapping together with mapping such that is a to path in , where and contains the collection of routes or paths in [2, 3]. The congestion of an edge of is measured by counting the routes in such that is in the route and denoted by . In other words, . The layout/wire length [4, 5] of by embedding in is defined as
Let be any subset of . If we represent , then , where is a partition. For , construct a set based on the edges of such that each edge in is duplicated -times. Such a set is denoted by . Then,
Furthermore, if , then . The correct layout of by embedding in is measured by
The main objective of parallel computing is to execute embeddings with the correct layout, and we certainly fix the accompanying function such that each edge of is to a shortest path under , see Figure 1. Apart from that, the important topological descriptor, Wiener index [6], which is used in the characterization of chemical compounds can be obtained through , where is the complete graph and is the considered molecular structure.
The minimum layout problem plays an important role in finding an optimal solution for very large-scale integration (VLSI) chips physical layout [2], minimizing time delay of simulations in parallel computer systems, computer aided designs, structural engineering, cloning and visual stimuli models, and parallel architecture [7, 8]. The computation of layout measure has been already studied in a variety of papers, see [9, 10] and the references cited therein, for more details. The present study continues the layout computation of circulant graphs into path- and cycle-of-complete graphs.
2. Congestion Lemma
The combinatorial isoperimetric problems have emerged with important applications in the fields of communication systems and computer and physical sciences related disciplines. Harper [11, 12] has discovered the primary significance of edge isoperimetric problem (EIP), and it has been categorized into two types as follows [13].
Definition 1. (EIP(1)). For a graph network , if , then . Given a positive integer , . Then, EIP(1) finds and such that .
Definition 2. (EIP(2)). For a graph network , if , then . Given a positive integer , . Then, EIP(2) finds and such that .
In such a case, is identified as optimal set corresponding to the EIP.
Lemma 1 (see [11]). (i) For a graph network , for all . (ii) If is an -regular graph, and, for any positive integer , .
The minimum layout of the hypercube network by embedding in a grid structure is derived using congestion lemma [14]. The generalized version of the congestion lemma appeared in [15] and the modified version in [16]. Here, we present a more general result that exemplifies the regularity and the optimality on the guest network.
In what follows, let and be two given networks and . Suppose the removal of from splits the network into components, namely, . A graph embedding of into is -repulsive if, when we let , , the following conditions hold:(i)If , , then (ii)If , such that and , for , then
Lemma 2. Let and be an -repulsive graph embedding of into . We have . Moreover, among all the graph embeddings of into , is minimum if and only if the value is maximum among all partitions of with , .
Proof. Let . Since any edge in either belongs to one of or in and in for some and , we obtainWe now easily compute bearing the conditions of the -repulsive embedding. By assumption (i), the contribution to from the edges of , , is zero. By assumption (ii), every edge of increases by 1. Thus, .
Assume that is minimum. Suppose we had a partition giving a larger value than , and we could define an embedding using this partition such that , a contradiction. Conversely, let be not minimum. Suppose there exists a graph embedding such that , and consequently, we can find a partition with a larger value than , which is not possible because of the -repulsive embedding under .
Lemma 3. Let and be an -repulsive graph embedding of into . Among all the graph embeddings of into , (a) is minimum if s are optimal with respect to EIP(1) and , , and (b) when is an r-regular network, is minimum if s are optimal with respect to EIP(2) and .
Proof. We assume that s are optimal sets with respect to EIP(1). Such a case results in is minimum. Hence, is minimum [15]. By extending the idea to EIP(2), we can easily derive the case of -regular network by applying the simple fact .
It is interesting as well as crucial to note that all s are not optimal, yet imply that is minimum. Furthermore, when , the above lemma is reduced to the modified congestion lemma [16], as in Case (a), and the congestion lemma [14], as in Case (b).
3. Layout Computation
The purpose of the section is to derive the layout of circulant networks into a few graph structures generated from the path and the cycle. We begin with the basic results on circulant networks [2, 17, 18].
Definition 3. (see [2]). A circulant network, denoted by , , , is constructed from such that .
With the optimum fault tolerance and best routing functionality, the circulant network is considered an excellent network over the years on account of its applications in the areas of computer binary code designs and telecommunication network systems. Particularly, circulant network is a natural generalized form of the double loop network, and in addition, a matrix representation generates the circulant if all its rows are periodic rotations of the first one. From the construction of circulant networks, one can easily see that and are, respectively, the cycle and the complete graph . In our study, we denote the cycle as a peripheral cycle. From the symmetry of circulant network, we have that , , is a regular network of degree .
Lemma 4 (see [19]). A set of consecutive vertices of , , is an optimal set with respect to EIP(2) in , , .
Lemma 5 (see [19]). For a circulant network , and , , we have
Definition 4. A path-of-complete graph is obtained by unifying a bone path and complete graphs such that the edge , , of the bone path shares an edge of the complete graph . We denote it by . In an analogous way, we can define a cycle-of-complete graph by combining a bone cycle of length and complete graphs. This graph is denoted by .
Clearly, the number of vertices in and are and , respectively. The different cases of path- and cycle-of-complete graphs are shown in Figure 2. In the literature, these structures are sometimes called necklace graphs and also sharing between graphs by vertices, see [20, 21].
(a)
(b)
(c)
In what follows, and , .
Theorem 1. The minimum layout of circulant network , , into path-of-complete graph such that is given by .
Proof. We begin with the embedding method of and . Label the peripheral cycle vertices of as and the bone path vertices , , of as in such a way that label the vertices (except on the path) of complete graph , , from to . We prove that the graph embedding of into defined by yields the minimum layout. For , let be the set of edges in the complete graph . Then, reduced to components with , , , and . By Lemma 4, the induced subgraph by on is an optimal set. We now conclude that is -repulsive embedding of into . By Lemma 3, is minimum and . Therefore, .
Theorem 2. The minimum layout of circulant network , , into cycle-of-complete graph such that is given by
Proof. We first give the labeling of into and followed by embedding algorithm. Label the peripheral cycle vertices of as and the bone cycle vertices , , of as along the vertices (except on the cycle) of complete graph , , from to . Let be an embedding from into defined by . Case 1 (m even): for , let be the set of edges in the complete graphs and . Then, reduced to number of components s in which components have cardinalities one each and the remaining components with and vertices. By Lemma 4, the induced subgraph by on is an optimal set. Therefore, is -repulsive embedding of into . By Lemma 3, is minimum and . By the construction of the edge cuts, we infer that is a partition of , and hence, . Case 2 (m odd): for , let be the set of edges in the complete graphs and . Then, reduced to number of components s in which components have cardinalities one each and the remaining components with and vertices. By Lemma 4, the induced subgraph by is an optimal set. Therefore, is -repulsive embedding of into . By Lemma 3, is minimum and . We note that is a partition of , and hence, .
4. Conclusion
In analyzing the structural properties of a connected network, the measure such as graph embedding is of greater interest. As we know that the path, cycle, complete, and circulant graphs are important interconnection networks due to their simplicity, in this article, we successfully embedded circulant graphs into path- and cycle-of-complete graphs with minimum layout by the way of characterizing the congestion lemma with respect to regularity and optimality on the guest network.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflict of interest.
Authors’ Contributions
M. A. and A. J. S. conceptualized the study; J. B. L and J. N. D. investigated the study; M. A. wrote and prepared the original draft; A. J. S. and J. N. D. reviewed and edited the manuscript; J. B. L. supervised the study