Guest Graph Research Articles

Optimal embedding of hypercube into exchanged hypercube and optical multi-mesh hypercube

Embedding a (guest) graph into (host) another graph can be utilized for structural simulation, processor allocation, and algorithm porting in the field of parallel architecture. Dilation and edge congestion are two prominent indicators which evaluate the quality of the embedding. In this paper, we consider the guest graph as hypercube network and the host graphs to be exchanged hypercube and optical multi-mesh hypercube. We propose new ordering which minimizes dilation and edge congestion. We also compute the layout of embedding hypercube into the aforementioned interconnection networks.

Minimum Congestion and Wirelength of Embedding the nth Cartesian Product of K4 into Various Kinds of Grid Networks

Embedding an interconnection network into another network is one of the main problems in parallel processing and computing systems. Interconnection networks in multiprocessing systems facilitate effective communication among various system components. Obtaining the minimum cutwidth, congestion, and wirelength in graph embedding problems is of great significance for network design and architecture, where the minimum is taken over all embedding of guest graphs into host graphs, and addressing these issues can reduce time and cost in embedded design. This work aims to accurately calculate the minimum congestion and wirelength for the [Formula: see text] (the [Formula: see text]th Cartesian product of [Formula: see text]) layout into [Formula: see text]-dimensional, [Formula: see text]-dimensional, and [Formula: see text]-dimensional grids by the optimal solution of edge isoperimetric problem on [Formula: see text].

Optimization of layout for embedding half hypercube into conventional tree architectures

SummaryEmbedding a graph into another graph can be utilized for structural simulation, processor allocation, and algorithm porting in the field of parallel architecture. This has the potential to enhance the physical layout of network‐on‐chip (NoC) devices as well as to investigate their virtualization possibilities. Layout is one of the many indicators of graph embedding. An optimal layout in NoC design can result in a decreased wiring area and cost, as well as the reduction in communication delay between parallel processing components. In this work, the guest graph is the half hypercube, which possess efficient routing, fault tolerance, and Hamiltonian features. The edge isoperimetric problems is solved for the half hypercube using a novel technique called the isochronal ordering. The host graphs considered in our work are complete binary trees, ‐rooted complete binary trees, and other few predominantly investigated tree based architectures.

Exact Wirelength of Embedding 3-Ary n-Cubes into Certain Cylinders and Trees

Graph embeddings play a significant role in the design and analysis of parallel algorithms. It is a mapping of the topological structure of a guest graph G into a host graph H, which is represented as a one-to-one mapping from the vertex set of the guest graph to the vertex set of the host graph. In multiprocessing systems, the interconnection networks enhance the efficient communication between the components in the system. Obtaining minimum wirelength in embedding problems is significant in the designing of networks and simulating one architecture by another. In this paper, we determine the wirelength of embedding 3-ary n-cubes into cylinders and certain trees.

Optimal Wirelength of Balanced Complete Multipartite Graphs onto Cartesian Product of {Path, Cycle} and Trees

In any interconnection network, task allocation plays a major role in the processor speed as fair distribution leads to enhanced performance. Complete multipartite networks serve well for this purpose as the task can be split into different partites which improves the degree of reliability of the network. Such an allocation process in the network can be done by means of graph embedding. The optimal wirelength of a graph embedding helps in the distribution of deterministic algorithms from the guest graph to other host graphs in order to incorporate its unique deterministic properties on that chosen graph. In this paper, we propose an algorithm to compute the optimal wirelength of balanced complete multipartite graphs onto the Cartesian product of trees with path and cycle. Moreover, we derive the closed formulae for wirelengths in specific trees like (1-rooted) complete binary tree and sibling graphs.

Embedding Complete Bipartite Graph into Quadrilateral Snake

This paper deals with the technique that maps a guest graph into a host graph, known as Graph embedding. This tool is used in the simulation of interconnection networks. The exact wirelength of embedding complete bipartite graphs into quadrilateral snakes is considered as a research problem in this paper.

Embedding of Complete graphs and Cycles into Grids with holes

Abstract An important feature of an interconnection network is its ability to efficiently simulate one architecture by another. Such a simulation problem can be mathematically formulated as a graph embedding problem. Although the definition of an embedding is an into mapping from Guest Graph to Host Graph, so far in the literature, the embedding has been considered as a mapping from G onto H. In other words, the number of processors in G and H are considered to be the same. In this paper, we increase the number of processors in H by 1. The question is to find the processor in H which does not have the pre-image under the embedding mapping, so that the wirelength of the embedding is minimum. We relate this problem to finding transmission of vertices in the host graph.

Fault-Tolerant Path-Embedding of Twisted Hypercube-Like Networks (THLNs)

It is known widely that an interconnection network can be denoted by a graph G = ( V , E ) , where V denotes the vertex set and E denotes the edge set. Investigating structures of G is necessary to design a suitable topological structure of interconnection network. One of the critical issues in evaluating an interconnection network is graph embedding, which concerns whether a host graph contains a guest graph as its subgraph. Linear arrays (i.e., paths) and rings (i.e., cycles) are two ordinary guest graphs (or basic networks) for parallel and distributed computation. In the process of large-scale interconnection network operation, it is inevitable that various errors may occur at nodes and edges. It is significant to find an embedding of a guest graph into a host graph where all faulty nodes and edges have been removed. This is named as fault-tolerant embedding. The twisted hypercube-like networks ( T H L N s ) contain several important hypercube variants. This paper is concerned with the fault-tolerant path-embedding of n-dimensional (n-D) T H L N s . Let G n be an n-D T H L N and F be a subset of V ( G n ) ∪ E ( G n ) with | F | ≤ n - 2 . We show that for two different arbitrary correct vertices u and v, there is a faultless path P u v of every length l with 2 n - 1 - 1 ≤ l ≤ 2 n - f v - 1 - α , where α = 0 if vertices u and v form a normal vertex-pair and α = 1 if vertices u and v form a weak vertex-pair in G n - F ( n ≥ 5 ).

Vertex decomposition method for wirelength problem and its applications to enhanced hypercube networks

In this study, the authors discuss the vertex congestion of any embedding from the guest graph into the host graph and outline a rigorous mathematical method to compute the wirelength of that embedding. Further, they show that the computation of the optimal wirelength depends on finding optimal solutions for another graph partition problem such as edge isoperimetric problem in that guest graph. On the other side, they consider an important variant of the popular hypercube network, the enhanced hypercube, and obtain the nested optimal solutions for the edge isoperimetric problem. As a combined output, they illustrate the authors' technique by embedding enhanced hypercube into a caterpillar and from that reducing the linear layout of the enhanced hypercube. As another application of their technique, they embed the hypercube as well as the enhanced hypercube on the two rows extended grid structure with optimal wirelength for the first time and showing that the existing edge congestion technique cannot be used to solve this problem.

A linear time algorithm for embedding locally twisted cube into grid network to optimize the layout

Given a guest graph G and a host graph H, the layout of an embedding from G into H is to find the sum of lengths of all the shortest paths in H corresponding to the edges in G. In this paper, we present an algorithm to embed a variant of the binary hypercube known as the locally twisted cube into a 2-dimensional grid architecture and obtain the exact layout of the embedding in linear time.

Wirelength of embedding complete multipartite graphs into certain graphs

Graph embedding is an important technique that maps a guest graph into a host graph, usually an interconnection network. Many applications can be modeled as graph embedding. In architecture simulation, graph embedding has been known as a powerful tool for implementation of parallel algorithms or simulation of different interconnection networks. The quality of an embedding can be measured by certain cost criteria. One of these criteria is the wirelength and has been well studied by many authors (Lai and Tsai, 2010 [18]; Fan and Jia, 2007 [10]; Han et al., 2010 [15]; Fang and Lai, 2005 [11]; Park and Chwa, 2000 [23]; Rajasingh et al., 2004; Yang et al., 2010; Yang, 2009 [27]; Manuel et al., 2009; Bezrukov et al., 1998; Rostami and Habibi, 2008 [26]; Choudum and Nahdini, 2004 [7]; Guu, 1997 [14]). In this paper, we compute the exact wirelength of embedding complete multipartite graphs into certain graphs, such as path, cycle, wheel, hypertree, cylinder, torus and 3-regular circulant graphs.

Fault-tolerant cycles embedding in folded hypercubes

The generalized conditional fault-tolerant embedding is investigated, in which the n-dimensional folded hypercube networks (denoted by FQ n ) acts as the host graph, and the longest fault-free cycle represents the guest graph. Under the conditions looser than that of previous works, it is shown that FQ n has a cycle with length at least 2 n -2|F v | when the number of faulty vertices and non-critical edges is at most 2n - 4; where |F v | is the number of faulty vertices. It provides further theoretical evidence for the fact that FQ n has excellent node-fault-tolerance and edge-fault-tolerance when used as a topology of large scale computer networks.

Optimal Edge Fault-Tolerant Bijective Embedding of a Complete Graph over a Cycle

An embedding of a guest graph G over a host graph H is an injective map Φ from the vertices of G to the vertices of H and a routing map ρ, which associates every edge e=xy in G to a Φ(x)-Φ(y) path ρ(e) in H. Given an edge f in H the number of edges e in G such that f belongs to ρ(e) is the (edge) congestion cong(f) of f. The length of ρ(e) is called the dilatation dil(e) of e. The sum of all th dilatations is the cost of the embedding. The removal of an edge f of H gives rise to a surviving graphGf, consisting of the guest graph without those edges that cross f, i.e., Gf=G−{e:f∈ρ(e)}. Given n and b, we are facing the problem of finding a minimum cost embedding Φ of a graph G with n vertices over the cycle Cn, such that for any surviving graph Gf, there is an embedding of the complete graph Kn over Gf whose congestions are not greater than b. This work presents a lower bound for the optimal cost of such problem and a family of embeddings that match this bound over a broad range of combinations of n and b.

Embedding hypercubes and folded hypercubes onto Cartesian product of certain trees

The hypercube network is one of the most popular interconnection networks since it has simple structure and is easy to implement. The folded hypercube is an important variation of the hypercube. Interconnection networks play a major role in the performance of distributed memory multiprocessors and the one primary concern for choosing an appropriate interconnection network is the graph embedding ability. A graph embedding of a guest graph G into a host graph H is an injective map on the vertices such that each edge of G is mapped into a path of H. The wirelength of this embedding is defined to be the sum of the lengths of the paths corresponding to the edges of G. In this paper we embed hypercube and folded hypercube onto Cartesian product of trees such as 1-rooted complete binary tree and path, sibling tree and path to minimize the wirelength.

Bothway embedding of circulant network into grid

Graph embedding is an important technique that maps a guest graph into a host graph, usually an interconnection network. In this paper, we compute the dilation and wirelength of embedding circulant network into grid and vice versa.

Separator-based graph embedding into multidimensional grids with small edge-congestion

We study the problem of embedding a guest graph with minimum edge-congestion into a multidimensional grid with the same size as that of the guest graph. Based on a well-known notion of graph separators, we show that an embedding with a smaller edge-congestion can be obtained if the guest graph has a smaller separator, and if the host grid has a higher but constant dimension. Specifically, we prove that any graph with N nodes, maximum node degree Δ, and with a node-separator of size s, where s is a function such that s(n)=O(nα) with 0≤α<1, can be embedded into a grid of a fixed dimension d≥2 with at least N nodes, with an edge-congestion of O(Δ) if d>1/(1−α), O(ΔlogN) if d=1/(1−α), and O(ΔNα−1+1d) if d<1/(1−α). This edge-congestion achieves constant ratio approximation if d>1/(1−α), and matches an existential lower bound within a constant factor if d≤1/(1−α). Our result implies that if the guest graph has an excluded minor of a fixed size, such as a planar graph, then we can obtain an edge-congestion of O(ΔlogN) for d=2 and O(Δ) for any fixed d≥3. Moreover, if the guest graph has a fixed treewidth, such as a tree, an outerplanar graph, and a series–parallel graph, then we can obtain an edge-congestion of O(Δ) for any fixed d≥2. To design our embedding algorithm, we introduce edge-separators bounding extension, such that in partitioning a graph into isolated nodes using edge-separators recursively, the number of outgoing edges from a subgraph to be partitioned in a recursive step is bounded. We present an algorithm to construct an edge-separator with extension of O(Δnα) from a node-separator of size O(nα).

Tabu search for the cyclic bandwidth problem

The Cyclic Bandwidth (CB) problem for graphs consists in labeling the vertices of a guest graph G by distinct vertices of a host cycle Cn (both of order n) in such a way that the maximum distance in the cycle between adjacent vertices in G is minimized. To the best of our knowledge, this is the first research work investigating the use of metaheuristic algorithms for solving this challenging combinatorial optimization problem in the case of general graphs.In this paper a new carefully devised Tabu Search algorithm, called TScb, for finding near-optimal solutions for the CB problem is proposed. Different possibilities for its key components and input parameter values were carefully analyzed and tuned, in order to find the combination of them offering the best quality solutions to the problem at a reasonable computational effort.Extensive experimentation was carried out, using 113 standard benchmark instances, for assessing its performance with respect to a Simulated Annealing (SAcb) implementation. The experimental results show that there exists a statistically significant performance amelioration achieved by TScb with respect to SAcb in 90 out of 113 graphs (79.646%). It was also found that our TScb algorithm attains 56 optimal solutions and establishes new better upper bounds for the other 57 instances. Furthermore, these competitive results were obtained expending reasonable computational times.

Layout of Embedding Circulant Networks into Linear Hexagons and Phenylenes

Circulant network has been used for decades in the design of computer and telecommunication networks due to optimal fault-tolerance and routing capabilities. Further, it has been used in VLSI design and distributed computation. Hexagonal chains are of great importance of theoretical chemistry because they are the natural graph representations of benzenoid hydrocarbons, a great deal of investigations in mathematical chemistry has been developed to hexagonal chains. Hexagonal chains are exclusively constructed by hexagons of length one. Phenylenes are a class of chemical compounds in which carbon atoms form 6 and 4 membered cycles. Graph embedding has been known as a powerful tool for implementation of parallel algorithms or simulation of different interconnection networks. An embedding f of a guest graph G into a host graph H is a bijection on the vertices such that each edge of G is mapped into a path of H. The wirelength (layout) of this embedding is defined to be the sum of the length of the paths corresponding to the edges of G. In this paper we obtain the minimum wirelength of embedding circulant networks into linear hexagonal chains and linear phenylenes. Further we discuss the embedding of faulty circulant networks into linear hexagonal chains and linear phenylenes.

Finding vertex-surjective graph homomorphisms

The Surjective Homomorphism problem is to test whether a given graph G called the guest graph allows a vertex-surjective homomorphism to some other given graph H called the host graph. The bijective and injective homomorphism problems can be formulated in terms of spanning subgraphs and subgraphs, and as such their computational complexity has been extensively studied. What about the surjective variant? Because this problem is NP-complete in general, we restrict the guest and the host graph to belong to graph classes G and H, respectively. We determine to what extent a certain choice of G and H influences its computational complexity. We observe that the problem is polynomial-time solvable if H is the class of paths, whereas it is NP-complete if G is the class of paths. Moreover, we show that the problem is even NP-complete on many other elementary graph classes, namely linear forests, unions of complete graphs, cographs, proper interval graphs, split graphs and trees of pathwidth at most 2. In contrast, we prove that the problem is fixed-parameter tractable in k if G is the class of trees and H is the class of trees with at most k leaves, or if G and H are equal to the class of graphs with vertex cover number at most k.

A special antidilation problem for meshes and Hamming graphs

The antidilation problem consists of a mapping from the set of vertices of the guest graph G into the set of vertices of the host graph H such that distance of images of adjacent vertices of G is maximized in H. This is a dual problem to the well known dilation problem. In this paper, we study an interesting special case of the antidilation problem when the guest and host graphs are the same. We prove exact results for d-dimensional meshes, tori and Hamming graphs. In all three cases, the antidilation is very close to radius(H), which is a desired property. As a consequence we solve an open problem of Lagarias about the antidilation of paths in d-dimensional meshes.