Cycle Embedding Research Articles

Enabling high fault-tolerant embedding capability of alternating group graphs

The Hamiltonian path/cycle serves as a robust tool for transmitting messages within parallel and distributed systems. However, the prevalent device-intensive nature of these systems often leads to the occurrence of faults. Tackling the critical challenge of tolerating numerous faults when constructing Hamiltonian paths and cycles in these systems is of utmost significance. The alternating group graph AGn is a suitable interconnection network for building parallel and distributed systems due to its outstanding topological properties. Many studies have been dedicated to the fault-tolerant embedding of Hamiltonian paths and cycles in AGn, but they all fail to reach the desired level of fault-tolerance. In this paper, we aim to enhance the edge fault-tolerant embedding capability of AGn by leveraging a newly emerged fault model, called the Partitioned Edge Fault (PEF) model. We prove the existence of Hamiltonian paths/cycles tolerating large-scale edge faults in AGn under the PEF model. Additionally, we propose a fault-tolerant Hamiltonian path embedding algorithm for AGn under the PEF model and validate its effectiveness through experiments. To gauge the significance of each missed edge, we conduct experimental calculations for the average path length of every edge along the Hamiltonian path produced by the proposed embedding algorithm. Furthermore, the fault-tolerance comparison and experimental analysis reveal that our results improve the fault-tolerant embedding capability of AGn from a linear level to an exponential one. To our knowledge, this is the first time that a Hamiltonian path/cycle embedding approach is proposed and actually implemented for AGn with large-scale edge faults.

Cycle Embedding in Enhanced Hypercubes with Faulty Vertices

The enhanced hypercube is a well-known variant of the hypercube and can be constructed from a hypercube by adding an edge to every pair of vertices with complementary addresses. Let Fv denote the set of faulty vertices in an n-dimensional enhanced hypercube Qn,k(1≤k≤n−1). In this paper, we conclude that if n≥2, then every fault-free edge of Qn,k−Fv lies on a fault-free cycle of every even length from 4 to 2n−2|Fv|, and if n(≥2) and k have the different parity, then every fault-free edge of Qn,k−Fv lies on a fault-free cycle of every possible odd length from n−k+4 to 2n−2|Fv|−1, where |Fv|≤n−2.

Hamiltonian cycle embedding with fault-tolerant edges and adaptive diagnosis in half hypercube

Hamiltonian cycle embedding with fault-tolerant edges and adaptive diagnosis in half hypercube

Embedding Spanning Disjoint Cycles in Hypercube Networks with Prescribed Edges in Each Cycle

One of the important issues in evaluating an interconnection network is to study the hamiltonian cycle embedding problems. A graph G is spanning k-edge-cyclable if for any k independent edges e1,e2,…,ek of G, there exist k vertex-disjoint cycles C1,C2,…,Ck in G such that V(C1)∪V(C2)∪⋯∪V(Ck)=V(G) and ei∈E(Ci) for all 1≤i≤k. According to the definition, the problem of finding hamiltonian cycle focuses on k=1. The notion of spanning edge-cyclability can be applied to the problem of identifying faulty links and other related issues in interconnection networks. In this paper, we prove that the n-dimensional hypercube Qn is spanning k-edge-cyclable for 1≤k≤n−1 and n≥2. This is the best possible result, in the sense that the n-dimensional hypercube Qn is not spanning n-edge-cyclable.

Embedding spanning disjoint cycles in augmented cube networks with prescribed vertices in each cycle

One of the important issues in evaluating an interconnection network is to study the Hamiltonian cycle embedding problems. For a positive integer k, a graph G is said to be spanning k-cyclable if for k prescribed vertices , there exist k disjoint cycles such that the union of spans G, and each contains exactly one vertex of . According to the definition, the problem of finding hamiltonian cycle focuses on k = 1. The notion of spanning cyclability can be applied to the problem of identifying faulty processors and other related issues in interconnection networks. The n-dimensional augmented cube is an important node-symmetric variant of the n-dimensional hypercube . In this paper, we prove that with is spanning k-cyclable for .

Classifying RNA Strands with A Novel Graph Representation Based on the Sequence Free Energy

ABSTRACT Ribonucleic acids (RNA) are macromolecules in all living cell, and they are mediators between DNA and protein. Structurally, RNAs are more similar to the DNA. In this paper, we introduce a compact graph representation utilizing the Minimum Free Energy (MFE) of RNA molecules' secondary structure. This representation represents structural components of secondary RNAs as edges of the graphs, and MFE of these components represents their edge weights. The labeling process is used to determine these weights by considering both the MFE of the 2D RNA structures, and the specific settings in the RNA structures. This encoding is used to make the representation more compact by giving a unique graph representation for the secondary structural elements in the graph. Armed with the representation, we apply graph-based algorithms to categorize RNA molecules. We also present the result of the cutting-edge graph-based methods (All Paths Cycle Embeddings (APC), Shortest Paths Kernel/Embedding (SP), and Weisfeiler - Lehman and Optimal Assignment Kernel (WLOA)) on our dataset [1] using this new graph representation. Finally, we compare the results of the graph-based algorithms to a standard bioinformatics algorithm (Needleman-Wunsch) used for DNA and RNA comparison.

Embedding spanning disjoint cycles in enhanced hypercube networks with prescribed vertices in each cycle

Embedding cycles into a network topology is crucial for the network simulation. In particular, embedding Hamiltonian cycles is a major requirement for designing good interconnection networks. A graph G is called k-spanning cyclable if, for any k distinct vertices v1,v2,…,vk of G, there exist k cycles C1,C2,…,Ck in G such that vi is on Ci for every i, and every vertex of G is on exactly one cycle Ci. If k=1, this is the classical Hamiltonian problem. In this study, we focus on embedding spanning disjoint cycles in enhanced hypercube networks and show that the n-dimensional enhanced hypercube Qn,m is k-spanning cyclable if k≤n and n≥4, and k-spanning cyclable if k≤n−1 and n=2,3. Moreover, the results are optimal with respect to the degree of Qn,m, and some experimental examples are provided to verify the theoretical results.

Vertex-pancyclicity of the (n,k)-bubble-sort networks

The cycle embedding is an important problem of networks, which can determine the fault tolerance of the networks. A network can be viewed as a graph. Let m be an integer with m≥4, G be a graph and w∈V(G) be an arbitrary vertex. The graph G is vertex-pancyclic if G has a cycle Cl of length l with w∈V(Cl) for every l∈{3,4,⋯,|V(G)|} and G is m-weak-vertex-pancyclic if G has a cycle Cl of length l with w∈V(Cl) for every l∈{m,m+1,⋯,|V(G)|}. Let G′ be a bipartite graph and w′∈V(G′) be an arbitrary vertex. The graph G′ is vertex-bipancyclic if G′ has a cycle Ch of length h with w′∈V(Ch) for any even integer h with 4≤h≤|V(G′)|. In this paper, we study the cycle embedding in the (n,k)-bubble-sort network Bn,k. We obtain that (1) Bn,1 is vertex-pancyclic for n≥3. (2) Bn,n−1 is vertex-bipancyclic for n≥4. (3) B4,2 and B5,2 are 6-weak-vertex-pancyclic and B5,3 is vertex-pancyclic. (4) Bn,k is vertex-pancyclic for n≥6 with 2≤k≤n−2 and every constructed cycle of Bn,k contains a residual edge for n≥4 with 2≤k≤n−2.

Every edge lies on cycles of folded hypercubes with a pair of faulty adjacent vertices

An n-dimensional folded hypercube FQn is a well-known extended structure of an n-dimensional hypercube Qn, which can be constructed from Qn by adding an edge to every pair of vertices with complementary addresses. FQn possesses many properties that are superior to those of Qn including the diameter, fault diameter, and connectivity. It has been shown that FQn is bipartite for every odd n≥3, and non-bipartite for every even n≥2. In this report, we let {f1,f2} denote the set of a pair of adjacent faulty vertices in FQn. We then show the cycle embedding properties as follows:1. For n≥3, every fault-free edge can lie on a cycle of any even length l with 4≤l≤2n−4 in FQn−{f1,f2};2. For every even n≥4, every fault-free edge can lie on a cycle of every odd length l with n+1≤l≤2n−3 in FQn−{f1,f2}.Our study improves the result of the previous study with respect to the embedding of the longest cycle in a non-bipartite FQn, and the quantity of the faulty vertices in FQ3.

Delivering sustainable design excellence: the potential role of holistic building performance evaluation

The argument for building performance evaluation (BPE) has, to date, largely been made by actors focused upon ensuring the achievement of quantitative standards of performance, such as energy performance and carbon emissions. However, it can also be argued that the value of an understanding of quantitative and qualitative performance and its role in evidence-informed design is much broader. Indeed, that the potential provided by the embedding of learning cycles from BPE stretches across both architecturally and sustainability led design paradigms. Thus providing opportunities to inform the significant changes in architectural practice required to achieve architectural and sustainable holistic design excellence. This paper aims to explore the role that BPE might play in this transition through reporting of performance of Stirling Prize winners, as proxies for Architectural Design Excellence; framework evaluation of the language associated with current BPE practice and establishment of an expanded holistic scope for BPE.

On Virtual Network Embedding: Paths and Cycles

Network virtualization provides a promising solution to overcome the ossification of current networks, allowing multiple Virtual Network Requests (VNRs) to be embedded on a common infrastructure. The major challenge in network virtualization is the Virtual Network Embedding (VNE) problem, which is to embed VNRs onto a shared substrate network and known to be $\mathcal {NP}$ -hard. The topological heterogeneity of VNRs is one important factor hampering the performance of the VNE. However, in many specialized applications and infrastructures, VNRs are of some common structural features, e.g., paths and cycles. To achieve better outcomes, it is thus critical to design dedicated algorithms for these applications and infrastructures by taking into accounting topological characteristics. Besides, paths and cycles are two of the most fundamental topologies that all network structures consist of. Exploiting the characteristics of path and cycle embeddings is vital to tackle the general VNE problem. In this paper, we investigate the path and cycle embedding problems. For path embedding, we prove its $\mathcal {NP}$ -hardness and inapproximability. Then, by utilizing Multiple Knapsack Problem (MKP) and Multi-Dimensional Knapsack Problem (MDKP), we propose an efficient and effective MKP-MDKP-based algorithm. For cycle embedding, we propose a Weighted Directed Auxiliary Graph (WDAG) to develop a polynomial-time algorithm to determine the least-resource-consuming embedding. Numerical results show our customized algorithms can boost the acceptance ratio and revenue compared to generic embedding algorithms in the literature.

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.

Three Types of Two-Disjoint-Cycle-Cover Pancyclicity and Their Applications to Cycle Embedding in Locally Twisted Cubes

Abstract A graph $G=(V,E)$ is two-disjoint-cycle-cover $[r_1,r_2]$-pancyclic if for any integer $l$ satisfying $r_1 \leq l \leq r_2$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ in $G$ such that the lengths of $C_1$ and $C_2$ are $l$ and $|V(G)| - l$, respectively, where $|V(G)|$ denotes the total number of vertices in $G$. On the basis of this definition, we further propose Ore-type conditions for graphs to be two-disjoint-cycle-cover vertex/edge $[r_1,r_2]$-pancyclic. In addition, we study cycle embedding in the $n$-dimensional locally twisted cube $LTQ_n$ under the consideration of two-disjoint-cycle-cover vertex/edge pancyclicity.

Cycles in folded hypercubes with two adjacent faulty vertices

It has been shown that the n-dimensional folded hypercube topology (FQn for short) is bipartite for every odd n≥3, and which is non-bipartite for every even n≥2[9]. In this paper, let {f1,f2} denote the set of a pair of adjacent faulty vertices in FQn. Then, we show the lengths of cycles embedding in FQn−{f1,f2} as follows:1.For n≥3, FQn−{f1,f2} contains a fault-free cycle of every even length from 4 to 2n−2;2.For every even n≥4, FQn−{f1,f2} contains a fault-free cycle of every odd length from n+1 to 2n−3. As a contribution, our study improves the result of [7] with respect to the longest length of cycle embedding in non-bipartite FQn, where the embedded longest length of cycle only up to 2n−2×2−1=2n−5 in FQn−{f1,f2}.

Embedding various cycles with prescribed paths into [formula omitted]-ary [formula omitted]-cubes

The k-ary n-cube has been one of the most popular interconnection networks for large-scale multi-processor systems and data centers. In this study, we investigate the problem of embedding cycles of various lengths passing through prescribed paths in the k-ary n-cube. For n≥2 and k≥5 with k odd, we prove that every path with length h (1≤h≤2n−1) in the k-ary n-cube lies on cycles of every length from h+(k−1)(n−1)/2+k to kn inclusive.

Cycles Embedding in Exchanged Crossed Cube

The (s+t+1)-dimensional exchanged crossed cube, denoted by ECQ(s, t), proposed by Li et al., combines the advantages of the hypercube and the crossed cube. It has been proven that ECQ(s, t) has better properties than the fundamental hypercube in the aspects of the fewer edges, lower cost factor and smaller diameter. This paper studies the embedding of cycles in ECQ(s, t). It is proved that ECQ(s, t) contains an l-cycle of every length l from 4 to 2s+t+1 except that ECQ(2, 3) and ECQ(3, 3) do not contain a cycle of length 9 where [Formula: see text] and [Formula: see text]. This result reveals the fact that ECQ(s, t) nearly remains the cycle embedding capability, while it only has about half edges of crossed cube.

Hamiltonian Cycle and Path Embeddings ink-Aryn-Cubes Based on Structure Faults

Hamiltonian Cycle and Path Embeddings in<i>k</i>-Ary<i>n</i>-Cubes Based on Structure Faults

Embedding cycles and paths on solid grid graphs

Despite many algorithms for embedding graphs on unbounded grids, only a few results on embedding graphs on restricted grids have been published. In this paper, we study the problem of embedding paths and cycles on solid grid graphs. We show that a cycle of length k is unit-length embeddable on a solid grid graph G if k is an even integer between four and the length of the longest cycle of G. In addition, our result shows that a path of length k is unit-length embeddable on G, between its two given vertices s and t, if $$k\le L$$k≤L and $$k\equiv L (\mathrm{mod}\ 2)$$kźL(mod2), in which L is the length of the longest path between s and t. Our presented two algorithms show that such embeddings can be found in linear time for cycles and quadratic time for paths, with respect to the size of graph G. In the case of rectangular grid graphs, the running time of the algorithms can be improved to O(k) and O$$(k^2)$$(k2), respectively. In addition, we extend our results to $$m\times n\times o$$m×n×o 3D grids. A application of our result is in the interconnection network mapping in parallel processing.

Fault-Tolerant Cycle Embedding in Cartesian Product Graphs: Edge-Pancyclicity and Edge-Bipancyclicity with Faulty Edges

A graph $G$ is called $k$ -edge-fault edge-bipancyclic ( $k$ -edge-fault edge- $r$ -pancyclic) if after deleting $k$ edges from $G$ , every edge in the resulting graph lies in a cycle of every even length from 4 to $|V(G)|$ (a cycle of every length from $r$ to $|V(G)|$ ), inclusively. In this paper, given two graphs $G$ and $H$ , which satisfy some specific properties, the edge-fault edge-bipancyclicity and edge-fault edge- $r$ -pancyclicity ( $r$ is decided on the properties of $G$ and $H$ ) of Cartesian product graphs $G\times H$ are efficiently evaluated. The obtained results are applied to two multiprocessor systems, the nearest neighbor mesh hypercubes and generalized hypercubes, both of which belong to Cartesian product graphs.

A Linear Algorithm for Embedding of Cycles in Crossed Cubes with Edge-Pancyclic

In this paper, we consider the problem of embedding a cycle containing number of nodes from 4 to 2^n through a prescribed edge in an n-dimensional crossed cube CQ_n. The main contribution of this paper is providing a systematic O(l) algorithm to find a cycle of length l containing (u, v) in CQ_n for any (u, v) ∈ E(CQ_n) nd any integer l with 4 ≤ l ≤ 2^n.