Properties Of Hypercube Research Articles

Hypercubes and Hamilton cycles of display sets of rooted phylogenetic networks

In the context of reconstructing phylogenetic networks from a collection of phylogenetic trees, several characterisations and subsequently algorithms have been established to reconstruct a phylogenetic network that collectively embeds all trees in the input in some minimum way. For many instances however, the resulting network also embeds additional phylogenetic trees that are not part of the input. However, little is known about these inferred trees. In this paper, we explore the relationships among all phylogenetic trees that are embedded in a given phylogenetic network. First, we investigate some combinatorial properties of the collection P of all rooted binary phylogenetic trees that are embedded in a rooted binary phylogenetic network N. To this end, we associated a particular graph G, which we call rSPR graph, with the elements in P and show that, if |P|=2k, where k is the number of vertices with in-degree two in N, then G has a Hamilton cycle. Second, by exploiting rSPR graphs and properties of hypercubes, we turn to the well-studied class of rooted binary level-1 networks and give necessary and sufficient conditions for when a set of rooted binary phylogenetic trees can be embedded in a level-1 network without inferring any additional trees. Lastly, we show how these conditions translate into a polynomial-time algorithm to reconstruct such a network if it exists.

Strong Menger Connectivity of Folded Hypercubes with Faulty Subcube

Menger-type problems in interconnection networks have received many attentions in recent years. A connected graph [Formula: see text] is strong Menger (edge) connected if there are [Formula: see text] vertex (edge)-disjoint paths joining any two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. Fault tolerance is an important criterion in the design of interconnection networks. The folded hypercube [Formula: see text] is an important variant of hypercube [Formula: see text] which remains many desirable properties of hypercube. We consider the strong Menger connectivity of folded hypercubes when part of the network is faulty. We show that [Formula: see text] [Formula: see text] is strong Menger (edge) connected. Which means that when a subcube [Formula: see text] is faulty, the surviving graph [Formula: see text] is strong Menger (edge) connected. This generalizes the result of [Formula: see text] in [J. Parallel Distrib. Comput. 138 (2020) 190–198].

Spectral properties of hypercubes with applications

In this paper, we study spectral properties of the hypercubes, a special kind of Cayley graphs. We determine explicitly all the eigenvalues and their corresponding multiplicities of the normalized Laplacian matrix of the hypercubes by a recursive method. As applications of these results, we derive the explicit formula to the eigentime identity for random walks on the hypercubes and show that it grows linearly with the network order. Moreover, we compute the number of spanning trees and the degree-Kirchhoff index of the hypercubes. Finally, we study the susceptible–infectious–susceptible (SIS) dynamics on the hypercubes and determine the epidemic threshold based on the spectral radius of the adjacency matrix. Throughout this paper, two numerical experiments are conducted based on the dynamics of complex networks, namely, random walks and epidemic spreading, and the results are consistent with our theoretical analysis.

Subgraph-based Strong Menger Connectivity of Hypercube and Exchanged Hypercube

Large scale multiprocessor systems or multicomputer systems, taking interconnection networks as underlying topologies, have been widely used in the big data era. Fault tolerance is becoming an essential attribute in multiprocessor systems as the number of processors is getting larger. A connected graph [Formula: see text] is called strong Menger (edge) connected if, for any two distinct vertices [Formula: see text] and [Formula: see text], there are [Formula: see text] vertex (edge)-disjoint paths between them. Exchanged hypercube [Formula: see text], as a variant of hypercube [Formula: see text], remains lots of preferable fault tolerant properties of hypercube. In this paper, we show that [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] are strong Menger (edge) connected, respectively. Moreover, as a by-product, for dual cube [Formula: see text], one popular generalization of hypercube, [Formula: see text] is also showed to be strong Menger (edge) connected, where [Formula: see text].

Hypercube-Based SNR-Adaptive Multidimensional Constellation Design for Uplink SCMA Systems

Designing multidimensional constellations (MdCs) is an integral part of sparse code multiple access (SCMA). Since the optimal maximum a posteriori (MAP) receiver for SCMA is too complex in most applications, one highly popular technique is the near-optimal message passing algorithm (MPA), where its performance improves with increasing the signal-to-noise ratio (SNR) and the number of iterations. When the number of MPA iterations has to be limited (e.g., low-latency and/or low-complexity and/or energy-sensitive applications), the performance gap between MAP and MPA becomes significant, especially at low-to-medium SNRs. Inspired by the promising features of hypercubes when used along with bit-interleaved coded modulation, we construct novel MdCs which are based on a unitary rotation of a hypercube by a rotation angle that aims to achieve the minimum frame-error-rate (FER). By exploiting special properties of hypercubes, we fit a second-order rational polynomial to a few measured FER samples, and find a close-to-optimal rotation angle at each SNR and MPA iteration. Our proposed MdCs provide substantial performance gains (as much as 2 dB) in comparison to the best known SCMA MdCs in the literature, especially in low-to-medium SNR regions when the number of MPA iterations has to be low, and in the presence of 5G-compliant LDPC codes.

Super Fibonacci Graceful Graphs and Fibonacci Cubes

The popularity of Fibonacci cubes is due to their wide range of uses. In mathematical chemistry, this concept is used in the study of hexagonal graphs. In computer science, Fibonacci cubes are interesting from an algorithmic point of view. V. Hsu introduced them in 1993 to simulate the connections of multiprocessor computer networks. He wanted to get graphs with hypercube properties, the order of which is not a power of two. Therefore, the problem of embedding other graphs in Fibonacci cubes is of interest.

Matching Preclusion for Exchanged Hypercubes

As spanning subgraphs of hypercubes, exchanged hypercubes contain less edges but maintain lots of desired properties of hypercubes. This paper considers matching preclusion, a kind of measures of edge-fault tolerance, of exchanged hypercubes EH(s, t). We show that EH(s, t) is maximally matched, that is, for s ≥ t, mp(EH(s, t)) = t + 1 and EH(s, t) is super matched if and only if (s, t) ≠ (1, 1). Comparing with results of matching preclusion for hypercubes, we conclude that exchanged hypercubes maintain the matching preclusion property of hypercubes, except for EH(1, 1).

Routing and wavelength assignment for augmented cubes in array-based wavelength-division-multiplexing optical networks

ABSTRACTThe augmented cube is a class of communication patterns that are employed by a number of typical parallel algorithms. It is one of the hypercube variations, and retains all favourable properties of hypercube. This paper addresses the routing and wavelength assignment (RWA) for the augmented cube communication pattern in a linear array wavelength-division-multiplexing optical network. By using congestion estimation and giving a RWA strategy, we prove that the optimal number of wavelengths is , where N denotes the size of the augmented cube.

Progressive Latin Hypercube Sampling: An efficient approach for robust sampling-based analysis of environmental models

Efficient sampling strategies that scale with the size of the problem, computational budget, and users’ needs are essential for various sampling-based analyses, such as sensitivity and uncertainty analysis. In this study, we propose a new strategy, called Progressive Latin Hypercube Sampling (PLHS), which sequentially generates sample points while progressively preserving the distributional properties of interest (Latin hypercube properties, space-filling, etc.), as the sample size grows. Unlike Latin hypercube sampling, PLHS generates a series of smaller sub-sets (slices) such that (1) the first slice is Latin hypercube, (2) the progressive union of slices remains Latin hypercube and achieves maximum stratification in any one-dimensional projection, and as such (3) the entire sample set is Latin hypercube. The performance of PLHS is compared with benchmark sampling strategies across multiple case studies for Monte Carlo simulation, sensitivity and uncertainty analysis. Our results indicate that PLHS leads to improved efficiency, convergence, and robustness of sampling-based analyses.

The $t/s$ -Diagnosability of Hypercube Networks Under the PMC and Comparison Models

A t/s-diagnosable system, a generalization of t/t-diagnosable system, refers to such a system that all the faulty nodes of the system can be isolated within a set of size at most s in the presence of at most t faulty nodes. In this paper, the t/s-diagnosability of the hypercubes under the PMC model (the comparison model) is evaluated. First, several novel properties of hypercube are proposed, which are previously unknown in the literatures. Second, based on the above properties of hypercubes, we show that an n-dimensional (n ≥ 5) hypercube is (kn - ((k(k + 1))/2) + 1)/(kn - ((k(k + 1))/2) + k - 1)-diagnosable in terms of both the PMC and the comparison models, where 2 ≤ k ≤ n - 2. Furthermore, we introduce a fast diagnosis algorithm to isolate the faulty nodes in a subset of the system under the PMC model (the comparison model). And the time complexity of the algorithm is O(n2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) for an n-dimensional hypercube.

Combining SAT Solvers with Computer Algebra Systems to Verify Combinatorial Conjectures

We present a method and an associated system, called MathCheck, that embeds the functionality of a computer algebra system (CAS) within the inner loop of a conflict-driven clause-learning SAT solver. SAT+CAS systems, a la MathCheck, can be used as an assistant by mathematicians to either find counterexamples or finitely verify open universal conjectures on any mathematical topic (e.g., graph and number theory, algebra, geometry, etc.) supported by the underlying CAS. Such a SAT+CAS system combines the efficient search routines of modern SAT solvers, with the expressive power of CAS, thus complementing both. The key insight behind the power of the SAT+CAS combination is that the CAS system can help cut down the search-space of the SAT solver, by providing learned clauses that encode theory-specific lemmas, as it searches for a counterexample to the input conjecture (just like the T in DPLL (T)). In addition, the combination enables a more efficient encoding of problems than a pure Boolean representation. In this paper, we leverage the capabilities of several different CAS, namely the SAGE, Maple, and Magma systems. As case studies, we study three long-standing open mathematical conjectures, two from graph theory regarding properties of hypercubes, and one from combinatorics about Hadamard matrices. The first conjecture states that any matching of any d-dimensional hypercube can be extended to a Hamiltonian cycle; the second states that given an edge-antipodal coloring of a hypercube there always exists a monochromatic path between two antipodal vertices; the third states that Hadamard matrices exist for all orders divisible by 4. Previous results on the graph theory conjectures have shown the conjectures true up to certain low-dimensional hypercubes, and attempts to extend them have failed until now. Using our SAT+CAS system, MathCheck, we extend these two conjectures to higher-dimensional hypercubes. Regarding Hadamard matrices, we demonstrate the advantages of SAT+CAS by constructing Williamson matrices up to order 42 (equivalently, Hadamard up to order $$4\times 42=168$$4×42=168), improving the bounds up to which Williamson matrices of even order have been constructed. Prior state-of-the-art construction was only feasibly performed for odd numbers, where possible.

推广立方连通圈网络的Hamilton分解的算法

立方连通圈网络是超立方体的有界度变形，它具有超立方体几乎所有的优良性质，而且克服了超立方体顶点度随网络规模增大而增大的缺点，是代替超立方体的一个具有强大竞争力的网络结构。但立方连通圈网络的结构是简单还是复杂呢？这是一个悬而未决的问题。带弦环网络是一类经典的互连网络，该网络具有结构简单等优点。在这篇文章中利用师海忠提出的正则图连通圈网络模型设计出了包含立方连通圈网络的一类网络——推广立方连通圈网络GCCC(n) (n > 2)，证明了GCCC(n) (n > 2)可分解为边不交的一个Hamilton圈和一个完美对集的并，即GCCC(n) (n > 2)是带弦环网络。并给出推广立方连通圈网络分解为边不交的一个Hamilton圈和一个完美对集的并的算法。 The cube-connected cycles network is a hypercube of bounded-degree derivative. It has almost all of the excellent properties of hypercube. It also overcomes the shortcoming of hypercube vertices of degrees which will increase with the augment of the network size. But the cube-connected cycles network is simple or complex? This is an open question. Chordal ring network is a kind of classical interconnection network, which has the advantages of simple structure and so on. In this article, a model of the regular graph connected cycle network which is proposed by Haizhong Shi is used to design a kind of network which contains the cube-connected cycles network—the generalized cube-connected network GCCC(n) (n > 2). The authors prove that GCCC(n) (n > 2) can be decomposed into union of edge-disjoint a Hamiltonian cycle and a perfect matching, namely, GCCC(n) (n > 2) is a chordal ring network. They also make a algorithm for GCCC(n) (n > 2) can be decomposed into union of edge-disjoint a Hamiltonian cycle and a perfect matching.

A new hypercube variant: Fractal Cubic Network Graph

Hypercube is a popular and more attractive interconnection networks. The attractive properties of hypercube caused the derivation of more variants of hypercube. In this paper, we have proposed two variants of hypercube which was called as “Fractal Cubic Network Graphs”, and we have investigated the Hamiltonian-like properties of Fractal Cubic Network Graphs FCNGr(n). Firstly, Fractal Cubic Network Graphs FCNGr(n) are defined by a fractal structure. Further, we show the construction and characteristics analyses of FCNGr(n) where r=1 or r=2. Therefore, FCNGr(n) is a Hamiltonian graph which is obtained by using Gray Code for r=2 and FCNG1(n) is not a Hamiltonian Graph. Furthermore, we have obtained a recursive algorithm which is used to label the nodes of FCNG2(n). Finally, we get routing algorithms on FCNG2(n) by utilizing routing algorithms on the hypercubes.

Average distance, surface area, and other structural properties of exchanged hypercubes

Exchanged hypercubes (Loh et al. in IEEE Trans Parallel Distrib Syst 16:866–874, 2005) are spanning subgraphs of hypercubes with about one half of their edges but still with many desirable properties of hypercubes. In this paper, it is shown that distance properties of exchanged hypercubes are also comparable to the corresponding properties of hypercubes. The average distance and the surface area of exchanged hypercubes are computed and it is shown that exchanged hypercubes have asymptotically the same average distance as hypercubes. Several additional metric and other properties are also deduced and it is proved that exchanged hypercubes are prime with respect to the Cartesian product of graphs.

The domination number of exchanged hypercubes

Exchanged hypercubes (Loh et al., 2005 [13]) are spanning subgraphs of hypercubes with about one half of their edges but still with many desirable properties of hypercubes. Lower and upper bounds on the domination number of exchanged hypercubes are proved which in particular imply that γ(EH(2,t))=2t+1 holds for any t⩾2. Using Hamming codes we also prove that γ(EH(s,2k−1))⩽(2s−2k)γ(Q2k−1)+22k−1(γ(Qs−)+1) holds for s⩾k⩾3.

DVcube: A novel compound architecture of disc-ring graph and hypercube-like graph

In this paper, we first introduce a family of interconnection network topologies, named disc-ring graphs. Disc-ring graphs possess many desirable topological properties in building parallel machines, such as fixed degree, small diameter, Hamiltonian decomposition, etc. We first examine the topological properties of disc-ring graphs and give an efficient routing algorithm for disc-ring graphs. We then bring forward a novel interconnection network of DQcube, which is a compound graph of disc-ring graph and hypercube. That is, it uses the hypercube as a unit cluster and connects many such clusters by means of a disc-ring graph at the cost that only one additional link is added to any node in each hypercube. The topological benefits of both basic graphs are preserved in the compound network. It utilizes the topological properties of hypercube to conveniently embed parallel algorithms into each cluster and the short diameter of disc-ring graph to support an efficient inter-cluster communication. The proposed methodologies for DQcube are further applied to many other compound graphs of disc-ring graph and hypercube-like graphs, including locally twisted cube, twisted cube, crossed cube, and Möbius cube, with diameter being a half of that of hypercube. We also give and analyze an efficient routing algorithm for the proposed compound networks in this article.

The 2-path-bipanconnectivity of hypercubes

In this paper, we introduce the concept of the k-path-(bi)panconnectivity of (bipartite) graphs. It is a generalization of the (bi)panconnectivity and of the paired many-to-many k-disjoint path cover. The 2-path-bipanconnectivity with only one exception of the n-cube Qn (n⩾4) is proved. Precisely, the following result is obtained: In an n-cube with n⩾4 given any four vertices u1, v1, u2, v2 such that two of them are in one partite set and the another two are in the another partite set. Let s=t=5 if C=u1u2v1v2 is a cycle of length 4, and s=d(u1,v1)+1 and t=d(u2,v2)+1 otherwise, where d(u,v) denotes the distance between two vertices u and v. And let i and j be any two integers such that both i−s⩾0 and j−t⩾0 are even with i+j⩽2n. Then there exist two vertex-disjoint (u1,v1)-path P and (u2,v2)-path R with ∣V(P)∣=i and ∣V(R)∣=j. As consequences, many properties of hypercubes, such as bipanconnectivity, bipanpositionable bipanconnectivity [18], bipancycle-connectivity [12], two internally disjoint paths with two given lengths, and the 2-disjoint path cover with a path of a given length [21], follow from our result.

Comparison of Topological Property of Perfect Difference Network and Hypercube

Perfect difference network are depend on the mathematical notion of perfect difference set.It constitute high performance interconnection networks for parallel and distribution system. Hypercube is loosely coupled parallel processor. The hypercube modeled as a discrete space with discrete time because the processors are in hamming distances. Here we discussed some basic properties of hypercube and PDN

Mutually independent bipanconnected property of hypercube

A graph is denoted by G with the vertex set V( G) and the edge set E( G). A path P = 〈 v 0, v 1, … , v m 〉 is a sequence of adjacent vertices. Two paths with equal length P 1 = 〈 u 1, u 2, … , u m 〉 and P 2 = 〈 v 1, v 2, … , v m 〉 from a to b are independent if u 1 = v 1 = a, u m = v m = b, and u i ≠ v i for 2 ⩽ i ⩽ m − 1. Paths with equal length { P i } i = 1 n from a to b are mutually independent if they are pairwisely independent. Let u and v be two distinct vertices of a bipartite graph G, and let l be a positive integer length, d G ( u, v) ⩽ l ⩽ ∣ V( G) − 1∣ with ( l − d G ( u, v)) being even. We say that the pair of vertices u, v is ( m, l)- mutually independent bipanconnected if there exist m mutually independent paths P i l i = 1 m with length l from u to v. In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices u and v in the n-dimensional hypercube, with d Q n ( u , v ) ⩾ n - 1 , is ( n − 1, l)-mutually independent bipanconnected for every l , d Q n ( u , v ) ⩽ l ⩽ | V ( Q n ) - 1 | with ( l - d Q n ( u , v ) ) being even. As for d Q n ( u , v ) ⩽ n - 2 , it is also ( n − 1, l)-mutually independent bipanconnected if l ⩾ d Q n ( u , v ) + 2 , and is only ( l, l)-mutually independent bipanconnected if l = d Q n ( u , v ) .

KCube: A novel architecture for interconnection networks

This paper proposes a novel architecture called KCube. KCube is a compound graph of Kautz digraph and hypercube. It employs the hypercube topology as a unit cluster and connects many such clusters by means of a Kautz digraph. It then utilizes the topological properties of hypercube to realize convenient embedding of parallel algorithms, and the short diameter of Kautz graph to support efficient inter-cluster communication. KCube possesses many attractive characteristics, such as modularity, expansibility, and regularity, while these benefits are achieved at the cost of only increasing the degree of any node by one, regardless of the network size. The methodology to construct KCube can also be applied to other compound networks after minimal modifications.