scholarly journals Design and Analysis of a Symmetric Log Star Graph with a Smaller Network Cost Than Star Graphs

Electronics ◽  
2021 ◽  
Vol 10 (8) ◽  
pp. 981
Author(s):  
Jung-Hyun Seo ◽  
Hyeong-Ok Lee
Keyword(s):  
Parallel Computing ◽  
Computer Science ◽  
Regular Graph ◽  
Connected Graph ◽  
Routing Algorithm ◽  
Star Graph ◽  
Topological Properties ◽  
Star Graphs ◽  
Network Cost ◽  
Network Costs

Graphs are used as models to solve problems in fields such as mathematics, computer science, physics, and chemistry. In particular, torus, hypercube, and star graphs are popular when modeling the connection structure of processors in parallel computing because they are symmetric and have a low network cost. Whereas a hypercube has a substantially smaller diameter than a torus, star graphs have been presented as an alternative to hypercubes because of their lower network cost. We propose a novel log star (LS) that is symmetric and has a lower network cost than a star graph. The LS is an undirected, recursive, and regular graph. In LSn, the number of nodes is n! while the degree is 2log2n − 1 and the diameter is 0.5n(log2n)2 + 0.75nlog2n. In this study, we analyze the basic topological properties of LS. We prove that LSn is a symmetrical connected graph and analyzed its subgraph characteristics. Then, we propose a routing algorithm and derive the diameter and network cost. Finally, the network costs of the LS and star graph-like networks are compared.

UNIDIRECTIONAL (n, k)-STAR GRAPHS

2002 ◽  
Vol 03 (01n02) ◽  
pp. 19-34 ◽  
Author(s):  
EDDIE CHENG ◽  
MARC J. LIPMAN
Keyword(s):  
Directed Graph ◽  
Connected Graph ◽  
Interconnection Networks ◽  
Routing Algorithm ◽  
Optimal Routing ◽  
Star Graph ◽  
Star Graphs

Arrangement graphs14 and (n, k)-star graphs11 were introduced as generalizations of star graphs1. They were introduced to provide a wider set of choices for the order of topologically attractive interconnection networks. Unidirectional interconnection networks are more appropriate in many applications. Cheng and Lipman5, and Day and Tripathi17 studied the unidirectional star graphs, and Cheng and Lipman7 studied orientation of arrangement graphs. In this paper, we show that every (n, k)-star graph can be oriented to a maximally arc-connected graph when the regularity of the graph is even. If the regularity is odd, then the resulting directed graph can be augmented to a maximally arc-connected directed graph by adding a directed matching. In either case, the diameter of the resulting directed graph is small. Moreover, there exists a simple and near-optimal routing algorithm.

THE SPANNING CONNECTIVITY OF THE (n,k)-STAR GRAPHS

2006 ◽  
Vol 17 (02) ◽  
pp. 415-434 ◽  
Author(s):  
HONG-CHUN HSU ◽  
CHENG-KUAN LIN ◽  
HUA-MIN HUNG ◽  
LIH-HSING HSU
Keyword(s):  
Regular Graph ◽  
Disjoint Paths ◽  
Star Graph ◽  
Star Graphs

A k-containerC(u, v) of a graph G is a set of k-disjoint paths joining u to v. A k-container C(u, v) is a k*-container if every vertex of G is incident with a path in C(u, v). A graph G is k*-connected if there exists a k*-container between any two distinct vertices u and v. A k-regular graph G is super spanning connected if G is i*-connected for all 1 ≤ i ≤ k. In this paper, we prove that the (n, k)-star graph Sn,k is super spanning connected if n ≥ 3 and (n-k) ≥ 2.

Fault Tolerance on Star Graphs

1997 ◽  
Vol 08 (02) ◽  
pp. 127-142 ◽  
Author(s):  
Shuo-Cheng Hu ◽  
Chang-Biau Yang
Keyword(s):  
Fault Tolerance ◽  
Fault Tolerant ◽  
Routing Algorithm ◽  
Constant Time ◽  
Multiprocessor Systems ◽  
Star Graph ◽  
Cycle Structure ◽  
Star Graphs ◽  
Broadcasting Algorithm

The capability of fault tolerance is one of the advantages of multiprocessor systems. In this paper, we prove that the fault tolerance of an n-star graph is 2n-5 with restriction to the forbidden faulty set. And we propose an algorithm for examining the connectivity of an n-star graph when there exist at most 2n - 4 faults. The algorithm requires O(n2 log n) time. Besides, we improve the fault-tolerant routing algorithm proposed by Bagherzadeh et al. by calculating the cycle structure of a permutation and the avoidance of routing message to a node without any nonfaulty neighbor. This calculation needs only constant time. And then, we propose an efficient fault-tolerant broadcasting algorithm. When there is no fault, our broadcasting algorithm remains optimal. The penalty is O(n) if there exists only one fault, and the penalty is O(n2) if there exist at most n - 2 faults.

ANALYSIS OF A CLASS OF NETWORKS BASED ON CUBIC STAR GRAPHS

1994 ◽  
Vol 04 (02) ◽  
pp. 191-222
Author(s):  
S.V.R. MADABHUSHI ◽  
S. LAKSHMIVARAHAN ◽  
S.K. DHALL
Keyword(s):  
Interconnection Networks ◽  
Fault Tolerant ◽  
Cayley Graphs ◽  
Binary Trees ◽  
Star Graph ◽  
Cubic Graphs ◽  
Topological Properties ◽  
Star Graphs ◽  
New Class ◽  
Edge Transitive

A new class of interconnection networks based on a family of graphs, called cubic graphs are introduced. These latter graphs arise as Cayley graphs of certain subgroups of the symmetric group. It turns out that these Cayley graphs are a hybrid between the binary hypercube and the star graph, and hence are called cubic star graphs, and are denoted by CS(m, n), m≥1 and n≥1. CS(m, n) inherits several of the properties of the hypercube and the star graph. In this paper, we present an analysis of the symmetric and topological properties. In particular, it is shown that CS(m, n) is edge transitive and hence maximally fault tolerant. We give an algorithm for finding the shortest path and provide an enumeration of the node disjoint paths. Optimal algorithms for single source and all-source broadcasting (also called gossiping) are derived. It is shown that CS(m, n) is Hamiltonian and interesting embeddings of several cycles, grids, and binary trees are derived. The paper concludes with a comparison of CS(m, n) with the binary hypercube and the star graph.

scholarly journals Recursively Divided Pancake Graphs with a Small Network Cost

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 844
Author(s):  
Jung-Hyun Seo ◽  
Hyeong-Ok Lee
Keyword(s):  
Computer Science ◽  
Sorting Algorithm ◽  
Topological Properties ◽  
Signed Permutation ◽  
Permutation Graphs ◽  
Sorting Problem ◽  
Network Cost

Graphs are often used as models to solve problems in computer science, mathematics, and biology. A pancake sorting problem is modeled using a pancake graph whose classes include burnt pancake graphs, signed permutation graphs, and restricted pancake graphs. The network cost is degree × diameter. Finding a graph with a small network cost is like finding a good sorting algorithm. We propose a novel recursively divided pancake (RDP) graph that has a smaller network cost than other pancake-like graphs. In the pancake graph Pn, the number of nodes is n!, the degree is n − 1, and the network cost is O(n2). In an RDPn, the number of nodes is n!, the degree is 2log2n − 1, and the network cost is O(n(log2n)3). Because O(n(log2n)3) < O(n2), the RDP is superior to other pancake-like graphs. In this paper, we propose an RDPn and analyze its basic topological properties. Second, we show that the RDPn is recursive and symmetric. Third, a sorting algorithm is proposed, and the degree and diameter are derived. Finally, the network cost is compared between the RDP graph and other classes of pancake graphs.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

THE CONNECTION MACHINE IN A UNIVERSITY ENVIRONMENT: GESAMTHOCHSCHULE WUPPERTAL

1993 ◽  
Vol 04 (01) ◽  
pp. 137-141
Author(s):  
KLAUS SCHILLING
Keyword(s):  
Parallel Computing ◽  
Computer Science ◽  
Large Scale ◽  
Early History ◽  
Short Account ◽  
University Environment ◽  
Connection Machine ◽  
The University

A short account is presented on the early history, the intentions and the development of large scale parallel computing at the University of Wuppertal. It might serve as an illustration how common activities between computational and computer science can be stimulated, in the university environment.

Dagstuhl-Seminar “Dynamically and Partially Reconfigurable Architectures” (Dynamisch und partiell rekonfigurierbare Architekturen)

2004 ◽  
Vol 46 (4) ◽  
Author(s):  
Jürgen Becker
Keyword(s):  
Information Technology ◽  
Parallel Computing ◽  
Computer Science ◽  
Mobile Communication ◽  
Data Stream ◽  
Electrical Engineering ◽  
Reconfigurable Architectures ◽  
Automotive Application ◽  
Partial Reconfiguration ◽  
Computing Technique

SummaryThe paper addresses people from information technology, electrical engineering, computer science, and related areas. It gives an introduction and classification to fine-, coarse-, as well as multi-grain reconfigurable architectures. This data-stream-based and transport-triggered parallel computing technique in combination with dynamical and partial reconfiguration features demonstrates promising perspectives for future CMOS-based microelectronic solutions in multimedia and infotainment, mobile communication, as well as automotive application domains, among others.

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