Honeycomb networks: Topological properties and communication algorithms
01 Oct 1997-IEEE Transactions on Parallel and Distributed Systems (IEEE Press)-Vol. 8, Iss: 10, pp 1036-1042
TL;DR: The honeycomb mesh, based on hexagonal plane tessellation, is considered as a multiprocessor interconnection network and honeycomb networks with rhombus and rectangle as the bounding polygons are considered.
Abstract: The honeycomb mesh, based on hexagonal plane tessellation, is considered as a multiprocessor interconnection network. A honeycomb mesh network with n nodes has degree 3 and diameter /spl ap/1.63/spl radic/n-1, which is 25 percent smaller degree and 18.5 percent smaller diameter than the mesh-connected computer with approximately the same number of nodes. Vertex and edge symmetric honeycomb torus network is obtained by adding wraparound edges to the honeycomb mesh. The network cost, defined as the product of degree and diameter, is better for honeycomb networks than for the two other families based on square (mesh-connected computers and tori) and triangular (hexagonal meshes and tori) tessellations. A convenient addressing scheme for nodes is introduced which provides simple computation of shortest paths and the diameter. Simple and optimal (in the number of required communication steps) routing, broadcasting, and semigroup computation algorithms are developed. The average distance in honeycomb torus with n nodes is proved to be approximately 0.54/spl radic/n. In addition to honeycomb meshes bounded by a regular hexagon, we consider also honeycomb networks with rhombus and rectangle as the bounding polygons.
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TL;DR: This paper proposes a suitable addressing scheme for nodes, derive a formula for distance between nodes, and presents a very simple and elegant routing algorithm for hexagonal interconnection.
Abstract: Nodes in a hexagonal network are placed at the vertices of a regular triangular tessellation, so that each node has up to six neighbors. The network is proposed as an alternative interconnection network to a mesh connected computer (with nodes serving as processors) and is used also to model cellular networks where nodes are the base stations. In this paper, we propose a suitable addressing scheme for nodes (with two variants), derive a formula for distance between nodes, and present a very simple and elegant routing algorithm. This addressing scheme and corresponding routing algorithm for hexagonal interconnection are considerably simpler than previously proposed solutions. We then apply the addressing scheme for solving two problems in cellular networks. With the new scheme, the distance between the new and old cell to which a mobile phone user is connected can be easily determined and coded with three integers, one of them being zero. Further, in order to minimize the wireless cost of tracking mobile users, we propose hexagonal cell identification codes containing three, four, or six bits, respectively, to implement a distance based tracking strategy. These schemes do not have errors in determining cell distance in existing hexagonal based cellular networks. Another application is for connection rerouting in cellular networks during a path extension process.
119 citations
Cites background or methods from "Honeycomb networks: Topological pro..."
...…determining cell distance in existing hexagonal based cellular networks, which is not the case with recently proposed 3 bit cell identification codes based on an artificial square mesh placed over a hexagonal network (moreover, the existing mesh schemes fail to address the diagonal moves that may…...
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...I. Stojmenovic is with SITE, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada....
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...We shall adapt the coordinate system that was proposed for a honeycomb network by Stojmenovic [21]....
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TL;DR: This paper determines minimum metric bases for hexagonal and honeycomb networks by using the duality of these networks to determine a minimum metric basis for these networks.
102 citations
Cites background from "Honeycomb networks: Topological pro..."
...Keywords: metric basis, metric dimension, hexagonal network, honeycomb network, dual of a graph....
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TL;DR: This paper proves that honeycomb and diamond networks are special subgraphs of complete 2D and 3D tori, respectively, and shows this viewpoint to hold important implications for their physical layouts and routing schemes.
Abstract: Honeycomb and diamond networks have been proposed as alternatives to mesh and torus architectures for parallel processing. When wraparound links are included in honeycomb and diamond networks, the resulting structures can be viewed as having been derived via a systematic pruning scheme applied to the links of 2D and 3D tori, respectively. The removal of links, which is performed along a diagonal pruning direction, preserves the network's node-symmetry and diameter, while reducing its implementation complexity and VLSI layout area. In this paper, we prove that honeycomb and diamond networks are special subgraphs of complete 2D and 3D tori, respectively, and show this viewpoint to hold important implications for their physical layouts and routing schemes. Because pruning reduces the node degree without increasing the network diameter, the pruned networks have an advantage when the degree-diameter product is used as a figure of merit. Additionally, if the reduced node degree is used as an opportunity to increase the link bandwidths to equalize the costs of pruned and unpruned networks, a gain in communication performance may result.
77 citations
Cites background or methods from "Honeycomb networks: Topological pro..."
...A honeycomb network (Fig. 1a, without the wraparound links) is formed by tiling the plane with regular hexagons and placing a degree-3 node at each vertex in the natural way [11], [19], [ 20 ]....
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...In addition to economy and ease of layout, to be discussed in Section 5, an advantage of treating the honeycomb network as a pruned 2D torus is that as in torus, we can base the routing algorithm on the offsets x and y in dimensions X and Y. The resulting algorithm is simpler than the one suggested in [ 20 ]....
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...A particularly attractive variant has a rectangular exterior shape and is known as a honeycomb rectangular mesh (HReM) or torus (HReT), depending on the absence or presence of wraparound links [ 20 ]....
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12 Dec 2008
TL;DR: A rapid-prototyping hardware is presented, which implements a unique hexagonal lattice topology of 55 tunable Gm cells for reconfigurable instantiation of Gm-C filters in a 0.13 mum CMOS technology.
Abstract: This paper reports on a field-programmable analog array (FPAA) for high-frequency continuous-time analog filters. A rapid-prototyping hardware is presented, which implements a unique hexagonal lattice topology of 55 tunable Gm cells for reconfigurable instantiation of Gm-C filters in a 0.13 mum CMOS technology. The typical power dissipation is 70 mW at a 1.2 V supply. The chip structure allows intuitive mapping of Gm-C filter schematics with up to seven independent nodes, immediate download to the hardware, and evaluation on the working prototype. Implementations of first- and second-order low-pass and sixth-order bandpass filters are presented and show the feasibility of the array for frequencies up to the order of one hundred MHz.
71 citations
Cites background from "Honeycomb networks: Topological pro..."
...Further advantages of honeycomb network structures can be found by a more formal exploration [12]....
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01 Jan 2002
TL;DR: The purpose of this chapter is to survey recent research on location management in cellular networks and presents some assumptions that are commonly used to evaluate a location management scheme in terms of network topology, call arrival probability, and mobility.
Abstract: It has been known for over one hundred years that radio can be used to keep in touch with people on the move. However, wireless communications using radio were not popular until Bell Laboratories developed the cellular concept to reuse the radio frequency in the 1960s and 1970s [31]. In the past decade, cellular communications have experienced an explosive growth due to recent technological advances in cellular networks and cellular telephone manufacturing. It is anticipated that they will experience even more growth in the next decade. In order to accommodate more subscribers, the size of cells must be reduced to make more efficient use of the limited frequency spectrum allocation. This will add to the challenge of some fundamental issues in cellular networks. Location management is one of the fundamental issues in cellular networks. It deals with how to track subscribers on the move. The purpose of this chapter is to survey recent research on location management in cellular networks. The rest of this chapter is organized as follows. Section 2.2 introduces cellular networks and Section 2.3 describes basic concepts of location management. Section 2.4 presents some assumptions that are commonly used to evaluate a location management scheme in terms of network topology, call arrival probability, and mobility. Section 2.5 surveys popular location management schemes. Finally, Section 2.6 summarizes the chapter.
67 citations
Cites background from "Honeycomb networks: Topological pro..."
...However, in [28] the authors have pointed out that the distance between two cells can be computed easily if the cell address can be assigned systematically using the coordinate system proposed for the honeycomb network in [36]....
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References
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Book•
01 Sep 1991
TL;DR: This chapter discusses sorting on a Linear Array with a Systolic and Semisystolic Model of Computation, which automates the very labor-intensive and therefore time-heavy and expensive process of manually sorting arrays.
Abstract: Preface Acknowledgments Notation 1 Arrays and Trees 1.1 Elementary Sorting and Counting 1.1.1 Sorting on a Linear Array Assessing the Performance of the Algorithm Sorting N Numbers with Fewer Than N Processors 1.1.2 Sorting in the Bit Model 1.1.3 Lower Bounds 1.1.4 A Counterexample-Counting 1.1.5 Properties of the Fixed-Connection Network Model 1.2 Integer Arithmetic 1.2.1 Carry-Lookahead Addition 1.2.2 Prefix Computations-Segmented Prefix Computations 1.2.3 Carry-Save Addition 1.2.4 Multiplication and Convolution 1.2.5 Division and Newton Iteration 1.3 Matrix Algorithms 1.3.1 Elementary Matrix Products 1.3.2 Algorithms for Triangular Matrices 1.3.3 Algorithms for Tridiagonal Matrices -Odd-Even Reduction -Parallel Prefix Algorithms 1.3.4 Gaussian Elimination 1.3.5 Iterative Methods -Jacobi Relaxation -Gauss-Seidel Relaxation Finite Difference Methods -Multigrid Methods 1.4 Retiming and Systolic Conversion 1.4.1 A Motivating Example-Palindrome Recognition 1.4.2 The Systolic and Semisystolic Model of Computation 1.4.3 Retiming Semisystolic Networks 1.4.4 Conversion of a Semisystolic Network into a Systolic Network 1.4.5 The Special Case of Broadcasting 1.4.6 Retiming the Host 1.4.7 Design by Systolic Conversion-A Summary 1.5 Graph Algorithms 1.5.1 Transitive Closure 1.5.2 Connected Components 1.5.3 Shortest Paths 1.5.4 Breadth-First Spanning Trees 1.5.5 Minimum Weight Spanning Trees 1.6 Sorting Revisited 1.6.1 Odd-Even Transposition Sort on a Linear Array 1.6.2 A Simple Root-N(log N + 1)-Step Sorting Algorithm 1.6.3 A (3 Root- N + o(Root-N))-Step Sorting Algorithm 1.6.4 A Matching Lower Bound 1.7 Packet Routing 1.7.1 Greedy Algorithms 1.7.2 Average-Case Analysis of Greedy Algorithms -Routing N Packets to Random Destinations -Analysis of Dynamic Routing Problems 1.7.3 Randomized Routing Algorithms 1.7.4 Deterministic Algorithms with Small Queues 1.7.5 An Off-line Algorithm 1.7.6 Other Routing Models and Algorithms 1.8 Image Analysis and Computational Geometry 1.8.1 Component-Labelling Algorithms -Levialdi's Algorithm -An O (Root-N)-Step Recursive Algorithm 1.8.2 Computing Hough Transforms 1.8.3 Nearest-Neighbor Algorithms 1.8.4 Finding Convex Hulls 1.9 Higher-Dimensional Arrays 1.9.1 Definitions and Properties 1.9.2 Matrix Multiplication 1.9.3 Sorting 1.9.4 Packet Routing 1.9.5 Simulating High-Dimensional Arrays on Low-Dimensional Arrays 1.10 problems 1.11 Bibliographic Notes 2 Meshes of Trees 2.1 The Two-Dimensional Mesh of Trees 2.1.1 Definition and Properties 2.1.2 Recursive Decomposition 2.1.3 Derivation from KN,N 2.1.4 Variations 2.1.5 Comparison With the Pyramid and Multigrid 2.2 Elementary O(log N)-Step Algorithms 2.2.1 Routing 2.2.2 Sorting 2.2.3 Matrix-Vector Multiplication 2.2.4 Jacobi Relaxation 2.2.5 Pivoting 2.2.6 Convolution 2.2.7 Convex Hull 2.3 Integer Arithmetic 2.3.1 Multiplication 2.3.2 Division and Chinese Remaindering 2.3.3 Related Problems -Iterated Products -Rooting Finding 2.4 Matrix Algorithms 2.4.1 The Three-Dimensional Mesh of Trees 2.4.2 Matrix Multiplication 2.4.3 Inverting Lower Triangular Matrices 2.4.4 Inverting Arbitrary Matrices -Csanky's Algorithm -Inversion by Newton Iteration 2.4.5 Related Problems 2.5 Graph Algorithms 2.5.1 Minimum-Weight Spanning Trees 2.5.2 Connected Components 2.5.3 Transitive Closure 2.5.4 Shortest Paths 2.5.5 Matching Problems 2.6 Fast Evaluation of Straight-Line Code 2.6.1 Addition and Multiplication Over a Semiring 2.6.2 Extension to Codes with Subtraction and Division 2.6.3 Applications 2.7 Higher-Dimensional meshes of Trees 2.7.1 Definitions and Properties 2.7.2 The Shuffle-Tree Graph 2.8 Problems 2.9 Bibliographic Notes 3 Hypercubes and Related Networks 3.1 The Hypercube 3.1.1 Definitions and Properties 3.1.2 Containment of Arrays -Higher-Dimensional Arrays -Non-Power-of-2 Arrays 3.1.3 Containment of Complete Binary Trees 3.1.4 Embeddings of Arbitrary Binary Trees -Embeddings with Dilation 1 and Load O(M over N + log N) -Embeddings with Dilation O(1) and Load O (M over N + 1) -A Review of One-Error-Correcting Codes -Embedding Plog N into Hlog N 3.1.5 Containment of Meshes of Trees 3.1.6 Other Containment Results 3.2 The Butterfly, Cube-Connected-Cycles , and Benes Network 3.2.1 Definitions and Properties 3.2.2 Simulation of Arbitrary Networks 3.2.3 Simulation of Normal Hypercube Algorithms 3.2.4 Some Containment and Simulation Results 3.3 The Shuffle-Exchange and de Bruijn Graphs 3.3.1 Definitions and Properties 3.3.2 The Diaconis Card Tricks 3.3.3 Simulation of Normal Hypercube Algorithms 3.3.4 Similarities with the Butterfly 3.3.5 Some Containment and Simulation Results 3.4 Packet-Routing Algorithms 3.4.1 Definitions and Routing Models 3.4.2 Greedy Routing Algorithms and Worst-Case Problems 3.4.3 Packing, Spreading, and Monotone Routing Problems -Reducing a Many-to-Many Routing Problem to a Many-to-One Routing Problem -Reducing a Routing Problem to a Sorting Problem 3.4.4 The Average-Case Behavior of the Greedy Algorithm -Bounds on Congestion -Bounds on Running Time -Analyzing Non-Predictive Contention-Resolution Protocols 3.4.5 Converting Worst-Case Routing Problems into Average-Case Routing Problems -Hashing -Randomized Routing 3.4.6 Bounding Queue Sizes -Routing on Arbitrary Levelled Networks 3.4.7 Routing with Combining 3.4.8 The Information Dispersal Approach to Routing -Using Information Dispersal to Attain Fault-Tolerance -Finite Fields and Coding Theory 3.4.9 Circuit-Switching Algorithms 3.5 Sorting 3.5.1 Odd-Even Merge Sort -Constructing a Sorting Circuit with Depth log N(log N +1)/2 3.5.2 Sorting Small Sets 3.5.3 A Deterministic O(log N log log N)-Step Sorting Algorithm 3.5.4 Randomized O(log N)-Step Sorting Algorithms -A Circuit with Depth 7.45 log N that Usually Sorts 3.6 Simulating a Parallel Random Access Machine 3.6.1 PRAM Models and Shared Memories 3.6.2 Randomized Simulations Based on Hashing 3.6.3 Deterministic Simulations using Replicated Data 3.6.4 Using Information Dispersal to Improve Performance 3.7 The Fast Fourier Transform 3.7.1 The Algorithm 3.7.2 Implementation on the Butterfly and Shuffle-Exchange Graph 3.7.3 Application to Convolution and Polynomial Arithmetic 3.7.4 Application to Integer Multiplication 3.8 Other Hypercubic Networks 3.8.1 Butterflylike Networks -The Omega Network -The Flip Network -The Baseline and Reverse Baseline Networks -Banyan and Delta Networks -k-ary Butterflies 3.8.2 De Bruijn-Type Networks -The k-ary de Bruijn Graph -The Generalized Shuffle-Exchange Graph 3.9 Problems 3.10 Bibliographic Notes Bibliography Index Lemmas, Theorems, and Corollaries Author Index Subject Index
2,895 citations
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TL;DR: The author, a well-known researcher in paralle l computing, once again has proved his expertise and authority on the materials covered and this book will certainly have an impact to the psychology of students and researchers alike.
Abstract: In the ever-expanding field of parallel computing, we have seen a number of textbooks , some emphasizing the design aspects of parallel algorithms based on abstract models of paralle l machines (such as PRAMs) and some others focusing on the topological properties of paralle l architectures . What is needed in this area is a book which provides a linkage between th e topological properties of a parallel network and its computational capabilities or limitations , as well as comparative analyses of parallel architectures, not only among the proposed ones but also in view of a desirable general-purpose parallel machine which is yet to be built . The book under review comes closest to this goal . The author, a well-known researcher in paralle l computing, once again has proved his expertise and authority on the materials covered . This book will certainly have an impact to the psychology of students and researchers alike, on ho w to correlate parallel architectures and algorithms . Physically, this book is organized around three categories of parallel architectures : Arrays and Trees, Meshes of Trees, and Hypercubic networks . Each category covers not only th e basic type of architectures but also other variants or related models . For example, Chapter 1 on Arrays and Trees encompasses linear arrays, two-dimensional arrays, trees, ring, torus, X tree, pyramid, multigrid networks, systolic and semisystolic networks, and higher-dimensional arrays as well . Similarly, Chapter 2 on Meshes of Trees shows different ways of looking at two-dimensional meshes of trees at the beginning and further extends to higher-dimensiona l meshes of trees, and shuffle-tree graphs at the end . The third chapter, Hypercubes and Related Networks, covers butterfly, cube-connected-cycles, Benes network, shuffle-exchange, de Bruij n network, butterfly-like networks (Omega network, flip network, baseline and reverse baselin e networks, Banyan and delta networks, and k-ary butterfy), and de Bruijn-type networks (k-ar y de Bruijn network, and generalized shuffle-exchange network) . Whereas the above parallel networks constitute the architectural domain of the hook as th e basis, the application domain — parallel computation problems and algorithms — threads th e chapters together and helps a reader to view the similarities and differences of each network , from algorithm design standpoint . In addition to the definitions and characterizations of th e topological properties of the parallel architectures, each chapter examines a carefully-chose n subset of fundamental computational problems such as integer arithmetic, prefix computation , list ranking, sorting and counting, matrix arithmetic, graph problems, Fast Fourier Transfor m and Discrete Fourier Transform, computational geometry, and image analysis etc . The solution s to these problems are explored from simple algorithms to more complicated ones until it achieve s optimality. This approach seems to be adequate to reveal the capability and limitations of eac h network . The problems and algorithms are not treated in an isolated context but provokes a reader to capture what is achievable in terms of speedup and efficiency, and what is the limi t in terms of lower hounds, in a particular parallel network under focus . The author pays special attention to the routing problem . Considering that routing is a common vehicle for solving most of the regular and irregular parallel computation problem s in a fixed-connection network, the general capability of each network against an abstract parallel machine model is properly exposed via routing problem . Also discussed are the containment/embedding of one network in another, i .e . mapping between networks and the simulatio n
665 citations
Book•
01 Mar 1997TL;DR: This chapter discusses models of Computation, Combinational Circuits, and Parallel Synergy, which aims to explain the construction of parallel circuits and their applications in medicine and engineering.
Abstract: 1. Introduction2. Models of Computation3. Combinational Circuits4. Parallel Prefix Computation5. Divide and Conquer6. Pointer-Based Data Structures7. Linear Arrays8. Meshes and Related Models9. Hypercubes and Stars10. Models Using Buses11. Broadcasting with Selective Reduction12. Parallel SynergyBibliographyIndex
388 citations
TL;DR: An elegant, distributed routing scheme is developed for wrapped H-meshes so that each node in an H-mesh can compute shortest paths from itself to any other node with a straightforward algorithm of O(1) using the addresses of the source-destination pair only, independent of the network's size.
Abstract: A family of six-regular graphs, called hexagonal meshes or H-meshes, is considered as a multiprocessor interconnection network. Processing nodes on the periphery of an H-mesh are first wrapped around to achieve regularity and homogeneity. The diameter of a wrapped H-mesh is shown to be of O(p/sup 1/2/), where p is the number of nodes in the H-mesh. An elegant, distributed routing scheme is developed for wrapped H-meshes so that each node in an H-mesh can compute shortest paths from itself to any other node with a straightforward algorithm of O(1) using the addresses of the source-destination pair only, i.e. independent of the network's size. This is in sharp contrast with those previously known algorithms that rely on using routing tables. Furthermore, the authors also develop an efficient point-to-point broadcasting algorithm for the H-meshes which is proved to be optimal in the number of required communication steps. The wrapped H-meshes are compared against some other existing multiprocessor interconnection networks, such as hypercubes, trees, and square meshes. The comparison reinforces the attractiveness of the H-mesh architecture. >
210 citations
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TL;DR: This book is an excellent practical guide in modern parallel scientific programming, and it can be used for self-instruction and also for all specialists who are interested in parallel scientific computing.
Abstract: This textbook on parallel scientific computing presents state-of-the-art material in scientific algorithm design for modem parallel computers. The first volume of this textbook (published in 1986) described the art of scientific computing in Fortran 77 on single-processor syslg, tems. The first edition of the second volume was published in 1996. In the current Second Edition, all codes have been corrected according to version 2.06 of Fortran 90. Volume 2 deals with Fortran 90 compilers, which are now widely available, and is devoted to parallel scientific computing. The book, written with support of the US National Science Foundation, can be very useful for graduate and postgraduate courses and also for all specialists who are interested in parallel scientific computing. It's well known that Fortran is excellent for scientific computing. But in scientific computing, multiprocessor systems are widely used now, instead of single-processor systems. Thus, the actual problem is to modify wellknown recipes according to parallel-programming ideas. This book has successfully solved this problem. First, the authors introduce Fortran 90, parallel programming, and parallel utility functions for Fortran 90. These functions include the move data, returning a location, argument checking and error handling, outer operations on vectors, scatter with combine, skew operations on matrices, polynomials, and recurrences routines. Next, the authors consider the most popular scientific numerical algorithms previously coded in Fortran 77 and present new codes elaborated by Fortran 90 with parallel facilities. They discuss the solution of linear algebra equations, interpolation and extrapolation problems, integration and evaluation of functions, computing of special functions and random numbers, sorting, solution of nonlinear sets of equation and eigenvalue problems, minimization and maximization of functions, Fourier transformation, statistical algorithms, integration of ODE and PDE, and less-numerical algorithms. By studying the presented Fortran 90 parallel codes, readers can get good experience in Fortran 90 and in parallel programming. T o read this book, you only need basic skills in numerical methods and in Fortran programming. Thus, it is an excellent practical guide in modern parallel scientific programming, and it can be used for self-instruction. Unfortunately, the mathematical background of all the algorithms described in the book are only presented in Volume 1. Therefore, to properly study all the routines, the reader must have Volume 1. The book's reference list is not large and contains only about 40 examples of Fortran textbooks, well-known textbooks on numerical methods, and a few books dedicated to parallel programming directly connected with the book's subject. All 3 SO routines considered in the book are available on diskettes or CD ROM for IBM PC, Macintosh, and Unix computers. Readers can purchase this software by mail.
147 citations
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