## Perfect Difference Networks and Related Interconnection Structures for Parallel and Distributed Systems (2005)

Venue: | IEEE Trans. Parallel and Distributed Systems |

Citations: | 7 - 3 self |

### BibTeX

@ARTICLE{Parhami05perfectdifference,

author = {Behrooz Parhami and Mikhail Rakov},

title = {Perfect Difference Networks and Related Interconnection Structures for Parallel and Distributed Systems},

journal = {IEEE Trans. Parallel and Distributed Systems},

year = {2005},

volume = {16},

pages = {714--724}

}

### OpenURL

### Abstract

Abstract—In view of their applicability to parallel and distributed computer systems, interconnection networks have been studied intensively by mathematicians, computer scientists, and computer designers. In this paper, we propose an asymptotically optimal method for connecting a set of nodes into a perfect difference network (PDN) with diameter 2, so that any node is reachable from any other node in one or two hops. The PDN interconnection scheme, which is based on the mathematical notion of perfect difference sets, is optimal in the sense that it can accommodate an asymptotically maximal number of nodes with smallest possible node degree under the constraint of the network diameter being 2. We present the network architecture in its basic and bipartite forms and show how the related multidimensional PDNs can be derived. We derive the exact average internode distance and tight upper and lower bounds for the bisection width of a PDN. We conclude that PDNs and their derivatives constitute worthy additions to the repertoire of network designers and may offer additional design points that can be exploited by current and emerging technologies, including wireless and optical interconnects. Performance, algorithmic, and robustness attributes of PDNs are analyzed in a companion paper.

### Citations

1308 |
Introduction to Parallel Algorithms and Architectures: Arrays
- Leighton
- 1992
(Show Context)
Citation Context ...e limitations inherent in each architecture. This, in turn, explains the proliferation of implemented and proposed connectivity schemes, sometimes characterized as the sea of interconnection networks =-=[19]-=-, [23]. At the application level, internode communication in a parallel or distributed system may occur via shared variables or message passing. In either case, a virtual complete connectivity is envi... |

529 |
Algebraic Graph Theory
- Biggs
- 1993
(Show Context)
Citation Context ... 1 nodes. The corresponding upper bound n d 2 þ 1 for undirected graphs is not much different [23]. Examples of diameter-2 networks of small sizes include the Petersen and HoffmanSingleton networks =-=[7]-=-. Perfect difference sets provide the mathematical tools for achieving this optimum number of nodes, in an asymptotic manner, within the framework perfect difference networks or PDNs (see Fig. 1 for t... |

462 | A survey of wormhole routing techniques in direct networks
- Ni, McKinley
- 1993
(Show Context)
Citation Context ... Ln. That the worst-case latency for messages in a network is highly dependent on D is obvious with store-and-forward routing. It is somewhat less clear that, even with wormhole switching [10], [21], =-=[22]-=-, network diameter plays a key role in communication latency, albeit in an indirect way. This is best understood by considering the case of short and long messages separately. For short worms, the tra... |

376 |
Some simplified NP-complete graph problems
- Garey, Johnson, et al.
- 1976
(Show Context)
Citation Context ...ultiplicity—For some values of , there exist more than one PDS. For example, we have the following PDSs of order 3: 0; 1; 3; 9 and 0; 1; 4; 6: It is easily verified that all numbers in the interval =-=[1, 12]-=- can be formed as the mod-13 difference of numbers in each of the two sets above: 1 1 0 1 0 2 3 1 6 4 3 3 0 4 1 4 0 9 4 0 5 1 9 6 1 6 9 3 6 0 7 3 9 0 6 8 9 1 1 6 9 9 0 0 4 10 0 3 1 4 11 1 3 4 6 12 0 1... |

297 | Performance Analysis of k-ary n-cube Interconnection Networks
- Dally
- 1990
(Show Context)
Citation Context ...r line-graph Ln. That the worst-case latency for messages in a network is highly dependent on D is obvious with store-and-forward routing. It is somewhat less clear that, even with wormhole switching =-=[10]-=-, [21], [22], network diameter plays a key role in communication latency, albeit in an indirect way. This is best understood by considering the case of short and long messages separately. For short wo... |

236 |
A Group Theoretic Model for Symmetric Interconnection Networks
- Akers, Krishnamurthy
- 1989
(Show Context)
Citation Context ...en the exploration of the design space depicted in Fig. 1, with particular emphasis on deriving networks with sublogarithmic degrees that can provide some of the desirable properties of the hypercube =-=[1]-=-, [9], [18], [29], [44]. Emphasis on sublogarithmic-degree networks was justified by VLSI area requirements and packaging constraints, including pin limitations; i.e., cost and realizability factors. ... |

207 |
The cubeconnected cycles: A versatile network for parallel computation
- Preparata, Vuillemin
- 1981
(Show Context)
Citation Context ...on of the design space depicted in Fig. 1, with particular emphasis on deriving networks with sublogarithmic degrees that can provide some of the desirable properties of the hypercube [1], [9], [18], =-=[29]-=-, [44]. Emphasis on sublogarithmic-degree networks was justified by VLSI area requirements and packaging constraints, including pin limitations; i.e., cost and realizability factors. Perfect differenc... |

152 |
Generalized hypercube and hyperbus structures for computer network
- Bhuyan, Agrawal
- 1984
(Show Context)
Citation Context .... 1. They offer the benefits of full connectivity at a much lower cost. The part of design space that falls between the hypercube and Kn is of little interest in architectures with wired connectivity =-=[6]-=-, [46], but becomes more practical and, thus, interesting with wireless or optical links. 2 EVALUATION CRITERIA FOR NETWORKS Network diameter D, defined as the longest of the internode distances, is a... |

113 |
Graph bisection algorithms with good average case behavior
- Bui, Chaudhuri, et al.
- 1987
(Show Context)
Citation Context ...ction width of an arbitrary PDN is not yet known. Note that calculation of bisection width for an arbitrary graph is an NP-complete problem [12] and it remains so even for the class of regular graphs =-=[8]-=-. This explains the dearth of results on network bisection width. 7 CONCLUSION We have introduced perfect difference networks and the mathematical underpinnings that make them desirable as robust, hig... |

98 | A survey of collective communication in wormhole-routed massively parallel computers
- McKinley, Tsai, et al.
- 1994
(Show Context)
Citation Context ...-graph Ln. That the worst-case latency for messages in a network is highly dependent on D is obvious with store-and-forward routing. It is somewhat less clear that, even with wormhole switching [10], =-=[21]-=-, [22], network diameter plays a key role in communication latency, albeit in an indirect way. This is best understood by considering the case of short and long messages separately. For short worms, t... |

90 |
A theorem in finite projective geometry and some applications to number theory
- Singer
- 1938
(Show Context)
Citation Context ...dimensional PDNs (introduced in Section 5), which was previously referred to as “hyperhub” [32], [33], was renamed for the same reasons. Perfect difference sets were first discussed by Singer in 1938 =-=[38]-=-. The formulation was in terms of points and lines in a finite projective plane. The theory of finite projective planes is highly developed [15], but these mathematical notions are not required to und... |

77 |
Introduction to Parallel Processing: Algorithms and Architectures
- Parhami
- 1999
(Show Context)
Citation Context ...tations inherent in each architecture. This, in turn, explains the proliferation of implemented and proposed connectivity schemes, sometimes characterized as the sea of interconnection networks [19], =-=[23]-=-. At the application level, internode communication in a parallel or distributed system may occur via shared variables or message passing. In either case, a virtual complete connectivity is envisaged ... |

54 | A combinatorial problem
- Guy
- 1967
(Show Context)
Citation Context ... number n that is of the form 2 þ þ 1, where ph and p is a prime number. It is suspected, though not yet proven for arbitrarily large values of n, that PDSs do not exist for other values of n [11], =-=[13]-=-. However, practically speaking, this is not alarming, given that primes and their powers are quite abundant, both in the range of practical interest for interconnection network size and asymptoticall... |

53 |
Analysis of Chordal Ring Networks
- Arden, Lee
- 1981
(Show Context)
Citation Context ...excellent performance and logarithmic diameter, is often used as a reference point for comparisons. bypass links or chords have been added in order to reduce the network diameter (e.g., chordal rings =-=[3]-=-) or richly connected graphs in which certain links are systematically removed through pruning [17], [24], [26] to reduce the network cost. Fig. 1 is an abstract view of the design space for interconn... |

43 |
Symmetry in interconnection networks based on Cayley graphs of permutation groups: a survey, Parallel Computing 19 (4
- Lakshmivarahan, Jwo, et al.
- 1993
(Show Context)
Citation Context ...loration of the design space depicted in Fig. 1, with particular emphasis on deriving networks with sublogarithmic degrees that can provide some of the desirable properties of the hypercube [1], [9], =-=[18]-=-, [29], [44]. Emphasis on sublogarithmic-degree networks was justified by VLSI area requirements and packaging constraints, including pin limitations; i.e., cost and realizability factors. Perfect dif... |

21 |
Projective Geometry: From Foundations to Applications
- Beutelspacher, Rosenbaum
- 1998
(Show Context)
Citation Context ...ferences. There is an alternate way in which we can formulate an interconnection structure based on the normal-form PDS f0; 1;s2; ...;s g of order [33]. This alternate scheme was briefly discussed in =-=[5]-=-, but the filing of the patent in [30] and the second author’s prior work that led to the patent predate [5]. The second author’s prior work in this area, and the ensuing patents, are based on the fol... |

21 |
Interconnection networks based on generalization of cube-connected cycles
- Carlsson, Cruthirds, et al.
- 1985
(Show Context)
Citation Context ...e exploration of the design space depicted in Fig. 1, with particular emphasis on deriving networks with sublogarithmic degrees that can provide some of the desirable properties of the hypercube [1], =-=[9]-=-, [18], [29], [44]. Emphasis on sublogarithmic-degree networks was justified by VLSI area requirements and packaging constraints, including pin limitations; i.e., cost and realizability factors. Perfe... |

21 | A Unified Formulation of Honeycomb and Diamond Networks
- Parhami, Kwai
- 2001
(Show Context)
Citation Context .... bypass links or chords have been added in order to reduce the network diameter (e.g., chordal rings [3]) or richly connected graphs in which certain links are systematically removed through pruning =-=[17]-=-, [24], [26] to reduce the network cost. Fig. 1 is an abstract view of the design space for interconnection networks (including perfect difference networks, introduced later in this paper) in terms of... |

18 |
Swapped networks: unifying the architectures and algorithms of a wide class of hierarchical parallel processors,”Proc
- Yeh, Parhami
- 1996
(Show Context)
Citation Context ...the design space depicted in Fig. 1, with particular emphasis on deriving networks with sublogarithmic degrees that can provide some of the desirable properties of the hypercube [1], [9], [18], [29], =-=[44]-=-. Emphasis on sublogarithmic-degree networks was justified by VLSI area requirements and packaging constraints, including pin limitations; i.e., cost and realizability factors. Perfect difference netw... |

16 | Periodically Regular Chordal Rings
- Parhami, Kwai
- 1999
(Show Context)
Citation Context ...ss links or chords have been added in order to reduce the network diameter (e.g., chordal rings [3]) or richly connected graphs in which certain links are systematically removed through pruning [17], =-=[24]-=-, [26] to reduce the network cost. Fig. 1 is an abstract view of the design space for interconnection networks (including perfect difference networks, introduced later in this paper) in terms of node ... |

14 |
A survey of difference sets
- Hall
- 1956
(Show Context)
Citation Context ...hat forms the basis of the definition of perfect difference sets and, then, the definition itself. All results in this section are from [38]. Additional information on difference sets can be found in =-=[14]-=-, [40], [43]. Theorem 1. A sufficient condition that there exist þ 1 integers s0;s1; ...;s , having the property that their 2 þ differences si sj;i6 j; 0 i; j , are congruent, modulo 2 þ þ 1, to the ... |

12 |
A class of binary recurrent codes with limited error propogation
- Robinson, Bernstein
- 1967
(Show Context)
Citation Context ... structures and networks, as discussed here and in Section 4, perfect difference sets can be applied to a variety of other design problems. Examples include highly efficient error control codes [20], =-=[37]-=-, block designs, which are related to orthogonal Latin squares and find applications in scheduling and design of experiments [4], and signal encoding to ensure negligible autocorrelation and cross-cor... |

11 | Incomplete k-ary n-cube and Its Derivatives
- Parhami, Kwai
- 2004
(Show Context)
Citation Context ...ks or chords have been added in order to reduce the network diameter (e.g., chordal rings [3]) or richly connected graphs in which certain links are systematically removed through pruning [17], [24], =-=[26]-=- to reduce the network cost. Fig. 1 is an abstract view of the design space for interconnection networks (including perfect difference networks, introduced later in this paper) in terms of node degree... |

10 |
and analysis of product networks
- Youssef
- 1995
(Show Context)
Citation Context ...t mixed-radix integer xq 1xq 2 ...x0 in the range 0 to nq 1 nq 2 ... n0 1, so that nodes x and y are connected iff their labels differ only in one digit, say xj 6 yj, and xj is connected to yj in Gj =-=[45]-=-. The node degree of a product graph is the sum dq 1 þ dq 2 þ ...þ d0 of the node degrees for the component graphs and its diameter is the sum Dq 1 þ Dq 2 þ ...þ D0 of the diameters. A wide variety of... |

5 |
The hyperstar interconnection network
- Al-Ayyoub, Day
- 1998
(Show Context)
Citation Context ...me “hyperstar” [30], [31], [34] was originally coined for what we call “PDN” in this paper. The change of name to the more descriptive PDN was triggered by our desire to avoid confusion with networks =-=[2]-=- and various commercial hardware and softPARHAMI AND RAKOV: PERFECT DIFFERENCE NETWORKS AND RELATED INTERCONNECTION STRUCTURES FOR PARALLEL AND... 717 ware products that already use the name “hyperst... |

4 | Performance, Algorithmic, and Robustness Attributes of Perfect Difference Networks
- Parhami, Rakov
- 2005
(Show Context)
Citation Context ...PDNs. Hence, many algorithms developed for basic PDN lead directly to a number of corresponding algorithms for H 2 with little or no additional effort. Some examples are provided in a companion paper =-=[27]-=- that deals with the performance, algorithmic, and robustness attributes of PDNs. 6 TOPOLOGICAL PROPERTIES In addition to the network diameter, which is 2 in the case of basic PDNs, various other topo... |

4 |
Method of Interconnecting Nodes and a Hyperstar Interconnection Structure
- Rakov
- 1998
(Show Context)
Citation Context ...s, in an asymptotic manner, within the framework perfect difference networks or PDNs (see Fig. 1 for the place of PDN in the spectrum of network choices). It should be noted that the name “hyperstar” =-=[30]-=-, [31], [34] was originally coined for what we call “PDN” in this paper. The change of name to the more descriptive PDN was triggered by our desire to avoid confusion with networks [2] and various com... |

3 |
Multidimensional Hyperstar and Hyperhub Interconnection Methods and Structures
- Rakov
- 1999
(Show Context)
Citation Context ...TION STRUCTURES FOR PARALLEL AND... 717 ware products that already use the name “hyperstar.” A class of multidimensional PDNs (introduced in Section 5), which was previously referred to as “hyperhub” =-=[32]-=-, [33], was renamed for the same reasons. Perfect difference sets were first discussed by Singer in 1938 [38]. The formulation was in terms of points and lines in a finite projective plane. The theory... |

3 |
Hyperstar and Hyperhub Optical Networks Interconnection Methods and Structures
- Rakov
- 2000
(Show Context)
Citation Context ...TRUCTURES FOR PARALLEL AND... 717 ware products that already use the name “hyperstar.” A class of multidimensional PDNs (introduced in Section 5), which was previously referred to as “hyperhub” [32], =-=[33]-=-, was renamed for the same reasons. Perfect difference sets were first discussed by Singer in 1938 [38]. The formulation was in terms of points and lines in a finite projective plane. The theory of fi... |

3 |
Method of Interconnecting Functional Nodes and a Hyperstar Interconnection Structure
- Rakov, Mackall
- 2001
(Show Context)
Citation Context ...mptotic manner, within the framework perfect difference networks or PDNs (see Fig. 1 for the place of PDN in the spectrum of network choices). It should be noted that the name “hyperstar” [30], [31], =-=[34]-=- was originally coined for what we call “PDN” in this paper. The change of name to the more descriptive PDN was triggered by our desire to avoid confusion with networks [2] and various commercial hard... |

2 |
Hyperstar: A New Interconnection Topology
- Rakov
- 1998
(Show Context)
Citation Context ...an asymptotic manner, within the framework perfect difference networks or PDNs (see Fig. 1 for the place of PDN in the spectrum of network choices). It should be noted that the name “hyperstar” [30], =-=[31]-=-, [34] was originally coined for what we call “PDN” in this paper. The change of name to the more descriptive PDN was triggered by our desire to avoid confusion with networks [2] and various commercia... |

2 | Viable Architectures for High-Performance Computing,” The Computer
- Ziavras, Wang, et al.
- 2003
(Show Context)
Citation Context ...They offer the benefits of full connectivity at a much lower cost. The part of design space that falls between the hypercube and Kn is of little interest in architectures with wired connectivity [6], =-=[46]-=-, but becomes more practical and, thus, interesting with wireless or optical links. 2 EVALUATION CRITERIA FOR NETWORKS Network diameter D, defined as the longest of the internode distances, is an impo... |

1 |
On Simple Difference Sets,” Sankhya
- Evans, Mann
- 1951
(Show Context)
Citation Context ...or any number n that is of the form 2 þ þ 1, where ph and p is a prime number. It is suspected, though not yet proven for arbitrarily large values of n, that PDSs do not exist for other values of n =-=[11]-=-, [13]. However, practically speaking, this is not alarming, given that primes and their powers are quite abundant, both in the range of practical interest for interconnection network size and asympto... |

1 |
On the Perfect r-Partitions of r2 r þ 1,” Trans
- Kirkman
(Show Context)
Citation Context ...nted that a PDS of order pz can be easily generated when required. Property 4. Relationship with perfect partitions—PDSs are closely related to perfect partitions, which have an even longer history =-=[16]-=-. Take any PDS in normal form and find the mod-n differences siþ1 si between consecutive numbers in it, including the difference s0 s . For example: PDS ð 3;n 13Þ : 0; 1; 3; 9 mod-n differences siþ... |

1 |
Why Network Diameter is Still Important
- Parhami, Yeh
- 2000
(Show Context)
Citation Context ...bility of deadlock or else forces us to use less aggressive routing algorithms. Either alternative implies lower performance. For a more detailed exposition of the importance of network diameter, see =-=[28]-=-. Average internode distance is defined as the average of the lengths of the distances between all nðn 1Þ pairs of nodes, or perhaps between all n2 pairs of nodes when the distance of each node to its... |

1 |
Computer Systems Design Using Apparatus of the Ideal Ring
- Rakov, Vakulskiy
- 1992
(Show Context)
Citation Context ... 2 þ 6 10 6 þ 4 11 6 þ 4 þ 1 12 2 þ 6 þ 4 Such a mod-n sequence, which is also known as a perfect partition, ideal code,orideal ring proportions, can be used in synthesizing PDN-type structures =-=[35]-=-, [36]. However, PDSs provide a more straightforward and efficient tool in this regard. Note that a PDS is transformed to a corresponding perfect partition via modular subtraction of consecutive terms... |

1 |
Using the Ideal Code Proportion Apparatus for Improving the Local Nets
- Rakov, Vakulskiy, et al.
- 1992
(Show Context)
Citation Context ... 10 6 þ 4 11 6 þ 4 þ 1 12 2 þ 6 þ 4 Such a mod-n sequence, which is also known as a perfect partition, ideal code,orideal ring proportions, can be used in synthesizing PDN-type structures [35], =-=[36]-=-. However, PDSs provide a more straightforward and efficient tool in this regard. Note that a PDS is transformed to a corresponding perfect partition via modular subtraction of consecutive terms, whil... |

1 |
Synthesis of Optimum Pulsed Sequences Having the Property of ‘No More than One Coincidence,’” Radio Eng
- Sverdlik, Meleshkevich
- 1974
(Show Context)
Citation Context ...nal Latin squares and find applications in scheduling and design of experiments [4], and signal encoding to ensure negligible autocorrelation and cross-correlation for ease of decoding and separation =-=[39]-=-, [41], [42]. These applications may be characterized by their need for provision of distance, variety, and/or Fig. 3. The chordal ring structure of the PDN with n 7 nodes based on the perfect diffe... |

1 |
Optimal Discrete Signals
- Sverdlik
- 1975
(Show Context)
Citation Context ...rms the basis of the definition of perfect difference sets and, then, the definition itself. All results in this section are from [38]. Additional information on difference sets can be found in [14], =-=[40]-=-, [43]. Theorem 1. A sufficient condition that there exist þ 1 integers s0;s1; ...;s , having the property that their 2 þ differences si sj;i6 j; 0 i; j , are congruent, modulo 2 þ þ 1, to the intege... |

1 |
Table of Optimal Sets with the Property of ‘No More than One Coincidence,’” Radio Eng
- Sverdlik, Meleshkevich
- 1975
(Show Context)
Citation Context ...tin squares and find applications in scheduling and design of experiments [4], and signal encoding to ensure negligible autocorrelation and cross-correlation for ease of decoding and separation [39], =-=[41]-=-, [42]. These applications may be characterized by their need for provision of distance, variety, and/or Fig. 3. The chordal ring structure of the PDN with n 7 nodes based on the perfect difference ... |

1 |
Synthesis of Ensembles of Pulse Sequences with Properties of ‘No More than One Coincidence,’” Radio Eng
- Sverdlik, Meleshkevich
- 1976
(Show Context)
Citation Context ...uares and find applications in scheduling and design of experiments [4], and signal encoding to ensure negligible autocorrelation and cross-correlation for ease of decoding and separation [39], [41], =-=[42]-=-. These applications may be characterized by their need for provision of distance, variety, and/or Fig. 3. The chordal ring structure of the PDN with n 7 nodes based on the perfect difference set f0... |

1 |
A Course in Combinatorics, (see chapter 27 entitled “Difference Sets and Automorphisms”), Cambridge Univ
- Lint, Wilson
- 1992
(Show Context)
Citation Context ...e basis of the definition of perfect difference sets and, then, the definition itself. All results in this section are from [38]. Additional information on difference sets can be found in [14], [40], =-=[43]-=-. Theorem 1. A sufficient condition that there exist þ 1 integers s0;s1; ...;s , having the property that their 2 þ differences si sj;i6 j; 0 i; j , are congruent, modulo 2 þ þ 1, to the integers 1; ... |