混合式弦環網路之距離相關問題

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(1)國立交通大學應用數學系博士論文混合式弦環網路之距離相關問題 On Distance-related Problems of Mixed Chordal Ring Networks. 研究生：藍國元指導教授：陳秋媛博士. 中華民國九十九年六月.

(2) 混合式弦環網路之距離相關問題 On Distance-related Problems of Mixed Chordal Ring Networks 研究生：藍國元 Student: James K. Lan 指導教授：陳秋媛博士 Advisor: Chiuyuan Chen. 國立交通大學應用數學系博士論文. A Dissertation Submitted to Department of Applied Mathematics College of Science National Chiao Tung University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Applied Mathematics June 2010 Hsinchu, Taiwan, Republic of China. 中華民國九十九年六月.

(3) Abstract This research covers the less trodden field of mixed chordal ring networks, in the attempt to discover the existence of efficient algorithms on distance-related problems, including the minimum distance diagram construction, the diameter computation, and the node-to-node shortest path routing. The extensively studied double-loop network has proven to hold efficient algorithms on the above specified distance-related problems. The significance of this research lies in mixed chordal ring network’s achievement of a better diameter, as well as the in-vertex-transitive feature of it, which makes its exploration on distance-related problems a lot more sophisticated. We first study and investigate the minimum distance diagram problem. We find that the minimum distance diagram of a mixed chordal ring network can be obtained by reassembling the pseudoMDD. This observation can be used to study other distance-related problems. For the diameter computation problem, we proposed an efficient algorithm to compute the diameter of a given mixed chordal ring network. For the optimization problem of finding optimal networks, we improve previous lower and upper bounds and successfully obtain a class of optimal mixed chordal ring networks. For the routing problem, two node-to-node routing algorithms are presented for flexible applications: the shortest-path-based routing algorithm and the dynamic routing algorithm. In addition, we also present an optimal faulttolerant routing algorithm for mixed chordal ring networks in the presence of up to one node or link failure. All the routing algorithms presented do not require routing tables and only very little computational overhead is needed.. Keywords: Mixed chordal ring network; Double-loop network; Algorithm; Diameter; Optimal routing; Fault-tolerant routing; Minimum distance diagram; Interconnection network; Parallel processing.. ii.

(4) 中文摘要本研究涵蓋混合式弦環網路中較少被討論的部份，並試圖發掘混合式弦環網路在與距離相關的問題中，是否存在有效率的演算法。這些問題包括最短距離圖的建造、直徑的計算、以及點與點之間的最優路由連線設計。上述與距離相關之問題在已被廣泛研究的雙環式網絡中，已經找得到有效率的演算法。本研究的重要性在於混合式弦環網路的直徑比雙環式網絡來得小，以及，混合式弦環網路沒有點對稱性質，因此使得上述與距離相關之問題複雜很多。我們首先研究混合式弦環網路的最短距離圖建造問題。我們發現混合式弦環網路的最短距離圖可經由重新組合「虛擬距離圖」得到。這樣的觀察可以讓我們研究其它與距離相關之問題。針對直徑計算問題，我們提出一個有效率的演算法可以計算出任一給定的混合式弦環網路的直徑。關於找出混合式弦網環路中最小直徑的最佳化問題，我們改進了前人針對此問題所提出的上下限，並且成功地得到一個無限最優混合式弦環網路族。針對網路路由連線設計問題，我們提出兩個可彈性應用的點對點最優路由連線設計演算法：基於最短路徑路由演算法及動態路由演算法。此外，我們也提出了一個最優容錯路由演算法。此演算法在網路壞掉一個點或一個邊時可以執行正確。上述所有路由演算法都不需要路由表格，並且只需要非常小的額外計算花費。. 關鍵字：混合式弦環網路；雙環式網路；演算法；直徑；最優路由；容錯路由；最短距離圖；連接網絡；平行處理。. iii.

(5) Acknowledgments 首先，我要以最誠摯的心意感謝我的指導老師：陳秋媛老師。當初會想攻讀博士有很大一部分因素是因為老師的關係。從碩士到博士跟老師這麼多年，深深地被老師對學生的熱心給感動到！任何疑難雜症都可以找老師商量、指點。而且不論對象是誰，她都很樂意幫助學生，實在是一位難能可貴的好老師。與老師 meeting 時，學習到老師對相關論文的看法及觀點；有論文上的想法或問題請益老師時，她都會非常用心的審視我的問題。此外當投稿不順利時，老師會鼓勵我、支持我，並幫助我渡過低潮。當與學弟妹有新的研究發現時，老師會很積極讓我們能團結一致，全力寫出結果。不儘如此，老師鼓勵我參加國際型研討會去發表論文，讓我增廣見聞，也學習別人是怎麼做研究的。有任何機會，老師都會主動詢問我。我只能說，能夠遇到這樣的老師，我真是幸運！另外也要感謝在攻讀博士期間所認識朋友。我們家族的學弟妹：柏澍、威雄、鈺傑、志文、子鴻、信菖、松育、宜君、士慶、慧棻、思賢、思綸、健峰、詩妤、恭毅，由於你們在每週 Group Meeting 的精彩表現，讓我獲益匯淺。還有學長姊學弟妹：君逸、宏賓、文祥、惠蘭、飛黃、業忠。有你們的參與下，讓我的生活多采多姿！！在此也要感謝攻讀博士期間打工家教的家長：陸媽媽及黃媽媽。由於家裡經濟的關係，平時需要在外面打工。而陸媽媽及黃媽媽對我非常地好，讓我比較沒有經濟上的壓力。此外，很幸運地在這段期間也遇到我人生中的另一半。人的一生不可能一輩子都只靠一個人活下去，感謝她的出現，讓我在人生的重要階段有了支持及陪伴，讓我的人生充滿彩色！！就讀外文系的她當然是我寫作請益的最佳對象。不儘如此，當有任何想法時，她能不厭其煩的聽我述說；遇到不順利的時候，能夠傾聽我、幫助我，與我一起分擔！！而且理工的我以及人文的她剛好形成最大的互補，讓我攻讀博士期間能夠有更美好的生活及無窮的支持！！我只能說，謝謝你，baby ～～最後當然最重要的還是感謝我的父母，沒有他們就沒有我。從小呵護我、栽培我唸書，始終在背後支持我。平常回家會一直進補我，讓我有健狀的身心。你們是我任何成就的最大推手！感謝的心，不止於此，僅以微薄紙筆，代表我心！. iv.

(6) Contents. Abstract (in English). ii. Abstract (in Chinese). iii. Acknowledgements. iv. Contents. vi. List of Tables. ix. List of Figures. ix. List of Algorithms. xii. 1 Introduction. 1. 1.1 Interconnection Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2 Evaluation Criteria for Networks. . . . . . . . . . . . . . . . . . . . . . . . .. 3. 1.3 Distributed Loop Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 1.5 Summary of the Contribution of This Research vi. . . . . . . . . . . . . . . . .. 11.

(7) 2 Background Material. 14. 2.1 Fundamental Concepts of Graph Theory . . . . . . . . . . . . . . . . . . . .. 14. 2.2 The Double-loop Network . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. 2.3 The Mixed Chordal Ring Network . . . . . . . . . . . . . . . . . . . . . . . .. 20. 3 The Minimum Distance Diagrams 3.1 Integer Lattice. 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24. 3.1.1. Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25. 3.1.2. The Labeling Function . . . . . . . . . . . . . . . . . . . . . . . . . .. 27. 3.1.3. The Interconnection Rules . . . . . . . . . . . . . . . . . . . . . . . .. 28. 3.1.4. The Distance-related Properties . . . . . . . . . . . . . . . . . . . . .. 28. 3.2 Finding an Optimal Copy . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31. 3.3 The MDDs of MCRNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36. 3.4 MDD Construction Algorithm for MCRNs . . . . . . . . . . . . . . . . . . .. 39. 4 The Diameter. 41. 4.1 The MAXDIST Subroutine . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41. 4.2 An Efficient Diameter-computing Algorithm . . . . . . . . . . . . . . . . . .. 44. 5 Optimal Networks. 48. 5.1 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49. 5.2 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49. 5.3 Optimal Mixed Chordal Ring Networks . . . . . . . . . . . . . . . . . . . . .. 54.

(8) 6 Routing. 56. 6.1 A Shortest-Path-Based Routing Algorithm . . . . . . . . . . . . . . . . . . .. 58. 6.1.1. Routing Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58. 6.1.2. Computing the Routing Parameter . . . . . . . . . . . . . . . . . . .. 59. 6.2 A Dynamic Routing Algorithm . . . . . . . . . . . . . . . . . . . . . . . . .. 65. 6.2.1. Finding a Shortest Route in the Plane . . . . . . . . . . . . . . . . .. 65. 6.2.2. A Dynamic Routing Algorithm . . . . . . . . . . . . . . . . . . . . .. 68. 6.3 Fault-tolerant Routing in MCRNs . . . . . . . . . . . . . . . . . . . . . . . .. 72. 6.3.1. Finding Alternative Paths . . . . . . . . . . . . . . . . . . . . . . . .. 74. 6.3.2. Finding the Lowest Cost Point . . . . . . . . . . . . . . . . . . . . . .. 80. 7 Experimental Results. 86. 7.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86. 7.2 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 89. 8 Conclusions. 91. 8.1 Summary of This Research . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 91. 8.2 Directions for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . .. 94. A Cheng-Hwang-Algorithm. 97. B Optimal Mixed Chordal Ring Networks and Double-Loop Networks. 99. References. 110.

(9) List of Tables. 1.1 Previous results on double-loop networks and mixed chordal ring networks. .. 11. 3.1 The nodes that can be reached from node u by using one link. . . . . . . . .. 28. 6.1 Comparing the SP-based routing algorithm with the dynamic routing algorithm. 57 7.1 The optimal MCRNs that are not achieved by setting s = 1 when N ≤ 5000.. 88. 8.1 Comparing the double-loop network with the mixed chordal ring network.. .. 93. B.1 Optimal MCRNs and DLNs for N = 6, 8, . . . , 256. . . . . . . . . . . . . . . .. 99. ix.

(10) List of Figures. 1.1 Examples of direct network topologies. . . . . . . . . . . . . . . . . . . . . .. 3. 1.2 A DLN and an MCRN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 2.1 MDDs of DLNs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17. 2.2 Chen-Hwang-Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18. 2.3 The (, h, p, n) determined by the Chen-Hwang-Rules in [17]. . . . . . . . . .. 18. 2.4 The inconsistency between Chen-Hwang-Rules and Cheng-Hwang-Algorithm.. 19. 3.1 Embedding a mixed chordal ring network into a double-loop network. . . . .. 25. 3.2 The tessellation of the plane formed by the pseudoMDD of MCR(22; 1, 7).. 26. 3.3 The labels for each point in the plane. . . . . . . . . . . . . . . . . . . . . .. 27. 3.4 The interconnection rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28. β 3.5 The illustrations of π u , copies of π u , Rα u and Ru for u = 20. . . . . . . . . .. 32. 3.6 Two possible ways to find an optimal copy of π u . The left figure is for the πu ∈ Γ+ case; the right figure is for the πu ∈ Γ− case. . . . . . . . . . . . .. 33. 3.7 The partitions of Z+ × Z+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34. 3.8 The partition of the pseudoMDD. . . . . . . . . . . . . . . . . . . . . . . .. 38. x.

(11) 3.9 The MDD0 and MDD1 of MCR(22; 1, 7). . . . . . . . . . . . . . . . . . . . .. 38. 3.10 The tessellation of the plane formed by the MDD0 of MCR(22; 1, 7). . . . .. 39. 3.11 The dimension of MDDλ .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39. 4.1 The five subcases of the case > h. . . . . . . . . . . . . . . . . . . . . . . .. 43. 5.1 A counterexample to the proof of Theorem 1.4.1. . . . . . . . . . . . . . . .. 50. 5.2 The improved ratio of our upper bound as compared to the previous upper bound for N = 6, 8, 10, . . . , 10004 (total 5000 N’s). . . . . . . . . . . . . . . .. 54. 6.1 Steps of finding the routing parameter. . . . . . . . . . . . . . . . . . . . . .. 59. 6.2 Finding the location of πμ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60. 6.3 An example of shortest path based routing. . . . . . . . . . . . . . . . . . . .. 64. 6.4 A shortest routing path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67. 6.5 Routing in MCR(22; 1, 7). . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70. 6.6 The paths correspond to the applying of the S-Link-First-Algorithm. . . . .. 75. 6.7 The paths correspond to the applying of the W-Link-First-Algorithm. . . . .. 75. 6.8 A detour: adding two more links to avoid a fault. . . . . . . . . . . . . . . .. 77. 6.9 The illustrations of the cases to the proof in Lemma 6.3.3. . . . . . . . . . .. 79. 6.10 The illustrations of the cases to the proof in Lemma 6.3.4. . . . . . . . . . .. 80. 6.11 Find the lowest cost point. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 81. 6.12 Optimal fault-tolerant in MCR(34; 1, 3). Two vectors that characterizing the pseudoMDD are α = (14, −5), β = (−10, 6). . . . . . . . . . . . . . . . . .. 82. 6.13 Optimal fault-tolerant in MCR(22; 1, 7). . . . . . . . . . . . . . . . . . . . .. 84.

(12) 7.1 An exhaustive computer search shows that 98.88% of optimal MCRNs MCR(N; s, w) can be obtained by setting s = 1 when N ≤ 5000. . . . . . . . . . . . . . . .. 87. 7.2 Comparing the minimum diameter between MCRNs and DLNs. . . . . . . .. 89. 7.3 Comparing the minimum average distance between MCRNs and DLNs. . . .. 90.

(13) List of Algorithms 1. MCRN-MDD-Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40. 2. MCRN-Diameter-Algorithm . . . . . . . . . . . . . . . . . . . . . . . .. 45. 3. SP-Based-Routing-Algorithm (SPBRA) . . . . . . . . . . . . . . . . .. 63. 4. Dynamic-Routing-Algorithm (DRA) . . . . . . . . . . . . . . . . . . .. 71. 5. S-Link-First-Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73. 6. W-Link-First-Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73. 7. Fault-Tolerant-Routing-Algorithm (FTRA) . . . . . . . . . . . . . . .. 85. xiii.

(14) Chapter 1 Introduction. 1.1. Interconnection Networks. In recent years, interconnection networks are applicable in many different fields, ranging from internal buses in very large-scale integration (VLSI) circuits to wide area computer networks. Among others, these applications include parallel computing, backplane buses and system area networks, telephone switches, internal networks for asynchronous transfer mode (ATM) and Internet Protocol (IP) switches, processor/memory interconnects for vector supercomputers, interconnection networks for multi-computers and distributed shared-memory multiprocessors, clusters of workstations and personal computers, local area networks, metropolitan area networks, wide area computer networks, and networks for industrial applications [26, 31, 37, 61]. To implement high performance parallel and distributed systems by designing interconnection architectures is a task both significant and challenging. [55, 56]. The choice of the interconnection network may affect several characteristics of the final system, including implementation cost (node complexity, VLSI area, wiring density), performance, ease of programming, reliability, and scalability. Throughout times, many different interconnec1.

(15) CHAPTER 1. INTRODUCTION. 1.1. INTERCONNECTION NETWORKS. tion networks had been applied in commercially available concurrent systems and numerous research prototypes [46, 54]; other alternatives are proposed and evaluated in theoretical studies [56]. Interconnection networks have been traditionally classified according to the operating mode (synchronous or asynchronous) and network control (centralized, decentralized or distributed) [31]. According to [31], there are four major classes based primarily on network topology: shared-medium networks, direct networks (router-based networks), indirect networks (switch-based Networks) and hybrid networks. In this research, our target networks, double-loop networks and mixed chordal ring networks, belong to direct networks. The direct network or point-to-point network is a popular interconnection network architecture that scales well to a large number of processors [31]. A direct network consists of a set of nodes, each node being directly connected to a subset of other nodes in the network. Theses nodes may have different functional capabilities. One common component of theses nodes is a router, which handles message communication among nodes. Direct networks have been a popular interconnection architecture for constructing large-scale parallel computers. Almost all direct network topologies studied in the literature have some degree of symmetry. Such a symmetric topology has many advantages: First, it allows the network to be constructed from simple building blocks and expanded in a modular fashion. Second, the regular topology facilitates the use of simple routing algorithms. Third, it is easier to develop efficient computational algorithms for multiprocessors interconnected by a symmetric network. Finally, it makes the network easier to model and analyze. For example, in a ring network of N nodes labeled from 0 to N − 1, each processor i is directly connected to processors (i − 1) mod N and (i + 1) mod N. Mathematical models for interconnection networks have played important roles in understanding, synthesizing, and comparing a multitude of network architectures. The architecture of an interconnection network can be represented by a graph or a digraph, where vertices 2.

(16) CHAPTER 1. INTRODUCTION. 13. 1. 0. 13. 2. 12. 4. 11 5. 10 6 7. 8. 1. 0. (a). 13. 2. 12. 3. 9. 1.2. EVALUATION CRITERIA FOR NETWORKS. 4. 9. 6 7. 8. 2 3 4. 11. 5. 10. 1. 12. 3. 11. 0. (b). 5. 10 9. 6 8. 7. (c). Figure 1.1: Examples of direct network topologies: (a) (Undirected) Ring network (b) Chordal ring network (c) Directed chordal ring network.. represent processors/nodes and edges represent links/channels between processors/nodes. Fig. 1.1 shows some direct network topologies.. 1.2. Evaluation Criteria for Networks. The topology of a direct network determines many architecture features of the network and affects several performance metrics. Although the actual performance of a network depends on many technology and implementations factors, several topological properties and metrics can be used to evaluate and compare different topologies in a technology-independent manner. Most of these properties are derived from the graph model of the network topology. • Symmetry and Regularity A regular network is defined as a network in which each node connects to the same number of other nodes. A symmetric network is a network in which the topology looks identical when viewed from every node or every edge. There are two types of symmetric: Node symmetric and edge symmetric. In graph-theoretic terms, a graph is node-symmetric (vertex-transitive) if, for every pair of vertices u and v, there is an automorphism which maps u to v. The definition of edge-symmetric is identical to the node-symmetric, except 3.

(17) CHAPTER 1. INTRODUCTION. 1.2. EVALUATION CRITERIA FOR NETWORKS. that the automorphism maps edges among themselves. References to symmetry without qualification usually imply node-symmetry. The main advantage of symmetric in a network lies in the ease of routing data in the network. This allows all nodes to use the same routing algorithm. The task of pathselection is also often simplified. Many popular direct interconnection networks are regular and symmetric. Clearly, all networks in Fig. 1.1 are regular. In addition, networks in Figs. 1.1(a) and 1.1(b) are also symmetric. • Connectivity The primary factor relating directly to the robustness of a graph-modeled interconnection structure is its connectivity or edge connectivity. From the graph theory viewpoint, the connectivity (resp., edge connectivity) of an undirected graph is the minimum number of vertices (resp., edges) whose removal causes the graph to be disconnected or to contain only one vertex. A digraph is strongly connected if for each ordered pair u, v of vertices, there is a path from u to v. In a directed graph, the connectivity (resp., edge connectivity) is defined as the minimum number of vertices (resp., edges) whose removal causes the graph to be non-strongly connected. For some symmetric networks, the connectivity is usually the same as the degree of a node. • Distance Measures In a direct network, communication between two nodes that are not directly connected must take place through other nodes. The network diameter (diameter for short) D, defined as the longest of the internode distances, is an important figure of merit for networks. The diameter D indicates the worst-case number of hops in sending a message from one node to another. If the message delay is proportional to the number of links traversed, this provides an upper bound on the delay in the absence of any interfering traffic. The diameter D may also be viewed as a lower bound on the delay between two 4.

(18) CHAPTER 1. INTRODUCTION. 1.2. EVALUATION CRITERIA FOR NETWORKS. nodes that are located farthest from each other. Although diameter does not completely characterizes the performance of an interconnection network, it is still useful in comparing networks with respect to their power to perform certain operations. Although the diameter is useful in comparing two interconnection networks with identical node degrees, it may not always be indicative of the actual performance of the networks. Since two nodes in a network do not always communicate with each other by traversing the length of the diameter D, it is more important to measure the average distance traveled by a message in practice. Average internode distance D is defined as the average lengths of the distance between all N 2 pairs of nodes. The average distance is representative of average or expected communication latencies, whereas D represents the worst case. • Efficient Routing As interconnection networks differ in the way they accommodate message traffic, routing performance is a primary indicator of the overall benefits of a particular topology. Efficient message routing can improve the network utilization. Many parameters including the length of the route, the computational overhead, the memory requirement at each node and the extra overhead information included in the message, can affect the routing performance. The first issue in the algorithmic aspect is to design efficient algorithms such that every message is sent along a shortest path from its source node to its destination node. Thus one of the most important features to be taken into account in the design of an interconnection network is the existence of efficient algorithms for routing messages. When some nodes or links in the network fail, some routes become unavailable. However, assuming that the network remains connected, communication is still possible by sending affected message along a sequence of surviving routes. Therefore, the design of algorithms for sending messages along the shortest route after detecting the faulty element is also an important issue. 5.

(19) CHAPTER 1. INTRODUCTION. 1.3. 1.3. DISTRIBUTED LOOP NETWORKS. Distributed Loop Networks. Loop networks have been widely considered in recent years as good network models for interconnection or communication networks due to their regularity, simple structure and symmetry; see Bermond et al. [9] for an exhaustive survey on this topic. The ring network (i.e., the single-loop network) is one of the most simple and frequently used loop network for interconnection networks, and has many attractive properties such as simplicity, extendibility, low degree, and ease of implementation. Although the ring network has many attractive properties, it has poor reliability (any failure in an interface or communication link destroys the function of the network) and it has high transmission delay. As a result, a lot of hybrid topologies utilizing the ring network as a basis for synthesizing richer interconnection schemes have been proposed to improve the reliability and reduce the transmission delay [6, 20, 27, 64]. One example of the commonly used extensions for the ring network is the multi-loop network ML(N; s1 , s2 , . . . , s ), which was first proposed by Wong and Coppersmith in [64] for organizing multi-module memory services. The most widely studied multi-loop network is perhaps the double-loop network (DLN for short). A DLN DL(N; s1 , s2 ) can be modeled by using a digraph with N nodes 0, 1, . . . , N − 1 and 2N links as follows i → (i + s1 ) mod N, i = 0, 1, 2, . . . , N − 1, i → (i + s2 ) mod N, i = 0, 1, 2, . . . , N − 1, where 0 < s1 = s2 < N. The double-loop network has been used for local area network [47] as well as the large local area optical network as SONET [7]. Another example of the commonly used extensions for the ring network is the chordal ring network, which is constructed by adding chords to the ring topology [6, 41]. Arden and Lee [6] first proposed and studied the chordal ring network. More specifically, an (undirected) 6.

(20) CHAPTER 1. INTRODUCTION. 1.3. DISTRIBUTED LOOP NETWORKS. chordal ring network CR(N; w), where N is even and w is odd, can be modeled by using a graph with N nodes 0, 1, . . . , N − 1 and 3N/2 links: (i, (i + 1) mod N),. i = 0, 1, 2, . . . , N − 1,. (i, (i + w) mod N), i = 1, 3, 5, . . . , N − 1. See Fig. 1.1(b) for an example of CR(14; 5). Since then, more than one hundred papers have been published on the topic of the chordal ring network and its variants. Especially, the chordal ring networks of degree 3, 4, and 6 have been widely discussed in the literature [8, 11, 24, 50, 65]. As was pointed out in [20], the chordal ring network is a 3-regular graph and it offers a happy medium between the (undirected) ring network and the undirected double-loop network in the amount of hardware. Also, it preserves the Hamiltonian cycle from the ring network and has a better diameter than the undirected ring network. In [41], Hwang and Wright considered the directed version of the chordal ring network and made a slight generalization on the ring links. More specifically, a directed chordal ring network DCR(N; s, w), where N is even and both s and w are odd, can be modeled by using a digraph with N nodes 0, 1, . . . , N − 1 and 3N/2 links: i → (i + s) mod N,. i = 0, 1, 2, . . . , N − 1,. i → (i + w) mod N, i = 1, 3, 5, . . . , N − 1. For an example, Fig. 1.1(c) is DCR(14; 1, 5). Recently, Chen et al. [20] introduced the mixed chordal ring network (MCRN for short) as a topology of interconnection networks. An MCRN MCR(N; s, w) can be modeled by. 7.

(21) CHAPTER 1. INTRODUCTION. 13. 1.4. MOTIVATION. 1. 0. 13. 2. 12. 4. 9. 4 5. 10. 6. 9. 7. 8. 3. 11. 5. 10. 2. 12. 3. 11. 1. 0. (a). 6 7. 8. (b). Figure 1.2: A DLN and an MCRN.. using a digraph with N nodes 0, 1, . . . , N − 1 and 2N links of the following types ring-links:. i → (i + s) mod N,. i = 0, 1, 2, . . . , N − 1,. chordal-links: i → (i − w) mod N, i = 0, 2, 4, . . . , N − 2, chordal-links: i → (i + w) mod N, i = 1, 3, 5, . . . , N − 1, where N is even, both s and w are odd. Figs. 1.2(a) and 1.2(b) illustrate DL(14; 1, 5) and MCR(14; 1, 5), respectively.. 1.4. Motivation. Since each node in the DLN or MCRN has two in-links and two out-links, the DLN and MCRN are very comparable1 . Throughout this thesis, N denotes the number of nodes in a communication network. For a fixed N, let DDL (N) and DM CR (N) denote the optimal (i.e., smallest) diameter of all DLNs and all MCRNs with N nodes, respectively. A well-known 1. When comparing the mixed chordal ring network with the double-loop network, we assume both networks have the same number of nodes.. 8.

(22) CHAPTER 1. INTRODUCTION. 1.4. MOTIVATION. lower bound on DDL (N) is as follows [64]: √ DDL (N) ≥ 3N − 2.. (1.4.1). For upper bounds on DDL (N), Hwang and Xu [42] managed to prove, using a heuristic method, that. √ DDL (N) ≤ 3N + 2(3N)1/4 + 5 for N ≥ 6348.. (1.4.2). In [57], Rödeseth further improved the above upper bound to be √ DDL (N) ≤ 3N + (3N)1/4 +. 5 2. for N ≥ 1200.. (1.4.3). For MCRNs, Chen et al. [20] showed the following result: Theorem 1.4.1. [20] There exists a choice of s and w such that the diameter of MCR(N; s, w) √ √ is no larger than 2N + 3. In other words, DM CR (N) ≤ 2N + 3. Since. √ 2N + 3 is severed as an upper bound, we have √ DM CR (N) ≤ 2N + 3.. (1.4.4). Note that there exist some erroneous cases in the proof of Theorem 1.4.1 and thus it is not known whether or not MCRNs can achieve a better diameter than DLNs. In spite of the erratum in the proof of Theorem 1.4.1, we confirm that MCRNs can achieve a better diameter than DLNs by giving an improved upper bound on DM CR (N) in Section 5.2 as DM CR (N) ≤ 2 N/2 + 1.. (1.4.5). From equations (1.4.1), (1.4.4) and (1.4.5), we can conclude definitely that the MCRN can achieve a better diameter than the DLN. One of the most important and fundamental optimization problem in designing interconnection networks is, for a given number of nodes N, how to find an optimal network with 9.

(23) CHAPTER 1. INTRODUCTION. 1.4. MOTIVATION. the smallest diameter and to give the construction of such a network. More precisely, for double-loop networks, DL(N; s1 , s2 ) is optimal if the diameter of DL(N, s1 , s2 ) is equal to DDL (N). This optimization problem for the double-loop network has been widely studied in the literature [2, 9, 10, 14, 15, 16, 30, 32, 42, 59]. However, to the best of our knowledge, there is no result about the exact value of DM CR (N) in the literature. Message routing is a fundamental and important function in interconnection networks. Efficient message routing not only can reduce the transmission delay but also can improve the network utilization. A routing algorithm is said to be optimal if every message is sent along a shortest path from its source node to its destination node. There has been a numerous amount of work on message routing in DLNs [22, 23, 35, 36, 40, 49]. In particular, it has been studied with respect to network applications such as message routing [35, 36, 49], permutation routing [40] and fault-tolerant routing [23, 49]. The minimum distance diagram (MDD for short), also called optimal routing region in [27], is a tool to encode distance-related information such as diameter and shortest route for multi-loop networks. It is well-known that the MDD of a DLN always forms an L-shape and one can compute the diameter and the average distance of a DLN from the lengths of segments on the boundary of an L-shape in constant time [33]. Cheng and Hwang [21] proposed an O(log N)-time algorithm to derive the lengths of segments on the boundary of the L-shape of DL(N; s1 , s2 ). Furthermore, many researchers addressed designing efficient routing algorithms or fault-tolerant routing by using the L-shapes [22, 23, 36, 49]. For further results of the DLN; see the excellent survey papers [9, 38, 39]. In contrast to the DLN, there has been little work reported in the literature on distancerelated problems of MCRNs. To the best of our knowledge, neither the diameter-computating strategy nor the message-routing strategy was found in the literature. A natural question arises, namely, whether the diameter computation and the message routing in MCRNs can be done efficiently as in DLNs. Table 1.1 shows a comparison of previous results between 10.

(24) CHAPTER 1. INTRODUCTION. 1.5. SUMMARY OF THE CONTRIBUTION OF THIS RESEARCH. DLNs and MCRNs. Table 1.1: Previous results on double-loop networks and mixed chordal ring networks.. MDD construction Diameter computation Optimal networks Node-to-node routing Fault-tolerant routing. 1.5. DLN. MCRN. [64] [21, [10, [22, [23,. ? ? ? ? ?. 66] 14, 15, 59] 35, 36] 36, 49]. Summary of the Contribution of This Research. In this section, we present a summary of the specific problems analyzed and the results derived in this thesis. The contribution of our research will be introduced in Chapters 3-7. In Chapter 3, we consider the problem of exploring and constructing the MDD of a MCRN MCR(N; s, w). Specifically, we introduce the pseudoMDD that helps study the distance-related problems in MCRNs. By mapping the nodes of a MCRN to the twodimensional integer lattice, one can study the distance properties between the nodes of a MCRN. Due to the tessellation of the plane formed by pseudoMDD, we successfully obtain the MDD of a given MCRN from the pseudoMDD in a simple manner. In the last section of this chapter, we give an algorithm to construct the MDD of a MCRN. The visualization tool established in this chapter will be used throughout this thesis. In Chapter 4, we consider the problem of computing the diameter of an MCRN. Instead of constructing the MDD of an MCRN first, we present a subroutine that can compute the maximum of distances of the nodes in the MDD to the node at the origin in constant time as long as we have the L-shape of the pseudoMDD. As an application, we obtain an algorithm that can compute the diameter of a given MCRN in O(log N) worst-case time in Section 4.2. 11.

(25) CHAPTER 1. INTRODUCTION. 1.5. SUMMARY OF THE CONTRIBUTION OF THIS RESEARCH. In Chapter 5, we discuss the problem of finding optimal MCRNs. In other words, we are interested in finding MCRNs which achieve the smallest diameter among all MCRNs with the same number of nodes. Due to the difficulty of this optimization problem, we aim at looking for bounds on DM CR (N) instead of finding optimal MCRNs directly. In Section 5.3, we successfully obtains a class of optimal MCRNs which matches the upper and lower bounds presented in Sections 5.1 and 5.2. In Chapter 6, we consider the problem of routing in MCRNs. In particular, routing of node-to-node message with at most one faulty element in MCRNs is considered. We design and present two optimal node-to-node routing algorithms and an optimal fault-tolerant routing algorithm for MCRNs. The two optimal node-to-node routing algorithms presented are shortest-path-based routing and dynamic routing. The shortest-path-based routing algorithm computes the routing parameter that can be used to determine a routing path. This algorithm takes O(log N)time for a source node to compute the routing parameter, and each node on the routing path can take constant time to determine the link (and therefore the node) to send messages according to the routing parameter. On the other hand, for the dynamic routing algorithm, after an O(log N)-time precalculation to determine the network parameters (only computed once and stored them in all nodes), it can route messages using constant time at each node along the routing path. The routing path is augmented on-the-fly at each routing step. A shortest-path-based routing algorithm is presented in Section 6.1. A dynamic routing algorithm is presented in Section 6.2. In Section 6.3, we present an optimal fault-tolerant routing algorithm for MCRNs. The algorithm does not require routing tables; it is efficient and it requires very little computational overhead. After an O(log N)-time precalculation, the algorithm can route messages to the destination using a constant time at each node along the route. Moreover, the faulttolerant algorithm presented is guaranteed to find the optimal route at the presence of up 12.

(26) CHAPTER 1. INTRODUCTION. 1.5. SUMMARY OF THE CONTRIBUTION OF THIS RESEARCH. to one node or link failure. Here we summarize the contribution of this thesis: it proposes (a) an algorithm to construct the MDDs of a mixed chordal ring network, (b) an efficient algorithm to compute the diameter of a mixed chordal ring network, (c) improved upper and lower bounds on DM CR (N), (d) two optimal node-to-node routing algorithms for mixed chordal ring networks, (e) an optimal fault-tolerant routing algorithm for mixed chordal ring networks.. 13.

(27) Chapter 2 Background Material In this chapter, we present some background material on the double-loop network and the mixed chordal ring network, as well as some previous results related to both networks. In addition, some fundamental concepts of graph theory are given first. Our terminologies and notations of graph theory are standard; see [63] and also [13].. 2.1. Fundamental Concepts of Graph Theory. A graph G with n vertices and m edges consists of the vertex set V (G) = { v1 , v2 , . . . , vn } and edge set E(G) = { e1 , e2 , . . . , em }, where each edge consists of two (possibly equal) vertices called, endpoints. An element in V (G) is called a vertex of G. An element in E(G) is called an edge of G. When vertices u and v are the endpoints of an edge e, they are adjacent and are neighbors. We write (u, v) when { u, v } ∈ E(G). A loop is an edge whose endpoints are equal. Multiple edges are edges having the same pair of endpoints. A simple graph is a graph having no loops or multiple edges. A directed graph or digraph G consists of a vertex set V (G) and an edge set (or arc set) E(G), where each edge is an ordered pair of vertices. The first vertex of the ordered pair is 14.

(28) CHAPTER 2. BACKGROUND MATERIAL. 2.2. THE DOUBLE-LOOP NETWORK. the tail of the edge, and the second is the head ; together, they are the endpoints. We say that an edge is an edge from its tail to its head. We write u → v when there is an edge from u to v. In a digraph, a loop is an edge whose endpoints are equal. Multiple edges are edges having the same ordered pair of endpoints. A digraph is simple if each ordered pair is the head and tail of at most one edge. For a vertex v of a digraph G, the outdegree d+ (v) the number of edges with tail v. The indegree d− (v) is the number of edges with head v. Unless otherwise specified, the following definitions and terms hold for both graphs and digraphs. A separating set or vertex cut of a graph G is a set S ⊆ V (G) such that G \ S has more than one component. A graph is k-connected if every separating set has at least k vertices. A digraph G is strongly connected or strong if there is a path from u to v in G for every ordered pair u, v ∈ V (G). A digraph G is strongly k-connected if |V (G)| ≥ k + 1 and every separating set of G has at least k vertices. An isomorphism from a simple graph G to a simple graph H is a bijection f : V (G) → V (H) such that { u, v } ∈ E(G) if and only if { f (u), f (v) } ∈ E(H). An automorphism of G is an isomorphism from G into G. A graph G is vertex-transitive if for every pair u, v ∈ V (G), there is an automorphism that maps u to v.. 2.2. The Double-loop Network. A double-loop network (DLN for short) DL(N; s1 , s2 ) can be modeled by using a digraph with N nodes 0, 1, . . . , N − 1 and 2N links i → (i + s1 ) mod N, i = 0, 1, 2, . . . , N − 1, i → (i + s2 ) mod N, i = 0, 1, 2, . . . , N − 1, where 0 < s1 = s2 < N. The integers s1 , s2 are called steps or hops or jumps. The connectivity of the DLN has been determined by Doorn [60] (or see [38]): 15.

(29) CHAPTER 2. BACKGROUND MATERIAL. 2.2. THE DOUBLE-LOOP NETWORK. Theorem 2.2.1. [38] DL(N; s1 , s2 ) is strongly 2-connected if and only if gcd(N, s1 , s2 ) = 1. It is well-known [34] that DL(N; s1 , s2 ) is a Cayley digraph of the cyclic group ZN with the set of generators {s1 , s2 }. Since Cayley digraphs are vertex-transitive, the distancerelated problems of DLNs can be reduced to the problem of studying paths originated at a fixed vertex1 , usually node 0. A visualization tool that allows studying distance-related problems of DLNs from a geometric point of view is set up as follows: Consider the twodimensional integer lattice Z × Z. Given DL(N; s1 , s2 ), label each lattice point (x, y) (i.e., x and y being integers) of Z × Z by (xs1 + ys2 ) mod N. Unless otherwise specified, we refer to a point as a lattice point. A minimum distance diagram (MDD) of DL(N; s1 , s2 ) is an array with node 0 at point (0, 0) and node u at point (x, y) if and only if xs1 + ys2 ≡ u (mod N) and x + y is the minimum among all (x , y ) satisfying the congruence. Namely, a shortest path from node 0 to node u is through taking x s1 -steps and y s2 -steps (in any order). Note that an MDD includes every node exactly once. Most authors [2, 12, 18, 19, 21, 33, 38] always “break ties” lexicographically (choose with smaller y) whenever there are two (x, y)’s satisfying xs1 + ys2 ≡ u (mod N). Without this convention, Sabariego and Santos [58] showed that every DLN has at most two MDD’s. Throughout this thesis, we follow the convention used in the literature, i.e., we assume a DLN has only one MDD constructed by using the convention. Fig. 2.1(a) illustrates the MDD of DL(14; 1, 5). It is well-known [64] that the MDD of a DLN is of a definite form: an L-shape. The L-shape is determined by four parameters (, h, p, n); these four parameters are the lengths of four of the six segments on the boundary of the L-shape; see Fig. 2.1(a). For example, the MDD in Fig. 2.1(a) has an L-shape (, h, p, n) = (5, 3, 1, 1). An L-shape is degenerate if its shape is a rectangle; for example, the MDD in Fig. 2.1(b) is degenerate. 1. Although a network and the graph modeling it are conceptually distinct, we shall use the terms “node” and “verex” interchangeably when there is no ambiguity. 16.

(30) CHAPTER 2. BACKGROUND MATERIAL. 2.2. THE DOUBLE-LOOP NETWORK. . . . . . . . . . . . . . . . . . . . . . . . . .

(31) . . Figure 2.1: MDDs of DLNs.. Fiol et al. [33] observed that the distribution of all points with the same label repeat periodically and an MDD always tessellates the plane regardless of whether its L-shape is degenerate or not. By considering the relative positions of point with the label 0, Fiol et al. derived the following congruences: s1 − ns2 ≡ 0 (mod N) −ps1 + hs2 ≡ 0 (mod N).. (2.2.1). Let vectors α = (, −n) and β = (−p, h). It is known that all the points with the label 0 can be generated by repeatedly adding ±α and ±β to each new point with the label 0. Moreover, if one location of node u is known, then the positions of all other points with the label u can be expressed in terms of α and β [27]. Chen and Hwang [17] used the observation (2.2.1) to prove that an L-shape is degenerate if and only if exactly one of the two congruences: s1 ≡ 0 mod N and hs2 ≡ 0 mod N is satisfied. They introduced the Chen-Hwang-Rules [17] to define the lengths of segments on the boundary of the L-shape when an L-shape is degenerate; see Fig. 2.2. As an example, the L-shape of DL(10; 2, 7) in Fig. 2.1(b) is (5, 2, 2, 0). Wong and Coppersmith [64] gave an O(N)-time algorithm to construct an MDD (hence the L-shape) diagonally starting from point (0, 0). Specifically, consider filling numbers in 17.

(32) CHAPTER 2. BACKGROUND MATERIAL. 2.2. THE DOUBLE-LOOP NETWORK. Rule 1. Suppose hs2 ≡ s1 ≡ 0 (mod N ). Let the zero immediately above the Lshape be at point (i, h). Then p = − i, n = 0. Rule 2. Suppose s1 ≡ hs2 ≡ 0 (mod N ). Let the zero immediately to the right of the L-shape be at point (, j). Then p = 0, n = h − j. Rule 3. Suppose s1 ≡ hs2 ≡ 0 (mod N ). If h > , follow Rule 1; otherwise, follow Rule 2. Note that for an L-shape (, h, p, n), we have > 0, h > 0, p ≥ 0, n ≥ 0, p and n not both zero. Figure 2.2: Chen-Hwang-Rules. . . . . .

(33). . . . . . . . . . . . . . . . .

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(36).

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(40). . . Figure 2.3: The (, h, p, n) determined by the Chen-Hwang-Rules in [17].. { (x, y) | x ≥ 0, y ≥ 0, x ∈ Z, y ∈ Z }. Start from the origin (0, 0), then the line (1, 0), (0, 1), and then the line (2, 0), (1, 1), (0, 2), and so on. At each lattice point (x, y) (i.e., x, y being integers), if the value u, where xs1 + ys2 ≡ u (mod N), has not appeared so far, we fill u at point (x, y), otherwise we just leave a blank. We stop when all values of u, i.e. u = 0, 1, . . . , N − 1, have been accounted for. Cheng and Hwang [21] gavn an O(log N)-time algorithm, we call it Cheng-HwangAlgorithm, based on the Euclidean algorithm, to compute the L-shape (, h, p, n). For the completeness of this thesis, the Cheng-Hwang-Algorithm is given in Appendix A. However, when an L-shape is degenerate, the solution of (, h, p, n) determined by ChenHwang-Rules [17] does not always coincide with the values determined by the Cheng-HwangAlgorithm [21]. One such example is that for DL(15; 4, 5), Chen-Hwang-Rules determines the L-shape (, h, p, n) = (5, 3, 0, 1), whereas Cheng-Hwang-Algorithm determines the L18.

(41) CHAPTER 2. BACKGROUND MATERIAL. 2.2. THE DOUBLE-LOOP NETWORK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(42) . . Figure 2.4: The inconsistency between Chen-Hwang-Rules and Cheng-Hwang-Algorithm.. shape (, h, p, n) = (5, 7, 5, 4); see Fig. 2.4. Clearly, the result determined by Chen-Hwang-Rules is more accurate and precise. In addition, our algorithms (diameter-computing algorithm, routing algorithm) for the distancerelated problems on the MCRNs highly rely on the correct information of the L-shapes. Thus, to overcome this problem, Lee, Lan and Chen [45] proposed a simple modification to ˆ ˆh, pˆ, n the Cheng-Hwang-Algorithm as follows: Let (, ˆ ) denote the solution of Cheng-Hwang¯ h, ¯ p¯, n Algorithm and (, ¯ ), the solution of Chen-Hwang-Rules. Theorem 2.2.2. [45] Given DL(N; s1 , s2 ), let d = gcd(N, s1 ), d = gcd(N, s2 ). Then − 1} such that 1. If DL(N; s1 , s2 ) satisfies d > 1 and there exists 1 ≤ j ≤ min{d − 1, N d d s1 ≡ js2 (mod N) with j <. N , 2d. ˆ h ¯=h ˆ−n then ¯ = , ˆ , p¯ = 0, n ¯ = j.. 2. If DL(N; s1 , s2 ) satisfies d > 1, d > 1 and d s1 ≡ ds2 ≡ 0 (mod N) and d < d , then ˆ h ¯=n ˆ−n ¯ = , ¯=h ˆ , p¯ = 0.. 19.

(43) CHAPTER 2. BACKGROUND MATERIAL. 2.3. 2.3. THE MIXED CHORDAL RING NETWORK. The Mixed Chordal Ring Network. A mixed chordal ring network (MCRN for short) MCR(N; s, w) can be modeled by using a digraph with N nodes 0, 1, . . . , N − 1 and 2N links of the following types ring-links:. i → (i + s) mod N,. i = 0, 1, 2, . . . , N − 1,. chordal-links: i → (i − w) mod N, i = 0, 2, 4, . . . , N − 2, chordal-links: i → (i + w) mod N, i = 1, 3, 5, . . . , N − 1, where N is even, both s and w are odd, and 0 < s = w < N. It should be noted that the parameters s and w should satisfy s + w = N in order to prevent the multiple links between two nodes of the digraph, which means a waste of the hardware. Chen, Hwang and Liu [20] proved the following theorem. Theorem 2.3.1. [20] MCR(N; s, w) is strongly 2-connected if and only if gcd(N, s, w) = 1. The proofs of equation (1.4.4) and Theorem 2.3.1 are based on the idea of embedding a MCRN into a DLN. Specifically, Chen, Hwang and Liu [20] showed that the MCRN , s+w ) by combining nodes 2k + 1 MCR(N; s, w) can be embedded into the DLN DL( N2 ; s−w 2 2 mod N2 , s+w = s−w = and 2k+1+w as supernode k ∗ for all k = 0, 1, . . . , N/2−1, where s−w 2 2 2 s+w mod N2 . They used this idea to obtain the connectivity and diameter information of 2 the MCRNs. However, we observe that this embedding sometimes fails. Take MCR(10; 1, 5) ; 1−5 , 1+5 ), i.e., DL(5; 3, 3), which is clearly as an example; its corresponding DLN is DL( 10 2 2 2 not a valid DLN, yet MCR(10; 1, 5) is a valid mixed chordal ring network. In general, MCR(2(2k + 1); 1, 2k + 1) can not be embedded into a valid DLN. The idea used in [20] to prove Theorem 2.3.1 is to show that MCR(N; s, w) is strongly 2-connected if and only if , s+w ) is strongly 2-connected. We now correct the proof. its corresponding DLN DL( N2 ; s−w 2 2 First, a lemma is needed. 20.

(44) CHAPTER 2. BACKGROUND MATERIAL. 2.3. THE MIXED CHORDAL RING NETWORK. Lemma 2.3.2. For MCR(N; s, w), 1. if w =. N , 2. 2. if w =. N , 2. then DL( N2 ; s−w , s+w ) is a double-loop network; 2 2 then DL( N2 ; s−w , s+w ) is not a double-loop network and MCR(N; s, N2 ) is 2 2. itself the double-loop network DL(N; s, N2 ). , s+w ) is not a valid double-loop network whenever Proof. DL( N2 ; s−w 2 2 or. s+w 2. ≡ 0 (mod. N ) 2. or. s−w 2. ≡. s+w 2. (mod. s = w and s + w = N, it is impossible that s−w 2. ≡. s+w 2. (mod. N ) 2. if and only if w =. N . 2. N ) 2. s−w 2. ≡ 0 (mod. N ) 2. or gcd( N2 , s−w , s+w ) = 1. Since we assume 2 2. s−w 2. ≡ 0 (mod. N ) 2. or. s+w 2. ≡ 0 (mod. N ). 2. Also,. In addition, we have assumed gcd(N, s, w) = 1;. , s+w ) = 1. Thus we have the first if-statement. When w = therefore gcd( N2 , s−w 2 2. N N , 2 2. ≡ − N2. (mod N) occurs and the chordal-links of MCR(N; s, w) become: i → (i +. N ) 2. mod N, i = 0, 1, . . . , N − 1.. Thus MCR(N; s, N2 ) is itself the double-loop network DL(N; s, N2 ) with steps s and N/2, and we have the second if-statement. Lemma 2.3.2 shows that DL( N2 ; s−w , s+w ) is a valid embedding if and only if w = 2 2. N . 2. In. [20], the following lemma is proved. Lemma 2.3.3. ([20]) MCR(N; s, w) is strongly connected if and only if gcd(N, s, w) = 1. Now we give a correct proof for Theorem 2.3.1. Proof of Theorem 2.3.1: Necessity. Since MCR(N; s, w) is strongly 2-connected, it is also strongly connected. Thus, this part follows directly from Lemma 2.3.3. Sufficiency. There are two cases. Case 1: w = w =. N s−w , 2 2. N . 2. =. Then by Lemma 2.3.2, DL( N2 ; s−w , s+w ) is a double-loop network. Since 2 2. s+w . 2. Since gcd(N, s, w) = 1, gcd( N2 , s−w , s+w ) = 1. Thus by Theorem 2.2.1, 2 2 21.

(45) CHAPTER 2. BACKGROUND MATERIAL. 2.3. THE MIXED CHORDAL RING NETWORK. DL( N2 ; s−w , s+w ) is strongly 2-connected. Since the two nodes in each super-node can reach 2 2 each other through the chordal-links between them, MCR(N; s, w) is strongly 2-connected. Case 2: w =. N . 2. By Lemma 2.3.2, MCR(N; s, w) is itself the double-loop network DL(N; s, w).. Thus by Theorem 2.2.1 and by the assumption that gcd(N, s, w) = 1, MCR(N; s, w) is strongly 2-connected. Being vertex-transitive (or vertex symmetric) is a desirable property of an efficient network topology. Intuitively, a vertex-transitive network looks the same from any node. This property reduces the complexity of distance-related problems. For example, it allows the use of an identical routing algorithm at every node. However, as was pointed out in [44], an MCRN may fail to be vertex-transitive. One such example is MCR(12; 3, 5), in which node 0 can reach any node within 4 steps, while it takes 5 steps for node 1 to reach node 8. Although an MCRN may fail to be vertex-transitive, it does satisfy the even-odd-vertextransitive property: for every pair of vertices u, v ∈ {0, 1, . . . , N − 1} with the same parity, there is an automorphism ϕ that maps u to v. In other words, in an MCRN, all evennumbered nodes are symmetric and all odd-numbered nodes are symmetric. By using this property, we may pay our attention to node 0 and node 1 without loss of generality. In Theorem 2.3.4, we further prove that node 1 can be regarded as an even-numbered node in another MCRN. Two MCRNs MCR(N; s, w) and MCR(N; s , w ) are said to be strongly isomorphic if there is a bijection ϕ from the nodes of MCR(N; s, w) to the nodes of MCR(N; s , w ) such that ϕ(v + s) = ϕ(v) + s for all nodes v and either . ϕ(v − w) = ϕ(v) − w , for even v and even ϕ(v); ϕ(v + w) = ϕ(v) + w , for odd v and odd ϕ(v).. or. . ϕ(v − w) = ϕ(v) + w , for even v and odd ϕ(v); ϕ(v + w) = ϕ(v) − w , for odd v and even ϕ(v). 22.

(46) CHAPTER 2. BACKGROUND MATERIAL. 2.3. THE MIXED CHORDAL RING NETWORK. Theorem 2.3.4. MCR(N; s, w) and MCR(N; s, N − w) are strongly isomorphic. Proof. Let the bijection from the nodes of MCR(N; s, w) to the nodes of MCR(N; s, N −w) be ϕ(v) = (v + w) mod N.. (2.3.1). It is not difficult to check that ϕ(v+s) = ϕ(v)+s for all nodes v and ϕ(v−w) = ϕ(v)+N −w for even v and odd ϕ(v); ϕ(v + w) = ϕ(v) − N + w for odd v and even ϕ(v). Therefore MCR(N; s, w) and MCR(N; s, N − w) are strongly isomorphic. For convenience, the function in (2.3.1) is called the renaming function. From the above discussion, throughout this thesis, we will assume that MCR(N; s, w) satisfies the following conditions: s = w, s + w = N, w = N/2, and gcd(N, s, w) = 1.. (2.3.2). The first two assumptions are from the definition of the MCRN in order to prevent multiple links between two nodes. The reason for the assumption w = N/2 is that since MCR(N; s, N2 ) is DL(N; s, N2 ) and many previous results of DLNs can apply on it. Besides, since we only consider connected graph, the last assumption ensure the MCRN being strongly connected. Furthermore, by the even-odd-vertex-transitive property of MCRNs, without loss of generality, we may restrict our discussion on node 0 and node 1 (to obtain the diameter and to obtain a routing path). Moreover, by Theorem 2.3.4, node 1 of MCR(N; s, w) can be regarded as the even-numbered node (1 + w) mod N in MCR(N; s, N − w); the node (1 + w) mod N can be further regarded as node 0 in MCR(N; s, N − w).. 23.

(47) Chapter 3 The Minimum Distance Diagrams of Mixed Chordal Ring Networks The purpose of this chapter is to explore and to investigate the minimum distance diagrams of mixed chordal ring networks. Results derived from this chapter have been submitted to [43]. The definition of the minimum distance diagrams of a mixed chordal ring network is given in Section 3.3.. 3.1. The Two-Dimensional Integer Lattice Environment. One approach to study the distance-related problems of MCRNs is as that of in DLNs: Maps (or labels) each point of the two-dimensional integer lattice Z+ × Z+ to a node of a given MCRN. However, since an MCRN is only even-odd-vertex-transitive, it is not clear how to label each point of Z+ × Z+ for a given MCRN. In other words, how to define the labeling function from the points of Z+ × Z+ to the nodes of a given MCRN is our first issue. In the following, the labeling function we defined is based on the pseudoMDD introduced in Section 3.1.1. 24.

(48) CHAPTER 3. THE MINIMUM DISTANCE DIAGRAMS . 3.1. INTEGER LATTICE. . . . . . . . . . .

(49) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.1: Embedding a mixed chordal ring network into a double-loop network.. 3.1.1. The Embedding Technique and the PseudoMDD. Graph embedding is an important technique as we can take the advantage of all the known results about the host graph and apply these results on the guest graph. Given an MCR(N; s, w) , s+w by combining nodes with w = N/2, we can embed MCR(N; s, w) into DL N2 ; s−w 2 2 2k and 2k − w as supernode k ∗ for all k = 0, 1, . . . , N/2 − 1. Note that, unless otherwise mod N2 , s+w mod N2 , nodes of an MCRN are taken means s−w means s+w specified, s−w 2 2 2 2 modulo N (thus node u means node u mod N), and nodes of a DLN with N/2 nodes are taken modulo N/2 (thus node v means node v mod N/2). Figs. 3.1(a) and 3.1(b) illustrate the embedding of MCR(14; 1, 5) into DL(7; 5, 3) and the bold rounded rectangles indicate the supernodes (host nodes). Since we can embed an MCRN into a DLN, we can embed an MCRN into the MDD of the corresponding DLN. Given MCR(N; s, w), the pseudoMDD is constructed as follows: (see Figs. 3.1(c) and 3.1(d)): pseudoMDD: Replace each node u in the MDD of DL. N 2. ; s−w , s+w with two 2 2. nodes 2u and 2u − w. If u is at point (x, y), then 2u and 2u − w are at points (2x, y) and (2x + 1, y), respectively. Recall that the MDD of a DLN always forms an L-shape, and this MDD tessellates the plane. Since a pseudoMDD provides a one-to-one correspondence between the corresponding DLN’s MDD’s and the pseudoMDD’s, it is obvious that a pseudoMDD is also an 25.

(50) CHAPTER 3. THE MINIMUM DISTANCE DIAGRAMS. 3.1. INTEGER LATTICE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.2: The tessellation of the plane formed by the pseudoMDD of M CR(22; 1, 7).. L-shape, but the length of the horizontal segment on the boundary of the pseudoMDD is twice of that of the corresponding DLN’s MDD. For example in Figs. 3.1(c)(d), the pseudoMDD has an L-shape (, h, p, n) = (4, 4, 2, 1), whereas the corresponding DLN’s MDD has an L-shape (, h, p, n) = (2, 4, 1, 1). We have the following fact. Fact. A pseudoMDD has the following properties. (i) It contains every node of the MCRN exactly once. (ii) The shape is always an L-shape with parameters (2, h, 2p, n) whenever the corresponding DLN’s MDD has an L-shape (, h, p, n). (iii) It always tessellates the plane (see Fig. 3.2 for an example). The name “pseudoMDD” comes from the reason that a pseudoMDD may fail to be a “minimum” distance diagram. For example, consider Fig. 3.2. Both points (8, 0) and (6, 2) represent node 20. However, the distance (minimum number of links) from point (0, 0) to (8, 0) is 8 (the unique shortest path is 0 → 15 → 16 → 9 → 10 → 3 → 4 → 19 → 20) while the distance from point (0, 0) to (6, 2) is 6 (one of the shortest path is 0 → 1 → 2 → 3 → 4 → 19 → 20), yet point (8, 0) is inside the pseudoMDD. Note that some pseudoMDD’s are indeed MDD’s. See Section 3.3 for more further discussion. 26.

(51) CHAPTER 3. THE MINIMUM DISTANCE DIAGRAMS. 3.1.2. 3.1. INTEGER LATTICE. The Labeling Function. Recall that a node u at point (x, y) of the MDD of DL . s−w x 2. . +y. s+w 2. ≡u. N 2. satisfies ; s−w , s+w 2 2. N mod . 2. By the construction of the pseudoMDD of MCR(N; s, w), nodes 2u and 2u − w of MCR(N; s, w) are at points (2x, y) and (2x + 1, y), respectively. As a result, the labeling function for point z = (x, y) is ⎧ ⎪ ⎪ ⎪ ⎨ l(z) =. x (s 2. − w) + y(s + w). (mod N) if x is even; (3.1.1). ⎪ ⎪ ⎪ ⎩ x−1 (s − w) + y(s + w) − w. (mod N) if x is odd.. 2. Or, equivalently l(z) =. x. x. +y s− −y w 2 2. . mod N.. . (3.1.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.3: The labels for each point in the plane.. 27. .

(52) CHAPTER 3. THE MINIMUM DISTANCE DIAGRAMS. 3.1. INTEGER LATTICE. Table 3.1: The nodes that can be reached from node u by using one link.. Node u at point (x, y) node u+s u−w u+s u+w. x is even x is odd. 3.1.3. at point (x + 1, y + 1) (x + 1, y) (x + 1, y) (x − 1, y). The Interconnection Rules. It should be noticed that the interconnection rules between adjacent points in the twodimensional integer lattice are quite different from those of DLNs (recall that in DLNs, each point can reach either an east or a north point). Roughly speaking, a point (x, y) can reach either a) east or northeast points or b) east or west points, depending on the parity of x. Nodes that can be reached from node u at point (x, y) are shown in Fig. 3.4 and Table 3.1. Note that we will only consider points in Z+ × Z+ = { (x, y) ∈ Z × Z | x ≥ 0, y ≥ 0 }. . . . . . . . . . . . . . . . .

(53) . Figure 3.4: The interconnection rules.. 3.1.4. The Distance-related Properties. Some distance-related properties will be investigated in this section. For convenience, some notations will be introduced first. Define the parity of an integer x to be 0 if (x mod 2) equals to 0 and 1 if (x mod 2) equals to 1. The parity of an integer x is denoted by parity(x).. 28.

(54) CHAPTER 3. THE MINIMUM DISTANCE DIAGRAMS. 3.1. INTEGER LATTICE. Partition Z+ × Z+ into Γ+ and Γ− as follows: def. Γ+ =. . (x, y) ∈ Z+ × Z+ | x ≥ 2y ≥ 0. . and Γ− =. def. . (x, y) ∈ Z+ × Z+ | 0 ≤ x < 2y .. Each point z = (x, y) of Z+ × Z+ is associated with a distance (or norm), denoted by Δ(z), which is the minimum number of links that needs to be traversed from point (0, 0) to (x, y). The distance for each point can be determined as follows. Lemma 3.1.1. The distance of point z = (x, y) is ⎧ ⎪ ⎨ x Δ(z) =. if z ∈ Γ+ ,. ⎪ ⎩ 2y − parity(x). (3.1.3). −. if z ∈ Γ .. Proof. We prove this lemma by induction on x and y. For the basis step, clearly, Δ((0, 0)) = 0, Δ((x, 0)) = x, Δ((0, y)) = 2y and thus (3.1.3) holds. For the induction step, suppose (3.1.3) holds for points (x − 1, y), (x − 1, y − 1) and (x, y − 1). Now consider point (x, y), where x ≥ 1 and y ≥ 1. Case 1: x is even. Then Δ((x, y)) = min { d1 , d2 }, where d1 = Δ((x − 1, y)) + 1 and d2 = Δ((x, y − 1)) + 2. Subcase 1.1 : x ≥ 2y. By the induction hypothesis, d2 = x + 2. If x − 1 ≥ 2y, then by the induction hypothesis, d1 = x; if x − 1 < 2y, then we have x = 2y and hence, by the induction hypothesis, d1 = x. Therefore Δ((x, y)) = x and (3.1.3) holds. Subcase 1.2 : x < 2y. By the induction hypothesis, d1 = 2y. If x ≥ 2(y − 1), then x = 2y − 2. By the induction hypothesis d2 = 2y; if x < 2(y − 1), then by the induction hypothesis d2 = 2y. Therefore Δ((x, y)) = 2y and (3.1.3) holds. Case 2 : x is odd. Then Δ((x, y)) = min { d1 , d2 }, where d1 = Δ((x − 1, y)) + 1 and d2 = Δ((x − 1, y − 1)) + 1. 29.

(55) CHAPTER 3. THE MINIMUM DISTANCE DIAGRAMS. 3.1. INTEGER LATTICE. Subcase 2.1 : x ≥ 2y. If x − 1 ≥ 2y, then by the induction hypothesis, d1 = d2 = x; if x − 1 < 2y, then we have x = 2y, a contradiction to odd x. Hence Δ((x, y)) = x and (3.1.3) holds. Subcase 2.2 : x < 2y. Clearly by the induction hypothesis, d1 = 2y +1. If x−1 ≥ 2(y −1) then x = 2y − 1 and, by the induction hypothesis, d2 = x = 2y − 1; if x − 1 < 2(y − 1) then by the induction hypothesis, d2 = 2y − 1. Therefore Δ((x, y)) = 2y − 1 and (3.1.3) holds. Note that the distance function for point (x, y) in the two-dimensional integer lattice is quite different from the standard one (i.e., |x| + |y|). A tool that can compare the distances of the two points is given as follows. For point z = (x, y) and vector v = (v1 , v2 ) with v1 , v2 being integers, let z + v denote the point (x + v1 , y + v2 ). Then: Lemma 3.1.2. Suppose v = (v1 , v2 ) with even v1 . Then we have Δ(z) ≤ Δ(z + v) if (i) z ∈ Γ+ and v1 ≥ 0, v2 ≤ 0 or (ii) z ∈ Γ− and v1 ≤ 0, v2 ≥ 0 or (iii) v1 ≥ 0 and v2 ≥ 0. Proof. Since v1 is even, parity(x) = parity(x+v1 ). The first two cases (i) and (ii) come from (3.1.3) undoubtedly. Now consider case (iii). If z and z + v are both in Γ+ ( or Γ− ), then the result is easy to see. Suppose z ∈ Γ+ and z + v ∈ Γ− . Let z = (x + v1 , y). Since v1 ≥ 0, clearly z ∈ Γ+ . By (3.1.3), Δ(z) = x, Δ(z ) = x + v1 and Δ(z + v) = 2y + 2v2 − parity(x). Since z + v ∈ Γ− , we have x + v1 < 2y + 2v2 . Therefore Δ(z) ≤ Δ(z ) ≤ Δ(z + v) holds. The case of z ∈ Γ− and z + v ∈ Γ+ is similar to obtain. Note that if point z + v is outside Z+ × Z+ , then we may simply let Δ(z + v) = ∞ to ensure the correctness of Lemma 3.1.2. 30.

(56) CHAPTER 3. THE MINIMUM DISTANCE DIAGRAMS. 3.2. 3.2. FINDING AN OPTIMAL COPY. Finding an Optimal Copy. Suppose the pseudoMDD of MCR(N; s, w) has an L-shape (2, h, 2p, n). The following two vectors that characterize the shape of the pseudoMDD are crucial in the remaining discussion and are defined by α = (2, −n), β = (−2p, h). def. def. Since a pseudoMDD consists of N points, for each node u ∈ { 0, 1, . . . , N − 1 }, there is exactly one point of the pseudoMDD with label u and we denote this point by π u . In Z+ × Z+ , points having the same label as π u are called copies (or relocations) of π u . The set of all points with label u is denoted by Πu . Since a pseudoMDD can tessellate the plane, by considering all points with label 0, we perspicuously have that all the other copies of π 0 can be expressed in terms of α and β. More generally, point π is a copy of point π u if and only if π = π u + iα + jβ for some integers i, j; see Fig. 3.5. Given a π u , define Rα u and Rβu as follows: β Rα u = { π u + kα | k ∈ Z, k ≥ 0 } and Ru = { π u + kβ | k ∈ Z, k ≥ 0 } . def. def. Example. The pseudoMDD of MCR(22; 1, 7) in Fig. 3.5 has an L-shape (2, h, 2p, n) = (12, 2, 2, 1) and its shape is characterized by vectors α = (12, −1) and β = (−2, −2). For node u = 20, π u is the point (8, 0) and the copies of π u are enclosed by a circle. Rα u = ∅ β + + (since the points in Rα u are outside Z × Z ) and Ru = { (8, 0), (6, 2), (4, 4), (2, 6), (0, 8) }.. The purpose of this section is to look for an optimal copy of π u for each node u ∈ { 0, 1, . . . , N − 1 } of a given MCRN. We denote the optimal copy of π u by π ∗u . Clearly, Δ(π ∗u ) ≤ Δ(π) for all π ∈ Πu . Although there are infinite number of copies of π u in the β two-dimensional integer lattice, in fact, we only need to consider those copies in Rα u and Ru. 31.

(57) CHAPTER 3. THE MINIMUM DISTANCE DIAGRAMS. 3.2. FINDING AN OPTIMAL COPY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . β Figure 3.5: The illustrations of πu , copies of π u , Rα u and Ru for u = 20.. since a copy π ∈ Πu \. . β Rα u ∪ Ru. . is either outside Z+ × Z+ or, by Lemma 3.1.2(iii), has. a larger distance than that of π u . + − Each π u is associated with two points π + u and π u defined as follows: If πu ∈ Γ , let β + − − + − + and π − π+ u and π u denote the point in Ru such that π u ∈ Γ , π u ∈ Γ u = π u + β. α + − − + − Similarly, If πu ∈ Γ− , let π + u and π u denote the point in Ru such that π u ∈ Γ , π u ∈ Γ − and π + u = π u + α; see Fig. 3.6 for illustrations. Take Fig. 3.5 for an example. Suppose − π u = (8, 0) ∈ Γ+ , then π + u and π u are the point (6, 2) and (4, 4), respectively; suppose + + − π u = (0, 1) ∈ Γ− , then π − u = (0, 1) and π u = (12, 0). Note that for π u , its π u or π u may. not exist. For example, suppose π u = (1, 0) ∈ Γ+ in Fig. 3.5, then π u + β = (−1, 2) which is outside Z+ × Z+ . In this case, we have Δ(π u + β) = ∞. The following lemma tells that − for each π u , π ∗u can be found by only considering π + u and π u . − Lemma 3.2.1. Δ(π ∗u ) = min { Δ(π + u ), Δ(π u ) }.. Proof. Suppose π u ∈ Γ+ . Since π u − α is outside the first quadrant and by Lemma 3.1.2(i), Δ(π u ) ≤ Δ(π u + α) ≤ Δ(π u + 2α) ≤ · · · , we only need to consider points in Rβu . By + + − Lemma 3.1.2(i), Δ(π + u ) ≤ Δ(π u − β) ≤ Δ(π u − 2β) ≤ · · · . By Lemma 3.1.2(ii), Δ(π u ) ≤ − ∗ − − Δ(π − u + β) ≤ Δ(π u + 2β) ≤ · · · . Hence Δ(π u ) = min { Δ(π u ), Δ(π u ) }. The case of. π u ∈ Γ− is similar to prove. 32.

(58) CHAPTER 3. THE MINIMUM DISTANCE DIAGRAMS . . . . . 3.2. FINDING AN OPTIMAL COPY. . . . . . . . . . . . . Figure 3.6: Two possible ways to find an optimal copy of π u . The left figure is for the πu ∈ Γ+ case; the right figure is for the πu ∈ Γ− case.. By using the lengths (2, h, 2p, n) in a pseudoMDD, we partition Γ+ and Γ− as follows (see Fig. 3.7 for an illustration):. +. Γ =. ∞ . Γ+ i. −. and Γ =. i=0. ∞ . Γ− i ,. i=0. where (x, y) ∈ Z+ × Z+ | 0 ≤ x − 2y < 2h , Γ− (x, y) ∈ Z+ × Z+ | −2 ≤ x − 2y < 0 , 0 =. Γ+ 0 =. . (3.2.1). and for i ∈ Z, i ≥ 1 (x, y) ∈ Z+ × Z+ | 2h + (i − 1) · (2h + 2p) ≤ x − 2y < 2h + i · (2h + 2p) , + + Γ− = (x, y) ∈ Z × Z | −2 − i · (2 + 2n) ≤ x − 2y < −2 − (i − 1)(2 + 2n) . i Γ+ i =. . (3.2.2). According to the relative position of π u in Γ+ or Γ− , we can find π ∗u by using the following three lemmas (Lemmas 3.2.2, 3.2.3 and 3.2.4). For convenience, the equal sign followed π ∗u means “can be chosen as”. − ∗ Lemma 3.2.2. If π u belongs to Γ+ 0 or Γ0 , then π u = π u . − Proof. Let π u = (x, y). If π u ∈ Γ+ 0 , then π u + β = (x − 2p, y + h) ∈ Γ . This implies. 33.

(59) CHAPTER 3. THE MINIMUM DISTANCE DIAGRAMS. . . . 3.2. FINDING AN OPTIMAL COPY. . . . . . . . . . . . . . . . . Figure 3.7: The partitions of Z+ × Z+ . − + − π+ u = π u and π u = π u + β. By (3.1.3), Δ(π u ) = x, Δ(π u ) = 2y + 2h − parity(x) and − − ∗ Δ(π + u ) ≤ Δ(π u ) holds. By Lemma 3.2.1, π u = π u . The case of π u ∈ Γ0 is similar and we. omit the proof. ∗ Lemma 3.2.3. If π u belongs to Γ+ i for some positive integer i, then π u = π u + i · β.. . Proof. Let π u = (x, y), then π u + (i − 1) · β = x − 2(i − 1)p, y + (i − 1)h , π u + i · β = . (x − 2ip, y + ih) and π u + (i + 1) · β = x − 2(i + 1)p, y + (i + 1)h . It is not difficult to check that points π u + (i − 1) · β, π u + i · β and π u + (i + 1) · β are inside Z+ × Z+ . Since + π u ∈ Γ+ i , we clearly have π u ∈ Γ .. Now we further partition Γ+ i into two smaller parts (possibly empty):. L Γi. +. R Γi. +. (x, y) ∈ Z+ × Z+ | 2h + (i − 1) · (2h + 2p) ≤ x − 2y < 2h + (i − 1) · (2h + 2p) + 2p , = (x, y) ∈ Z+ × Z+ | 2h + (i − 1) · (2h + 2p) + 2p ≤ x − 2y < 2h + i · (2h + 2p). =. . Suppose π u ∈. L Γi. +. , then π u +(i−1)·β ∈ Γ+ and π u +i·β ∈ Γ− . Hence π + u = π u +(i−. + − 1)·β and π − u = π u +i·β. By (3.1.3), Δ(π u ) = x−2(i−1)p, Δ(π u ) = 2y+2ih−parity(x) and. 34.