兩種連接網路：三環式網路及 Log2(N, m, p) 交換網路之研究

全文

(1)國立交通大學應用數學系博士論文. 兩種連接網路：三環式網路及 Log2(N, m, p) 交換網路之研究 On Two Interconnection Networks: Triple-loop Networks and Switching Log2(N, m, p) Networks. 研究生：林琲琪指導教授：黃光明教授. 中華民國九十四年六月.

(2) 兩種連接網路：三環式網路及 Log2(N, m, p) 交換網路之研究 On Two Interconnection Networks: Triple-loop Networks and Switching Log2(N, m, p) Networks 研究生：林琲琪. Student：Bey-Chi Lin. 指導教授：黃光明教授. Advisor：Professor Frank K. Hwang. 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 2005 Hsinchu, Taiwan, Republic of China 中華民國九十四年六月.

(3) 兩種連接網路：三環式網路及 Log2(N, m, p) 交換網路之研究學生：林琲琪. 指導教授：黃光明. 國立交通大學應用數學系. 摘. 要. 本篇論文主要討論二種類型的網路：一是電腦網路(computer networks)；另一是應用在通訊上的交換網路(switching networks)。對於前者，我們主要針對三環式網路做研究；對於後者，我們則針對Log2(N, m, p)網路做研究。首先，我們先介紹三環式網路：一個記為ML(N; s1, …, sl)的多環式網路，若以一具N個點(0, 1, …, N − 1)，lN 條邊的有向圖來表示，其有向邊的連接方式為：i → i + s1, i → i + s2, …, i → i + sl, (mod N), i = 0,1, …, N −1。其中s1, …, sl這l個整數被稱做是多環式網路的“步＂。當l的值確定時，我們也可稱此多環式網路為l-環式網路。尤其當l = 2 時，此多環式網路又被稱為雙環式網路，記為DL(N; s1, s2)；當l = 3 時，此多環式網路則又被稱為三環式網路，記為TL(N; s1, s2, s3)。近期，雖然有許多的三環式網路已被提出，並且它們的效能也被研究，但是真實存在的此類網路，就數量來說仍是非常的稀少，因此，在此篇論文中，我們將會把已有的三環式網路推廣以增加其類型的數量，同時，我們也會提出一個啟發式(heuristic)的方式來最佳化我們所提出的三環式網路所需的參數，以增進其效能。在本篇論文中，我們將研究三個特定三環式網路的 3-直徑(3-diameter)，其中，我們用建構的方式來做此研究，亦即，我們在任意二點間建立三條點互斥 (node-disjoint)，且長度不超過直徑加 2 的最短路徑。 i.

(4) 接下來，我們將介紹Logd(N, m, p)網路： Lea 和 Shyy [32] 首先提出含有N = 2n條進線(inputs)和出線(outputs)的Log2(N, m, p)網路，其建構方式為將p個BY-1(n, m) 的複製網路垂直堆疊在某一進線層 (input stage)和出線層(output stage)中，其中 0 ≤ m ≤ n−1，並且每一進(出)線層含有 N個 1 × p (或 p × 1)的閂(crossbar)。之後，Hwang [24]將Log2(N, m, p)網路中，每個 2 × 2 的閂由d × d的閂取代，於是把它推廣為Logd(N, m, p)網路。一個網路若目前送來的訊息，必須在所有的訊息皆依某一給定的演算法連接傳送，才可以保證被連接傳送時，這種不阻塞的程度稱為 wide-sense nonblocking。網路的交流量被分類為點對點(point-to-point)，例如傳統電話連接；另一為廣播式 (broadcast)，亦即點對所有(one to all)。假如每一訊息的最多接收者有所限制，那麼廣播式亦稱為多重傳播(multicast)，亦即點對多(one to many)；如果接收者被限制為 f，則稱為 f-cast。 Tscha和Lee [44] 對於多重傳播(multicast) Log2(N, 0, p)網路提出了fixed-size window演算法，並表明期望可以將此演算法推廣至Log2(N, m, p)網路。之後， Kabacinski 和 Danilewicz [29] 將 fixed-size window 演算法推廣至 variable size window演算法。在這篇論文中，我們更進一步地把variable size window演算法的結果，由Log2(N, 0, p)網路推廣至Log2(N, m, p)網路。. ii.

(5) On Two Interconnection Networks: Triple-loop Networks and Switching Log2(N, m, p) Networks Student: Bey-Chi Lin. Advisor: Frank K. Hwang. Department of Applied Mathematics National Chiao Tung University HsinChu 30050, Taiwan, Republic of China. Abstract This thesis is divided into two types of networks: computer networks and switching networks used in communication. In particular, we will study a class of computer networks called the triple-loop network, and a class of switching networks called Log2(N, m, p). We first introduce the former. A multi-loop network, denoted by ML(N; s1, …, sl), can be represented by a digraph on N nodes, 0, 1, …, N − 1 and lN links of l types: i → i + s1, i → i + s2, …, i → i + sl, (mod N), i = 0,1, …, N −1. The integers s1, …, sl are called the steps of the multi-loop network. When l is specified, we can also call it an l-loop network. In particular, when l = 2, the multi-loop network is usually called the double-loop network and is denoted by DL(N; s1, s2). When l = 3, the multi-loop network is usually called the triple-loop network and is denoted by TL(N; s1, s2, s3). Several triple-loop networks have been recently proposed and their efficiency studied. However, the number of cases for which one of these networks exist is sparse. In this thesis, we extend these networks to larger classes to enhance their realizability. We also give a heuristic method to optimize the network parameters to increase their efficiency. In this thesis, we study the k-diameters of three specific triple-loop networks. In particular, we construct three node-disjoint shortest paths no longer than the diameter iii.

(6) plus 2 for any pair of nodes. Next we introduce the Log2(N, m, p) network. Lea and Shyy [32] first proposed the Log2(N, m, p) network with N = 2n inputs (outputs), which consists of a vertical stacking of p copies of BY-1(n, m), 0 ≤ m ≤ n−1, sandwiched between and connected to an input stage and an output stage, each with N 1 × p (or p × 1) crossbars. Later, Hwang [24] extended the Log2(N, m, p) network to Logd(N, m, p) network by replacing the 2 × 2 crossbars with d × d crossbars. A network is wide-sense nonblocking (WSNB) if the connection of the current request is assured only when all connections are routed according to a given algorithm. Traffic can be classified as point-to-point, like 2-party phone calls, or broadcast, which is one to all. If there is a restriction on the maximum number of receivers per request, then broadcast is called multicast (one to many), or f-cast, if that number is specified to be f. Tscha and Lee [44] proposed a fixed-size window algorithm for the multicast Log2(N, 0, p) network and expressed a desire to see its extension to the Log2(N, m, p) network. Later, Kabacinski and Danilewicz [29] generalized the fixed-size window to variable size to improve the results. In this thesis, we further extend the variable-size results from the Log2(N, 0, p) network to Log2(N, m, p).. iv.

(7) 誌. 謝. 這本論文對我來說不僅僅只是呈現研究上的些許結果，更是蘊涵著這五年來，自己用心走過的每一個足跡。論文的完成，過程中有著許多人的支持和鼓勵，心裡著實有許多的感謝對每一個曾經參與過我生命的人，雖然只能用隻字片語去表達，但希望當中滿滿的感激你們都能明白。首先，最要感謝我的指導教授－黃光明老師，是老師深厚的學術實力，帶領我接觸各樣的領域，讓我得以站在他的肩膀上，看到更寛廣的世界，卻也同時深深地體會學海無涯；是老師的循循善誘、提攜後進，讓我發現如此渺小的自己，原來也可以解決國際性的問題，體驗過挑戰未知的興奮，才逐漸明白推動一個研究者不斷前進的動力來源；是老師對學術的認真投入，讓我看到一位學者的風範，當老師用教育的傳承來回饋這片孕育他成長的土地時，我更明白在研究這條路上，自己可以堅持些什麼，以及可以回報這個社會些什麼，雖然未來會有很多的困難等著我去面對，但老師亦教導我，看到問題，等同於看到了希望！即使最初是很意外地踏入組合數學這個領域，但很謝謝陳秋媛、傅恆霖、黃大原、張鎮華、翁志文等老師們的教導，和許多生活上的幫忙，讓我這個不是學習表現最好的學生，經過這些年從您們身上的學習，也能覺得是滿載而歸。謝謝林文鐽學長在我剛接觸連接網路這塊陌生的領域時，給予很多的指導和幫忙，讓我有勇氣踏入這個領域探索更多。謝謝君逸和飛黃這二位天才學弟，還有李珠矽老師和惠蘭共同在研究上的相互砌磋，你們的存在，讓我知道走在這條路上，我並不孤單，甚至可以享受研究的樂趣。謝謝莊慧如老師的舞蹈教學，讓我在學校生活中，找到另一片自己的小天地，得以悠遊其中，我承認對這每個禮拜僅僅一小時練舞時間的重視，不亞於對系上的必修課程，甚至對於有機會站上耗資數十萬元的華麗舞台表演，享受當藝人的快感，更是畢生難忘！並深感榮幸！謝謝阿珮、Alice 這二個影響我至深的好朋友，在每一個階段，用愛澆灌著我，不間斷地支持、鼓勵和教誨，陪伴我走過許多的風雨。如果今天我學得一絲一亳如何去愛身邊的人，明白絲毫愛的真諦，都是她們的功勞！共同經歷過的每一個過程，都讓我刻骨銘心！謝謝幼婷、豆豆、秀琴一路來的陪伴，每一份關係從衝突到溝通，到現在的 v.

(8) 彼此欣賞，這些漫長的過程或許當下很煎熬，但我很珍惜每一份浴火重生的友誼。謝謝陳依、雅卿、美玉、昇達、柏盛、人星、世學、Alan 這群一同在校園團契奮鬥的戰友，在你們身上我得到的遠比我能給予的多。謝謝建廷和我曾經一同經營男女朋友關係，即使關係回到原點，但這半年的點滴，依舊讓我感激在心。謝謝新竹教會這個大家庭裡的每一個弟兄姊妹。在這裡，我學到最多的是認識神，以及對生命的堅持；在這裡，我看到愛如何真真實實地活在我們當中。新竹，是我屬靈的故鄉，是我新生命的起點！謝謝同在一研究室的吟衡，一起經歷準備資格考的熬煉，體驗十年寒窗的艱辛，一同歡呼通過資格考的狂喜。在這間小小的研究室裡，有我們的交心、彼此打氣、天南地北的聊天，當然也有互吐苦水的真實片段。SA331 是我的另外一個家，裡頭有著這些年在新竹滿滿的回憶。榮譽室友－班榮超學長的加入，更讓這間研究室增色不少！謝謝過去在台中師院的同窗好友－玉仙和慧如，即使大學畢業後我們各奔東西，但是你們不間斷地為我打氣，所給的支持，是我最感窩心的。謝謝曾經在女二舍的歷屆室友：秀貞、姿瑩、曾翊、淑娟、敏慈、彥君，雖然只是短暫的交集，卻豐富了我的生活許多，不論是期待聽趣聞的心情、或是躺在床上閉著眼睛聊到三更半夜的片段、或是生平第一次的寢聚，都讓我對女二這棟冰冷的建築物，添進了許多的情感。最後，謝謝我的家人，他們總是在背後默默地支持我，尤其是我的父母，含辛茹苦地建立這個家，在我的成長過程中，給我絕對的信任，全力的支持以及廣大的自由度去做任何的嚐試。我知道當我可以無後顧之憂地在這個世界闖蕩時，是因為他們願意放手讓我去飛；我知道當我失意、遭遇挫折，可以隨時回家好好休息時，是因為有他們對我無條件的關愛！何其有幸，伴隨我成長的老師和朋友很多很多，感謝的話，怎麼說都說不完，紙短情長，僅將這本論文獻給我所摯愛的你們！結束了二十多年來的學生生涯，畢業，是我人生另一個階段的開始。期許自己不斷抱持著被磨練的態度去學習，懷抱著回饋這個社會的使命去努力，讓未來的每一步都能走得踏實。. vi.

(9) Contents i. 摘要. iii. Abstract. v. 致謝 Contents. vii. List of Figures. viii. List of Tables. x. Chapter 1 Introduction. 1. 1.1 Motivation ……………………………………………………………. 1. 1.2 Overview of the thesis ……………………………………………….. 4. Chapter 2 Preliminary and Classical Results of Multi-loop Networks. 5. 2.1 Architecture …...…………………………………………………….... 5. 2.2 Minimum Distance Diagram ………………………………………... 8. 2.3 Existence Conditions ……………………………………………….... 13. Chapter 3 Further Research on Triple-loop Networks. 15. 3.1 Generalizing and Fine Tuning H1 and H2 …………..…..…………... 15. 3.2 Wide-Diameter of H0 ……………….………………………………... 22. 3.3 Wide-Diameter of H1′ …………….………………………………….. 30. 3.4 Wide-Diameter of H2′ ……………….……………………………….. 41. Chapter 4 WSNB on Log2(N, m, p) Networks. 48. 4.1 Architecture …….……………………………………………………. 50. 4.2 Blockingness …………………………….………..………………….. 53. 4.3 Classical Multicast WSNB Results ……..………..…………………. 54. 4.4 WSNB Log2(N, m, p) ..……………………………………………. …. 57. 4.5 Optimization …………………………………………………………. 65. Chapter 5 Conclusions. 69. Reference. 70. vii.

(10) List of Figures 2.1.1. Single Loop Network ……………………………………………………. 5. 2.1.2 2.2.1. Distributed Double Loop Computer Network-DDLCN ………………… An MDD(0) of DL(16; 1, 7) ……………………………………………... 7 8. 2.2.2. An L-shape ………………………………………………………………. 8. 2.2.3. H0(l, m, n) ………………..………………………………………………. 9. 2.2.4. MDD of TL(134; 33, 15, 19) ……………………………………………. 9. 2.2.5. H1(h, m, n) ………………………….……………………………………. 9. 2.2.6. MDD of TL(2277; 12, −250, 51) ……………………………………….... 9. 2.2.7. H2(l, m, n) ……………………………………...………………………..... 10. 2.2.8. MDD of TL(4097; −59, −110, 256) ……………………………………... 10. 2.2.9. L-shape tessellates the plane …………………………………………….. 10. 2.2.10 Generical 3D tessellation of H0 ………………………………………….. 10. 3.1.1. H1′, H2′ ………………………………………………………………….... 17. 3.2.1. H0(0) and H0*(0) …………………………………………………………. 23. 3.2.2. Dimension routing for v1 > 0, v2 > 0, v3 > 0 …………………………….... 24. 3.2.3. (a) and (b) are H0(0) and H0(26), respectively, for l − m − n = 1, where N = 31, s1 = 6, s2 = −1, s3 = −5, l = 4, m = 2, n = 1 ……………………….... 25. 3.2.4. H0(7; 2, 1, 4) with v = 2, where u = 5, w = 1 …………………………….. 29. 3.3.1. H1′(0) and its copies …………………………………………………….... 31. 3.4.1. H2′(0) and its copies …………………………………………………….... 41. 4.1.1. Some self-routing networks …………………………………………….... 49. 4.1.2. Decomposition of BY−1(4, 2) …………………………………………..... 50. 4.1.3. Log2(8, 1, 3) …………………………………………………………….... 51. 4.2.1. A channel graph of BY−1(n, m) ………………………………………….. 53. −1. 4.3.1. A 2-window of BY (4, 2) ……………………………………………….. 4.4.1. Input 4 generates a 3-intersecting connection (4, 4) to (a) a 1-cast request (0, 0) and (b) a 2-cast request (0, {0, 8}) ……………………..…………... 4.4.2. Assume θ = 2 and (0, 0) is the request. r = 1 in the first output crossbar and connection (6, 1) blocks 1/2 copy, while r = 0 in the third output crossbar and connections (4, 4) and (5, 5) each blocks 1/4 copy. Dotted lines indicate channel graph between the first input and the first output viii. 55 59.

(11) crossbar …………………………............................................................. 4.4.3. 60. Connection (1, 8) blocks 1/2 copy if counted from the input side, but only 1/4 copy from the output side. Dotted lines indicate channel graph between the first input and the first output crossbar ……………………... ix. 63.

(12) List of Tables 4.4.3 Best choice of θ and corresponding value of p for m = 2 and some n….... x. 67.

(13) Chapter 1 1.1. Introduction. Motivation This thesis is divided into two types of networks: computer networks and. switching networks used in communication. In particular, we will study a class of computer networks called the triple-loop network, and a class of switching networks called Log2(N, m, p). We first introduce the former. A fundamental limitation of high-performance computer systems is the low rate at which data can be accessed and restored in the high-speed memory. To overcome this limitation, it is current practice to increase the parallelism of operation of the high-speed memory by incorporating several independent memory modules into the memory system. In [45], Stone describes a particular organization of a multimodule memory, designed to facilitate parallel block transfers. A device called the memory circulator is utilized. It consists of a bank of interconnected register, one for each memory, and control circuitry. Each register is connected to l other registers, and the connection pattern has cyclical symmetry. A pattern is completely determined by the selection of l different links. The problem is to select a set of links that will minimize the maximum and/or average number of register-to-register transfers required to achieve an arbitrary circulation. One can assume that one of the l links always connects the original register to an adjacent register. (See [41].) A multi-loop network, denoted by ML(N; s1, …, sl), can be represented by a digraph on N nodes, 0, 1, …, N − 1 and lN links of l types: i → i + s1, i → i + s2, …, i → i + sl, (mod N), i = 0,1, …, N −1. The integers s1, …, sl are called the steps of the multi-loop network. When l is specified, we can also call it an l-loop network. In particular, when l = 2, the network is usually called the double-loop network. When l = 3, the multi-loop network is usually called the triple-loop network. The double-loop network has been extensively studied in the literature (see [25] for a recent survey) as an interconnecting network for either processors 1.

(14) or memories in parallel computing [20], or as a local area computer networks [38], or as a large area communication network like SONET [39]. It is known that if gcd(N, s1, …, sl) = 1, then an l-loop network is l-connected, hence (l − 1)-fault tolerant, has relatively short diameter and other desirable properties (to be described in chapter 2). Several triple-loop networks have been recently proposed and their efficiency studied. However, they exist only under very restrictive conditions on network parameters. In this thesis, we extend these networks to larger classes to enhance their realizability. We also give a heuristic method to optimize the network parameters to increase their efficiency. Traditionally, connectivity and diameter were studied separately. Then various approaches have been proposed to study these two parameters together. One such approach led to the notion of k-diameter which was formalized and popularized in Hsu [21] and Hsu and Luczak [22]. The k-diameter of a digraph is the minimum length l such that there exist k node-disjoint paths no longer than l. In this thesis, we will study the k-diameters of these networks. In particular, we construct three node-disjoint shortest paths no longer than the diameter plus 2 for any pair of nodes. Next we introduce the Log2(N, m, p) network. In an s-stage network, crossbars are lined up into s columns, each called a stage. Switching networks composed of log2N stages are of great interest in both high-speed electronics and photonic switching. Define the states of a network as the set of all possible routings of all legitimate frames, legitimate means the load generated by each input and output terminal does not exceed its capacity; a frame means all requests are in a given session. A set of requests is routable if there exists a set of link-disjoint paths connecting the requests. A state is blocking if there exists a legitimate new request not routable in the current state, and is nonblocking otherwise. To obtain nonblocking characteristics, two methods have been proposed: horizontal cascading (HC) and vertical stacking (VS) [5, 31]. The HC method results in greater number of stages between each inlet–outlet 2.

(15) pair. More stages in a switching network induce greater signal attenuation in the case of photonic switching or greater delay in the case of electronic switching. For the VS method the question is how many copies of Log2N switching networks are to be connected in parallel to obtain nonblocking operation of the whole switching network. The number of copies needed in the case of space-division switching networks and point-to-point connections was given in [32, 40]. Lea and Shyy [32] first proposed the Log2(N, m, p) network (when m = 0, we denote it as a multi-Log2N network) with N = 2n inputs (outputs), which consists of a vertical stacking of p copies of BY-1(n, m), 0 ≤ m ≤ n−1, sandwiched between and connected to an input stage and an output stage, each with N 1 × p (or p × 1) crossbars. Apart from point-to-point connections, many services, for instance video-conference, video-distribution, multi-party communications, etc., will require connections from one input to many or even all outputs [35, 33, 23]. Nonblocking multicast multi-Log2N networks were first considered in [43]. Later, this result was improved in [44], where nonblocking operation of multi-Log2N switching networks was given, provided a special control algorithm, called a window algorithm, is used. Tscha and Lee [44] stated in conclusion that whether their approach could be extended to Log2(N, m, p) (to be defined in chapter 4) was unclear. Kabacinski and Danilewicz [29] generalized the window algorithm from fixed size to variable sizes. Danilewicz and Kabacinski [13, 14] also made an attempt to extend their results to Log2(N, m, p), but encountered some difficulties. In this thesis, we will give such an extension for the variable window-size algorithm by adopting a channel graph blockage analysis first used by Shyy and Lea [40] on a single-cast network. We also determine the optimal window size for given m, and then compare the performance among different m.. 3.

(16) 1.2. Overview of the thesis In chapter 2, we will give the architecture of multi-loop networks. Some most. studied topics of multi-loop networks: minimum distance diagram (MDD) and the tesselatibility of MDD shapes are also introduced. Later, we present some known classical results of existence conditions between L-shape (hyper-L) tile and double-loop (triple-loop) networks, respectively. In chapter 3, we first generalize the three classes of triple-loop networks studied in the literature to larger classes. Later, we construct the wide-diameters for each of these enlarged classes. In chapter 4, we first give the architecture of Logd(N, m, p) networks. Then the blockingness and channel graph are introduced. Next, we present the classical WSNB results for multicast Logd(N, m, p) networks. Later, we provide a new result using window algorithm which was first proposed by Tscha and Lee [44]. At last, we determine the optimal window size and the optimal number of extra stages.. 4.

(17) Chapter 2 Preliminaries and Classical Results of Multi-loop Networks 2.1. Architecture Multi-loop networks were first proposed by Wong and Coppersmith [47] for. organizing multimodule memory services. Fiol et al. [20] slightly extended its definition in their study of the data alignment problem in SIMD processors. Nowadays, it is used for both local area computer networks [36, 38] and large area communication networks like SONET [15, 39]. Multi-loop network architectures present an attractive topology for local networks [18, 36, 37], since they require simple control software and interfaces. They permit effective operation at higher data rates and over larger distances than broadcast busses since they do not suffer from carrier sense limitations. In a unidirectional single loop network with N nodes, (see Fig. 2.1.1) the host computers are connected to the networks via loop interface hardware. Each node i is connected to node i + 1 (mod N) to form a completer loop, and messages are passed from node to node along unidirectional links. There are no routing decisions to be made and there is thus no need for central control. A node simply transmits its message to the next node in the loop, and the message circulates around the network until it reaches the destination node. The interface hardware must be able to identify messages intended for its host.. Fig. 2.1.1 Single Loop Network.. 5.

(18) An important issue in loop networks is the control mechanism used for message transmission. This mechanism can be centralized or distributed. A distributed control mechanism seems to be more advantageous in terms of performance and reliability as there is no single central node responsible for networks operation. Newhall loop [18] and Pierce loop [37] are two access control mechanisms in common use for loop networks, and the delay insertion register mechanism [36, 45, 46] combines the best features of the first two schemes. There are several important issues to be studied in the design and analysis of loop networks architectures. The important characteristics of loop networks include the maximum delay for any message, the average delay, reliability, node processing overhead, and the saturation throughput. These performance measures are all interdependent and are related to the network topology. In particular, the three performance measures: reliability, delay, and nodal processing limitation, are affected by network size. There are two approaches to improve reliability. One is to bring all the interfaces to a central point. The other is to introduce link redundancy, i.e. there exist several alternate paths for communication between a pair of nodes. Raghavendra and Silvester [38] studied various loop networks architectures. Here, we take two architectures for 2-loop and 3-loop networks, respectively, for example. Distributed Double Loop Computer Network (DDLCN) was proposed by Liu [36], and is the topology of the SONET ring (see Fig. 2.1.2). In this network with N nodes, each node i is connected to i + 1 (mod N) and i – 1 (mod N) nodes. With these redundant links, the network can sustain single interface failures.. 6.

(19) Fig. 2.1.2 Distributed Double Loop Computer Network-DDLCN.. In terms of mathematical form, a multi-loop network, denoted by ML(N; s1, …, sl), can be represented by a digraph on N nodes, 0, 1, …, N − 1 and lN links of l types: i → i + s1, i → i + s2, …, i → i + sl, (mod N), i = 0,1, …, N −1. The integers s1, …, sl are called the steps of the multi-loop network. When l is specified, we can also call it an l-loop network. In particular, when l = 2, the multi-loop network is usually called the double-loop network and is denoted by DL(N; s1, s2). Thus, DDLCN is denoted by DL(N; 1, N – 1). When l = 3, the multi-loop network is usually called the triple-loop network and is denoted by TL(N; s1, s2, s3).. 7.

(20) 2.2. Minimum Distance Diagram A minimum distance diagram MDD(v) for DL(N; s1, s2) is a two-dimensional. array which gives the shortest paths from node v to every other node. Since DL(N; s1, s2) is node-symmetric, we need only study MDD(0), or simply, MDD. Let node 0 occupies cell (0, 0) in an MDD. Then node v occupies cell (i, j) (i is the column index and j the row index) if and only if is1 + js2 ≡ v (mod N) and i + j is the minimum among all (i′, j′) satisfying the congruence, equality is broken by minimizing i. Namely, a shortest path from 0 to v is through taking i s1-steps and j s2-steps (in any order). Fig. 2.2.1 gives the MDD of DL(16; 1, 7). Wong and Coppersmith [47] gave an O(N) time construction of MDD by sequentially adding nodes to the diagram which can be reached from node 0 in k steps for k = 0, 1, ..., until every node appears exactly once. They also proved that an MDD for a double-loop network is an L-shape which includes the degenerate form of a rectangle. It can be characterized by six parameters l, h, m, n, p, q (4 of them independent) (see Fig. 2.2.2). Thus, we denote it by L(l, h, n, p). This L-shape plays a crucial role in proving many desirable properties for DL(N; s1, s2). m 12. 13. 5. 6. 14. 15. 7. 8. 9. 10. 11. 0. 1. 2. 3. 4. n h. p q l. Fig. 2.2.1 An MDD(0) of DL(16; 1, 7).. Fig. 2.2.2 An L-shape.. The MDD for a triple-loop network is a three-dimensional array with each step in the xi-axis signifying an si-step. Unfortunately, the MDD does not have a uniform nice shape like the L-shape (see Fig. 2.2.4, Fig. 2.2.6, Fig. 2.2.8) and this fact has 8.

(21) hampered the study of triple-loop networks. Aguilό et al. [3] overcame this difficulty by skipping the triple-loop network and going directly to a nice three-dimensional shape which they called hyper-L tile. Later, Aguilo-Gost [4] identified two other shapes which she named H1 and H2 (see Fig. 2.2.5 and Fig. 2.2.7). For convenience, we use H0 (see Fig. 2.2.3) to denote the hyper-L shape. Note that H0 is characterized by three parameters l, m, n, and is highly structured and symmetrical, where l, m, n are integers, m ≥ n ≥ 0 and l > m + n. H1 and H2 are characterized by three parameters {h, m, n} and {l, m, n}, respectively, where l, h, m, n are positive integers. Thus, we also use H0(l, m, n), H1(h, m, n) and H2(l, m, n) to denote H0, H1 and H2, respectively.. Fig. 2.2.3 H0(l, m, n).. Fig. 2.2.5 H1(h, m, n).. Fig. 2.2.4 MDD of TL(134; 33, 15, 19).. Fig. 2.2.6 MDD of TL(2277; 12, −250, 51).. 9.

(22) s. Fig. 2.2.7 H2(l, m, n).. Fig. 2.2.8 MDD of TL(4097; −59, −110, 256).. Besides, suppose that ℜd is divided into unit hypercubes and a shape is a connected set of hypercubes. A shape is said to tessellate ℜd if any number of it can be fitted together with neither gaps nor overlapping (rotation or reflection not allowed). Fiol et al. [20] observed that an L-shape always tessellates the plane (see Fig. 2.2.9) regardless of the L-shape is degenerate or not. Aguliό-Gost [4] showed the 3D tessellation of hyper-L (see Fig. 2.2.10).. Fig. 2.2.9 L-shape tessellates the plane.. Fig. 2.2.10 Generical 3D tessellation of H0. 10.

(23) Chen et al. [11] gave a sufficient condition for a shape to tessellate. The following result follows as a special case. Theorem 2.2.1 Every MDD tessellates ℜd. For H0(l, m, n), Aguilό et al. [3] used the tesselatibility of the MDD shape to yield an 3 × 3 matrix which characterizes the interrelation among the locations of the same node (say, node 0) in several adjacent copies of the MDD. We use M0 to denote this characterizing matrix.. ⎛ l −m −n ⎞ ⎜ ⎟ M 0 = ⎜ −n l −m ⎟ ⎜ −m −n l ⎟⎠ ⎝ Namely, each row vector represents the steps to go from one node 0 to another. For example, the first row represents that after we use l s1-steps, −m s2-steps (− denotes the opposite direction) and −n s3-steps, we can go from one node 0 to another. By the same way, we define the characterizing matrices of H1(h, m, n) and H2(l, m, n) as follows:. n 2h ⎞ l+n ⎞ ⎛ n ⎛ 2l + n l + m ⎜ ⎟ ⎜ ⎟ M1 = ⎜ − m n + m h −2l l ⎟ , M 2 = ⎜ 3l + n ⎟. ⎜ −m −m h + m − n ⎟ ⎜ −2l − n l l + m + 2n ⎟⎠ ⎝ ⎠ ⎝. The diameter of a triple-loop network is the maximum distance among pairs of nodes in the network. Let N(D) denote the maximum number of nodes in a triple-loop network with diameter D. Hyper-L tiles were proven to be an effective tool to obtain lower bounds for N(D). In particular, Aguilό et al. [3] used the H0 to obtain N(D) ≥. 2 27. Aguliό-Gost [4] used the H1 to obtain 11. ( D + 3). 3. ≈ 0.074D3..

(24) N(D) ≥. D 3 ≈ 0.075D3,. 1485 273. and used the H2 to obtain N(D) ≥. 860 223. D 3 ≈ 0.08D3.. For convenience of comparison, the efficiency of a triple-loop network TL is defined [4] as E ( TL ) =. 12. N . D3.

(25) 2.3. Existence Conditions. Unfortunately, not every L-shape (hyper-L) tile has a double-loop (triple-loop) network realizing it; see [10] for examples. Thus it becomes important to determine when a L-shape (hyper-L) tile has a double-loop (triple-loop) network realizing it. Fiol et al. [20] (also see Chen and Hwang [9]) proved Theorem 2.3.1 Necessary and sufficient conditions that L(l, h, n, p) can be. implemented is that l > n, h ≥ p and gcd(l, h, n, p) = 1. By noting the locations of cells containing node 0 (as specified by M), they obtained the following equations: ls1 – ns2 ≡ 0 (mod N), – ps1 + hs2 ≡ 0 (mod N).. (2.3.1). Note that (2.3.1) can also be written as ⎛ s1 ⎞ ⎛ h n ⎞ ⎛ α ⎞ ⎛ l − n ⎞ ⎛ s1 ⎞ ⎛α ⎞ ⎜ ⎟ ⎜ ⎟ = N ⎜ ⎟ , or ⎜ ⎟ = ⎜ ⎟⎜ ⎟ ⎝ − p h ⎠ ⎝ s2 ⎠ ⎝β ⎠ ⎝ s2 ⎠ ⎝ p l ⎠ ⎝ β ⎠. for some integers α, β. Fiol et al. [2, 17] proposed the Smith normalization method to solve for s1 and s2. They proved: Theorem 2.3.2 There exists unimodular, integral 2 × 2 matrices L and R such that. ⎛ l −p⎞ ⎛1 0 ⎞ L⎜ ⎟R = S =⎜ ⎟ (the Smith normal form). ⎝ −n h ⎠ ⎝0 N ⎠ Furthermore, let ⎛w x⎞ L=⎜ ⎟ ⎝ y z⎠ Then DL(N, y, z) implements L(l, h, n, p) and (y, z) is unique up to isomorphism. The computation of L and R involves solving for q1, q2 in q1u – q2v = 1 for various pairs of (u, v).. 13.

(26) For general L(l, h, n, p), Chen and Hwang [9] gave the following method to find s1 and s2. For k = 0, 1, …, defines. s1k = h + kn,. s2k = p + kl.. Let Fk denote the set of prime factors of gcd ( s1k , s2k ) and F the set of prime factors of N. They used the sieve method in number theory to show the existence of a k such that f ∉ Fk for all f ∈ F. Then ( s1k , s2k ) is a solution of (2.3.1). For L(6, 4, 3, 2), we easily find the solution s1 = h = 4 and s2 = p =3. Next, we discuss the existence conditions for some triple-loop networks. A triple-loop network with a hyper-L shape is called a hyper-L triple-loop. Fiol [19] proposed two necessary conditions for the existence of an H0(l, m, n) triple-loop: (i) gcd (N, l, m, n) = 1, and (ii) gcd (2 × 2 minors of M0) = 1. Chen et al. [10] showed that (ii) implies (i) for H0 and gave a necessary and sufficient condition. Theorem 2.3.3 A necessary and sufficient condition for the existence of an H0(l, m,. n) triple-loop network is gcd(l2 – mn, m2 + ln, n2 + lm) = 1.. Furthermore, for a TL(N; s1, s2, s3) with H0(l, m, n) shape, if it satisfies the conditions of Theorem 2.3.3, then the solution of (s1, s2, s3) is (l2 − mn, m2 + ln, n2 + lm) unique up to the equivalence defined by a permutation of (s1, s2, s3) or a multiplication of (s1, s2, s3) by a scalar. Let M be a 3 × 3 integral matrix with |det(M)| = N > 0. Fiol [19] defined G(M) as the Cayley diagraph of the group Ζ3/MΖ3 with the generator set {e1, e2, e3}, where e1 = (1, 0, 0)T, e2 = (0, 1, 0)T, e3 = (0, 0, 1)T. Chen and Hung [8] used Cayley diagraph to derive the necessary and sufficient conditions for the existence of H1(h, m, n) and H2(l, m, n) triple-loops as follows. 14.

(27) Lemma 2.3.4 G(M) is isomorphic to a triple-loop network TL(N; s1, s2, s3) with ⎛ s1 ⎞ ⎜ ⎟ M ⎜ s2 ⎟ = 0 (mod N ) ⎜s ⎟ ⎝ 3⎠. if and only if gcd(all the 2 × 2 minors of M) = 1. Apply Lemma 2.3.4 to H1 and H2, they obtained Theorem 2.3.5 A necessary and sufficient condition for the existence of an H1(h, m,. n) triple-loop is gcd(m, n) = 1 and 3. m – n.. Theorem 2.3.6 A necessary and sufficient condition for the existence of an H2(l, m,. n) triple-loop is gcd(l, m, n) = 1.. 15.

(28) Chapter 3 3.1. Further Research on Triple-loop Networks. Generalizing and Fine Tuning H1 and H2 Aguilo, Fiol and Garcia [3] used the computer search to find some good MDDs. for l-loop networks. Of course, the computer search works only for very small N. Then they looked at those good MDDs and tried to identify their shapes to grow it to larger N but keeping the shape. The method of growing is to use the tesselatibility of the MDD shape to yield an l × l matrix M which characterizes the interrelations of the locations of the same node in several adjacent copies of the MDD. For a given shape S, we define F(S) as a family of all shapes obtained from S by varying the parameters of S. Such an approach encounters three problems. The first is that although the original shape is derived from a triple-loop network, there is no guarantee a member of F(S) also corresponds to a triple loop. Thus one has to check the existence of such a triple-loop. Necessary and sufficient conditions for existence were given in section 2.4 in principle. The second problem is that there are not many known good shapes to work with, and the existence of a given shape is sparse. The third problem is that there is no systematic way to optimize the parameters of a given shape. In this section, we [34] propose ways to alleviate problems 2 and 3. We will represent H1 and H2 each by a 6-parameter family, thus significantly enhancing the chance of finding H1 or H2 in the neighborhood of a given N. We also propose a method for sub-optimal selection of parameters. The price we pay is that the necessary and sufficiency condition for the existence of a corresponding triple-loop network becomes messy. We generalize H1 and H2 to H1′ and H2′ by allowing some line segments which have the same length to have different lengths. We mark the new parameters in Fig.. 16.

(29) 3.1.1. Note that all parameters of H1′ and H2′ are larger than or equal to 1. For H1′, m ≥ n and m′ ≥ n′.. H1′. H2′ Fig. 3.1.1 H1′, H2′.. It can be verified that H1′ tessellates R3 with 2h ⎞ n' ⎛ n ⎜ ⎟ M1 ' = ⎜ − m n '+ m ' h ⎟. ⎜ −m −m ' h + h ' ⎟ ⎝ ⎠. H1 is the special case of H1′ by setting m′ = m, n′ = n = h′. We apply the necessary and sufficient conditions given in [8] for the existence of a triple-loop network to H1′: gcd (determinants of the nine minors of M1′) = gcd ((n′ + 2m′)h + (n′ + m′)h′, (n′ + 2m′)h + n′h′, (n′ + 2m′)h, mh′, (n′ + 2m′)h + nh′, (n + 2m)h, (n′ + 2m′)m, nm′ − n′m, (n′ + m′)n + mn′) = gcd (m′h′, n′h′, (n′ + 2m′)h, mh′, nh′, (n + 2m)h, (n′ + 2m′)m, nm′ − n′m, (n′ + 2m′)n). (3.1.1). =1 ⇒ gcd (m′, n′, m, n) = 1,. (3.1.2) 17.

(30) gcd (h, h′, m, n) = 1,. (3.1.3). (3.1.1) is reduced to gcd (h′, (n′ + 2m′)h, (n + 2m)h, (n′ + 2m′)m, nm′ − n′m, (n′ + 2m′)n) = gcd (h′, n′ + 2m′, (n + 2m)h, nm′ − n′m) = 1 by (3.1.3). by (2) (3.1.4). The farthest nodes from the base node of H1′ must be at one of the circled node. Their distances are: d(A) = n + n′ + 3h + h′, d(B) = n + m + n′ + 2h + h′, d(C) = n + m + m′ + 2h, d(D) = m + n′ + m′ + 2h, d(E) = n + n′ + m′ + 2h + h′, d(F) = 2m + n + m′ + h, d(G) = m + 2m′ + n′ + h. Our heuristic method sets all these distances equal. Thus d(A) = d(B) ⇒ h = m, d(B) = d(C) ⇒ h′ = m′ − n′, d(C) = d(D) ⇒ n = n′, d(D) = d(E) ⇒ h′ = m − n. Summarizing, we have h = m = m′, n = n′ and h′ = m − n. , there are only two independent Therefore in the suboptimal setting H 1 parameters m and n, and the diameter is 4m + n. Note that for this suboptimal version, necessary and sufficient conditions for the existence of a corresponding triple-loop network is induced from (3.1.2), (3.1.3), (3.1.4) to gcd (m, n) = 1. is Efficiency of H 1 18.

(31) 3 2 3 ) = N = 4m + 6m n − n . E(H 1 3 D3 ( 4m + n ). Setting m = kn, then n can be canceled out and N 4k 3 + 6k 2 − 1 = . 3 D3 ( 4k + 1). (. ). 3 2 d ⎛ N ⎞ 12k ( k + 1) 4k + 6k − 1 ⋅12 − =0 ⎜ ⎟= 4 dk ⎝ D 3 ⎠ ( 4k + 1)3 ( 4k + 1). ⇒ k (k + 1)(4k + 1) = 4k 3 + 6k 2 − 1 ⇒ k0 =. 1+ 5 ≈ 1.5 2. Hence we choose n = 2 and m = 3 for integrality,. = 26 > 0.07580. E H 1 343. ( ). Setting n = 3 and m = 5 yields a slightly better efficiency 923 233 ≈ 0.07856 . Recall that the efficiency of H1 is 1485 273 ≈ 0.075 .. It can be verified that H2′ tessellates R3 with ⎛ 2l + n ' l '+ m ' m + 2n ⎞ ⎜ ⎟ M 2 ' = ⎜ 3l + n ' −2l ' m+n ⎟. ⎜ −2l − n ' 2m + 3n ⎟⎠ l' ⎝. H2 is the special case of H2′ by setting m′ = m, n′ = n, and l = l′ = m′ + n′. Again, we apply the necessary and sufficient conditions given in [8] for the existence of a triple-loop network to H2′: gcd (determinants of the nine minors of M2′) = gcd (−(8m′ + 11n′)l′ − (3m′ + 4n′)n, (2l′ + n)(3m′ + 5n′), m′l′ + 4n′l′ + nn′, −(5m′ + 7n′)l, −(n′ + m′)l − (2m′ + 3n′)m, (3m′ + 5n′)l + (n′ + m′)m, (n + l′)l, (n + 2l′)(m + 2l), 7ll′ + 3nl + 3ml′ + nm) = gcd ((8m′ + 11n′)l′ + (3m′ + 4n′)n, (2l′ + n)(3m′ + 5n′), m′l′ + 4n′l′ + nn′, (5m′ + 7n′)l, (n′ + m′)l + (2m′ + 3n′)m, (3m′ + 5n′)l + (n′ + m′)m, (n + l′)l, (n +. 19.

(32) 2l′)(m + 2l), 7ll′ + 3nl + 3ml′ + nm) = gcd (3nm′ − 5n′l′ + 4m′l′, 3nm′ − 14n′l′ − nn′, m′l′ + 4n′l′ + nn′, 7n′l + 5m′l, 4n′l + 2m′l − mm′ − 2mn′, 4n′l + 5mm′ + 5mn′, nl + ll′, nl + 3ll′ + ml′, 4ll′ + 3ml′ + mn) = gcd (3nm′ − 5n′l′ + 4m′l′, 3nm′ + m′l′ − 10n′l′, m′l′ + 4n′l′ + nn′, 7n′l + 5m′l, 6n′l − 5mm′ − 10mn′, 4n′l + 5mm′ + 5mn′, nl + ll′, 2ll′ + ml′, 3nl + 5ll′ − mn) = gcd ((5n′ + 3m′)l′, 3nm′ + m′l′ − 10n′l′, m′l′ + 4n′l′ + nn′, (7n′ + 5m′)l, 10n′l − 5mn′, 4n′l + 5mm′ + 5mn′, (n + l′)l, (2l + m)l′, 2nl + mn) = gcd ((5n′ + 3m′)l′, (3n + 7l′)m′, (7l′ + 3n)n′, (7n′ + 5m′)l, 5(2l − m)n′, 5(7n′ + 5m′)m, (n + l′)l, (2l + m)l′, (2l + m)n). (3.1.5). =1. The farthest nodes from the base node of H2′ must be at one of the circled node. Their distances are: d(A) = l′ + l + 6n + 5m, d(B) = l′ + 2l + 3n + 3m, d(C) = l′ + 3l + n′ + 3n + 3m, d(D) = l′ + 3l + n′ + 3n + m′ + 2m, d(E) = 2l′ + 2l + n′ + 3n + m′ + 2m, d(F) = 3l′ + l + n′ + 3n + m′ + 2m, d(G) = 2l′ + 3l + n′ + 2n + m′ + m, d(H) = 5l + 2n′ + n + m′ + m, d(I) = l′ + 4l + 2n′ + n + m′ + m, d(J) = 2l′ + 3l + 2n′ + n + m′ + m, d(K) = 5l′ + l + n′ + n + m′.. Our heuristic method sets all these distances equal except d(B). Thus d(A) = d(C) ⇒ 3n + 2m = 2l + n′, d(C) = d(D) ⇒ m′ = m′, d(D) = d(E) ⇒ l′ = l, 20.

(33) d(F) = d(G) ⇒ l = n + m.. Summarizing, we have l = l′ = m + n, n = n′ and m = m′.. , there are only two independent Therefore in the suboptimal setting H 2 parameters m and n, and the diameter is 8n + 7m. Note that for this suboptimal version, necessary and sufficient conditions for the existence of a corresponding triple-loop network is induced from (3.1.5) to gcd (m, n) = 1.. is Efficiency of H 2 N 40n3 + 110n 2 m + 96nm 2 + 27m3 E(H 2 ) = 3 = . 3 D ( 8n + 7 m ). Setting m = kn, then n can be canceled out and N 27 k 3 + 96k 2 + 110k + 40 = . 3 D3 ( 7k + 8 ). (. ) (. ). 2 27k 3 + 96k 2 + 110k + 40 ⋅ 21 d ⎛ N ⎞ 81k + 192k + 110 − =0 ⎜ ⎟= 3 4 dk ⎝ D 3 ⎠ ( 7k + 8 ) ( 7k + 8). ⇒ 6k 2 + k − 10 = 0 ⇒ k0 =. −1 + 241 ≈ 1.2 12. Hence we choose n = 5 and m = 6 for integrality,. = 44612 > 0.08091. E H 2 551368. ( ). Recall that the efficiency of H2 is 860 223 ≈ 0.08 .. 21.

(34) 3.2. Wide-Diameter of H0. Traditionally, connectivity and diameter were studied separately. Then various approaches have been proposed to study these two parameters together. One such approach led to the notion of k-diameter which was formalized and popularized in Hsu [21] and Hsu and Luczak [22]. The k-diameter of a digraph is the minimum length l such that there exist k node-disjoint paths non longer than l. Clearly, the 1-diameter is just the usual diameter D. Note that the k-diameters give a complete description of the interplay between the connectivity and the diameter. It also automatically provides the information if f faults occur for 1 ≤ f < k, then the diameter of the surviving graph, the fault-tolerant diameter, does not exceed the k-diameter.. In this section, we [27] will prove that H0 is 3-connected by constructing 3 node-disjoint paths from any node i to any other node j. A set P of k node-disjoint paths from i to j with lengths l1 ≤ l2 ≤ … ≤ lk is called a minimum-k-routing if for any such set of paths with lengths l1′ ≤ l2′ ≤ … ≤ lk′ we have li ≤ li′ for i = 1, …, k. P is called a weak minimum-k-routing if (l1, l2, …, lk) is lexicographically shorter than (l1′, l2′, …, lk′). Further, P is oblivious if the routing from i to j depends only on i and j. In this paper we give an oblivious weak minimum-3-routing for an arbitrary pair (i, j) and show that a minimum-3-routing does not exist. From the weak minimum-k-routing, we derive an upper bound of the k-diameter. In particular, the 3-diameter is at most D + 2. For convenient, let H0(N; s1, s2, s3) denote the TL(N; s1, s2, s3) with H0(l, m, n) shape. Let H0(0) denote the MDD(0) of H0(N; s1, s2, s3). By Theorem 2.3.1, we have known that every MDD(0) of triple-loop networks always tessellates ℜ3. One consequence is that there exists another shape H0*(0) with base 0 located at cell (l − m − n, l − m − n, l − m − n), which is adjacent to H0(0) in the tessellation (see Fig.. 3.2.1). 22.

(35) H0*(0). H0(0) Fig. 3.2.1 H0(0) and H0*(0).. A dimension routing from node u to node v means first taking all steps in one dimension (same si), then all steps in a second dimension, then all steps in a third dimension. For example, a dimension routing from node 0 to a node at (x1, x2, x3) with the dimension order (3, 1, 2) takes the x3 s3-steps first, then the x1 s1-steps and finally the x2 s2-steps. Note that a dimension routing always yields a shortest path. Since TL(N; s1, s2, s3) is node-transitive, it suffices to consider paths from node 0 to an arbitrary node v with coordinates (v1, v2, v3) in H0(0). Theorem 3.2.1 There exists an oblivious weak minimum-3-routing from node 0 to. an arbitrary node v in H0. Proof. Suppose v occupies cell (v1, v2, v3) in H0(0). We consider three cases:. (i). vi > 0 for i = 1, 2, 3. We use dimension routing. The dimension order for. path 1 is (1, 2, 3), for path 2 is (2, 3, 1) and for path 3 is (3, 1, 2) (see Fig. 3.2.2). Then clearly, the three paths are node-disjoint and each has length v1 + v2 + v3 which is the distance from 0 to v. Since the lengths of these 3 paths are equal to the distance from 0 to v, it’s obvious that the paths we construct constitute a minimum-3-routing.. 23.

(36) x3. Path 3 Path 2. Path 1. u v. x1. x2 Fig. 3.2.2 Dimension routing for v1 > 0, v2 > 0, v3 > 0.. (ii). Exactly one vi = 0 (say v3). We use dimension routing in the x3 = 0 plane (where v lines) with orders (1, 2) and (2, 1), respectively, to obtain two node-disjoint paths to v. The third path will be routed through the node u ≡ v − s3 (mod N) as a penultimate node. Suppose u is not in the x3 = 0 plane. Then path 3 is obtained by a dimension routing from node 0 to u starting with s3-steps. Since path 3 uses only nodes not in the x3 = 0 plane in H0(0), it is node-disjoint from paths 1 and 2. Call a node x occupying cell (x1, x2, x3) in H0(0) 1-maximal if cell (x1+1, x2, x3) is not in H0(0). Similarly we can define 2-maximal and 3-maximal.. Then u must be 3-maximal in H0(0) or v would lie in a plane x3 = k > 0 in H0(0), contradicting our assumption that v3 = 0. Suppose u is in the x3 = 0 plane. From the fat that u is 3-maximal, necessarily, l − m − n = 1. Hence v occupies cell (v1 + 1, v2 + 1, v3 + 1) in H0*(0). Path 3 starts with an s3-steps and enter cell (0, 0, 1), which can be treated as the base of H0(s3). It is easily verified that H0(s3) can be obtained from H0(0) by moving nodes on the boundary of the x1 = 0 and x2 = 0 planes (see Fig. 3.2.3). It u is not in the x3 = 1 plane (the floor plane in Fig. 3.2.3 (b)), implying 24.

(37) u is a boundary node of the x3 = 0 plane, then path 3 uses only nodes not in. the x3 = 0 plane, except u, which is not on paths 1 or 2. Hence path 3 is node-disjoint from paths 1 and 2.. x3. 23 28. 14 19 24 29 3. x2. x3. 16 15 22 21 20 27 26 25 2 1 0 8 30 7 6 13 5 12 11 18 4 17 10 9. 18. x1 x2. 9 14 19 24. 23. 10 15 20 25 30. 29. (a). 4. 11 16 21 26 0 5. 17 22 27 1 6. 28 2 7. 3 8 13. 12. (b). Fig. 3.2.3 (a) and (b) are H0(0) and H0(26), respectively, for l − m − n = 1, where N = 31, s1 = 6, s2 = -1, s3 = −5, l = 4, m = 2, n = 1.. Suppose u is in the x3 = 1 plane. Since v occupies cell (v1 + 1, v2 + 1, v3 + 1), u must occupy cell (v1 + 1, v2 + 1, v3) in H0(0) and hence also cell (v1 + 2, v2 + 2, v3 + 1) in H0*(0), which is also in H0(s3). Note that paths 1 and 2 enclose a rectangle 1 ≤ x1 ≤ v1+1, 1 ≤ x2 ≤ v2 + 1 in H0(s3), and u is outside of it. Hence a path from s3 to u using either the (1, 2) or the (2, 1) dimension routing bypasses the rectangle and consequently is node-disjoint with paths 1 and 2. Path 3 is completed by adding the steps from 0 to s3 and from u to v.. Since the lengths of paths 1 and 2 are equal to the distance from 0 to v, these two paths are shortest. Further, all shortest paths must start and end either with an s1-step or an s2-step (any combination allowed). Therefore a third disjoint path must start and end with an s3-step, i.e., the second node of the path is s3 and the penultimate node is u. Since our proposed third path 25. x1.

(38) uses dimension routing from s3 to u, it is shortest among the set of third disjoint paths given that the first second paths are shortest. Hence the proposed routing is a weak minimum-3-routing.. (iii). Exactly two vi = 0 (say, v3 = v2 = 0). Path 1 is the unique shortest path from node 0 to v along the x1-axis. Let u ≡ v − s3 (mod N) and w ≡ v − s2 (mod N). We will show that in H0(0) one of u and w has x2 > 0 and the other x3 > 0. Then we let path 2 go from 0 to s2, followed by a dimension routing to the node in {u, w} with x2 > 0 (in fact, the dimension routing starts with dimension 2, hence is also a dimension routing from 0). Similarly, path 3 goes from 0 to s3 followed by a dimension routing (starting from dimension 3) to the other node in {u, w}. Let Li, i ∈ {2, 3} denote the set of paths whose last step is a si-step. Then a weak minimum-3-routing must have one path from L2 and one from L3. But our proposed paths constitute a shortest pair from L2 and L3 since they use dimension routing. This proves weak minimum-3-routing. To prove the existence of the desirable u and w, we first prove a lemma which locates u and w in H0(0). Among the six permutations of (s1, s2, s3) mentioned in Theorem 3.2.1, call (s1 = a, s2 = b, s3 = c), (s1 = b, s2 = c, s3 = a), (s1 = c, s2 = a, s3 = b) type 1 and the other three permutations type 2,. where a = l2 − mn, b = m2 + ln, c = n2 + lm. Lemma 3.2.2 Let v = (v1, 0, 0).. (i). Suppose 0 ≤ v1 < m + n. Then u = (v1 + l − m − n, l − m − n, l − m − n − 1), w = (v1 + l − m − n, l − m − n − 1, l − m − n).. (ii). Suppose m + n ≤ v1 < l and n > 0. Then u = (v1 − m − n, l − n, l − m − 1) and w = (v1 − m − n, l − n − 1, l − m) if (s1, s2, s3) is of the first type. Otherwise, u. = (v1 − m − n, l − m, l − n − 1) and w = (v1 − m − n, l − m − 1, l − n) 26.

(39) (iii). Suppose m + n ≤ v1 < l, n = 0 and l − m − n = 1. Then u = (0, 0, l − 1) and w = (0, l − 1, 1) if (s1, s2, s3) is of type 1. Otherwise, u = (0, 1, l − 1) and w = (0, l − 1, 0).. Proof.. (i). v also occupies (v1 + l − m − n, l − m − n, l − m − n) in H0*(0). So u occupies. (v1 + l − m − n, l − m − n, l − m − n − 1) and w occupies (v1 + l − m − n, l − m − n − 1, l − m − n). Since v1 < m + n, the above two locations of u and w are in H0(0).. (ii). We first check v ≡ v1s1 (mod N) also occupies (v1 − m − n, l − n, l − m) if (s1, s2, s3) is of type 1.. (−m − n)(l2 − mn) + (l − n)(m2 + ln) + (l − m)(n2 + lm) = 0, (−m − n)(m2 + ln) + (l − n)(n2 + lm) + (l − m)(l2 − mn) = l3 − m3 − n3 − 3lmn ≡ 0 (mod N), (−m − n)(n2 + lm) + (l − n)(l2 − mn) + (l − m)(m2 + ln) = l3 − m3 − n3 − 3lmn ≡ 0 (mod N). It is easily checked that u = (v1 − m − n, l − n, l − m − 1) and w = (v1 − m − n, l − n − 1, l − m) are in H0(0). The proof is similar if (s1, s2, s3) is of. type 2.. (iii). By the given conditions, we have v1= m = l − 1. Therefore a = l2, b = (l − 1)2, c = l(l − 1). Note that. (l − 1)a ≡ lc ≡ lb + c (mod N), (l − 1)b ≡ la ≡ lc + a (mod N), (l − 1)c ≡ lb ≡ la + b (mod N). If (s1, s2, s3) is of type 1, then v, which occupies cell (l − 1, 0, 0) in H0(0), also occupies (0, 0, l) and (0, l, 1). Therefore u occupies (0, 0, l − 1). 27.

(40) and w occupies (0, l − 1, 1) in H0(0). The proof is similar if (s1, s2, s3) is of type 2.. We now prove that paths 2 and 3 are node-disjoint (their disjointness from path 1 is obvious). We consider three cases: 1. 0 ≤ v1 < m + n or m + n ≤ v1 < l and n > 0. The locations of u and w in H0(0) are given in Lemma 3.2.2. Since x2 > 0 for u and x3 > 0 for w, a (2, 1, 3) dimension routing exists from 0 to u and a (3, 1, 2) from 0 to w. Node-disjointness is easily verified. 2. m + n ≤ v1 < l, n = 0, l − m − n > 1. Since l − m − n > 1, u is 3-maximal and w 2-maximal in H0(0). Hence x2 > 0 for w and x3 > 0 for u. Use the (2, 1, 3) dimension routing from 0 to w, and the (3, 1, 2) dimension routing from 0 to u. Node-disjointness holds just as the previous two cases. 3. m + n ≤ v1 < l, n = 0, l − m − n = 1. Suppose (s1, s2, s3) is of type 1. By Lemma 3.3.2, u = (0, 0, l − 1) and w = (0, l − 1, 1) in H0(0). Since x3 > 0 for u and x2 > 0 for w, a (3, 1, 2) dimension routing (which degenerates into a dimension routing of (3)) exists from 0 to u, and a (2, 1, 3) dimension routing (which degenerates into a dimension routing of (2, 3)) exists from 0 to w. It is easily seen that the two paths are node-disjoint. Suppose (s1, s2, s3) is of type 2. Then we switch he dimension routings between u and w. Obliviousness is clear from the construction.. We give an example that a minimum-3-routing does not exist. For H0(31; 9, 8, 14) and v = 26, the proposed routing yields length (3, 3, 7) while the routing: P1′: 0-9-17-26, P2′: 0-8-16-25-3-12-26, P3′: 0-14-23-1-10-18-26 yields length (3, 6, 6). Since l3 > l3′, (P1, P2, P3) is not a minimum-3-routing. On the other hand, it is easily 28.

(41) seen that if a minimum-3-routing exists, then (P1, P2, P3), a weak minimum-3-routing, must be it. Corollary 3.2.3 The connectivity of H0 is 3. Theorem 3.2.4 The k-diameter of H0 is at most D + k − 1 for k = 1, 2, 3. Proof.. That the k-diameter for k = 1, 2, 3 does not exceed D + k − 1 is easily. verified by our construction. It is also easily checked that the 1-diameter is indeed D since only dimension routing is used for path 1. For k = 2, the worst case is case (iii) in which a path may take D + 1 steps. We take H0(7; 2, 1, 4) (see Fig. 3.2.4) with v = 2 for example to show that D + 1 is realizable. Here path 2 is (0, 4, 5, 2) of length 3 = D + 1. For k = 3, the worst case is case (ii) in which a path may take D + 2 steps. We take H0(31; 6, 30, 26) (see Fig. 3.2.3) with v = 4 for example to show that D + 2 is realizable. Here path 3 is (0, 26, 21, 16, 11, 10, 9, 4) of length 7 = D + 2.. x3. 5 1. 4 6 0 2 3 x1. x2. Fig. 3.2.4 H0(7; 2, 1, 4) with v = 2, where u = 5, w = 1.. Corollary 3.2.5 The 3-diameter of H0 is at most D + 2. Corollary 3.2.6 The diameter of H0 is at most D + 2 after two arbitrary failures. (nodes or links).. 29.

(42) 3.3. Wide-Diameter of H1′. In section 3.1, we have generalized H1 and H2 to H1′ and H2′ by allowing some line segments which have the same length to have different lengths. In this section, we also use oblivious weak minimum-3-routing to prove that H1′ is 3-connected by constructing 3 node-disjoint paths from any node i to any other node j. For 3-diameter of H2′, we will prove it in next section 3.4 by similar method. For convenient, let H1′(0) (H2′(0)) denote the MDD(0) of H1′(H2′). We define H1′(a, b, c)(0) as the copy of H1′(0), which is obtained by adding the a(n, n′, 2h) + b(−m, n′ + m′, h) + c(−m, −m′, h + h′) vector, to each nodes of H1′(0), where a, b, c. ∈ Ζ.(See Fig. 3.3.1) Similarly, we define H2′(a, b, c)(0) as the copy of H2′(0), which is obtained by adding the a(2l + n′, l′ + m′, m + 2n) + b(3l + n′, −2l′, m + n) + c(−2l −n′, l′, 2m + 3n) vector, to each nodes of H2′, where a, b, c ∈ Ζ.(See Fig. 3.4.1) We. call a node x occupying cell (x1, x2, x3) in H1′(0) or H2′(0) 1-maximal if cell (x1+1, x2, x3) is not in H1′(0) or H2′(0). Similarly we can define 2-maximal and 3-maximal.. Besides, we define that t ≡ v − s1 (mod N), w ≡ v − s2 (mod N), u ≡ v − s3 (mod N), t′ ≡ t − s1 (mod N), w′ ≡ w − s2 (mod N), and u′ ≡ u − s3 (mod N).. H1′(0) H1′(0, 1, 1)(0). H1′(1, −1, 1)(0). H1′(1, 0, 1)(0) H1′(1, −1, 0)(0). H1′(0, 1, 0)(0). H1′(1, 0, 0)(0) H1′(0) H1′(0, 0, −1)(0). 30.

(43) H1′(0, 0, 2)(0). H1′(0, 0, 1)(0) H1′(1, 0, −1)(0). H1′(0). H1′(0, 0, −1)(0). Fig. 3.3.1 H1′(0) and its copies.. Theorem 3.3.1 There exists an oblivious weak minimum-3-routing from node 0 to an. arbitrary node v in H1′. Suppose v occupies cell (v1, v2, v3) in H1′(0). Let l1, l2, l3 be the distances from 0 to t, w, u in H1′(0), respectively. The lengths of the three paths are (i). v1 + v2 + v3, v1 + v2 + v3 and v1 + v2 + v3, when vi > 0 for i = 1, 2, 3.. (ii). vj + vk, vj + vk and li + 2, when exactly one vi = 0 for i ∈ {1, 2, 3}, where j, k ∈. {v1, v2, v3} / {vi} and j ≠ k. (iii). vk, li + 1 and li + 1, when vi = vj = 0 for i, j ∈ {1, 2, 3}, and i ≠ j, where k = {v1, v2, v3} / {vi, vj}.. Proof. We consider three cases:. (i). vi > 0 for i = 1, 2, 3. We use dimension routing. The dimension order for path 1. is (1, 2, 3), for path 2 is (2, 3, 1) and for path 3 is (3, 1, 2). Then clearly, the three paths are node-disjoint and each has length v1 + v2 + v3 which is the distance from 0 to v. Since the lengths of these 3 paths are equal to the distance from 0 to v, it’s obvious that the paths we construct constitute a minimum-3-routing.. 31.

(44) (ii). Exactly one vi = 0. We consider three cases: a. v1 = 0. We use dimension routing in the x1 = 0 plane (where v lies) with orders (2, 3) and (3, 2), respectively, to obtain two node-disjoint paths to v. The third path will be routed through node t as a penultimate node. Suppose t is not in the x1 = 0 plane. Then path 3 is obtained by a dimension routing from node 0 to t starting with s1-steps. Since path 3 uses only nodes not in the x1 = 0 plane in H1′(0), it is node-disjoint from paths 1 and 2. Besides, we know that t is 1-maximal in H1′(0) or v would lie in a plane x1 = k > 0 in H1′(0), contradicting our assumption that v1 = 0.. Suppose t is in the x1 = 0 plane. From the fact that t is 1-maximal, necessarily, n = 1 or m = 1. For n = 1, we only need to consider the condition that t is located in the following two regions R1 and R2: 1. R1: x1 = 0, 0 ≤ x2 < n′, 2h + h′ ≤ x3 < 3h + h′. It occurs when v1 = 0, 2m′ ≤ v2 < 2m′ + n′, 0 ≤ v3 < h for v also occupies cell (n, v2 − 2m′, v3 + 2h + h′) in H1′(1, −1, 1)(0). Thus we have that t occupies cell (0, v2 − 2m′, v3 + 2h + h′) in H1′(0). Since t also occupies cell (m, v2 − m′, v3 + h) in H1′(0, 0, −1)(0). Therefore, t′ occupies cell (m − 1, v2 − m′, v3 + h) in H1′(0), and v occupies cell (m + 1, v2 − m′, v3 + h) in H1′(1, −1, 0)(0).. Path 3 starts with an s1-step and enter cell (1, 0, 0), followed by a. dimension routing to t′ in H1′(0), and then add an s1-step to t in H1′(0, 0, −1)(0).. Path 3 is completed by an s1-step to v in H1′(1, −1, 0)(0).. 2. R1: x1 = 0, n′ ≤ x2 < n′ + m′, 2h ≤ x3 < 2h + h′. It occurs when v1 = 0, 0 ≤ v2 < m′, 0 ≤ v3 < h′ for v also occupies cell (n, v2 + n′, v3 + 2h) in H1′(1, 0, 0)(0). Thus we have that t occupies cell (0, v2 + n′, v3 + 2h) in H1′(0). Since t also occupies cell (m, v2 + n′ + m′, v3 + h − h′) in H1′(0, 0, −1)(0). Therefore, t′ occupies cell (m − 1, v2 + n′ + m′, v3 + h − h′) in H1′(0), and v occupies cell (m + 1, v2 + n′ + m′, v3 + h − h′) in H1′(1, 0, −1)(0).. Path 3 starts with an s1-step and enter cell (1, 0, 0), followed by a 32.

(45) dimension routing to t′ in H1′(0), and then add an s1-step to t in H1′(0, 0, −1)(0).. Path 3 is completed by an s1-step to v in H1′(1, 0, −1)(0).. Hence, path 3 is node-disjoint from paths 1 and 2, and it has length at most D + 2.. For m = 1, we only need to consider the condition that t is located in the following four regions R1, R2, R3 and R4: 1. R1: x1 = 0, m′ ≤ x2 < m′ + n′, h + h′ ≤ x3 < 2h. (if h′ < h) It occurs when v1 = 0, 0 ≤ v2 < n′, 2h + 2h′ ≤ v3 < 3h + 2h′ for v also occupies cell (m, v2 + m′, v3 − h − h′) in H1′(0, 0, −1)(0). Thus we have that t occupies cell (0, v2 + m′, v3 − h − h′) in H1′(0). Since t also occupies cell (m, v2 + 2m′, v3 − 2h − 2h′) in H1′(0, 0, −1)(0). Therefore v occupies cell (m + 1, v2 + 2m′, v3 − 2h − 2h′) in H1′(0, 0, −2)(0). Path 3 starts with an s1-step and enter cell (1, 0, 0), followed by a dimension routing to t in H1′(0, 0, −1)(0). Path 3 is completed by an s1-step to v in H1′(0, 0, −2)(0). 2. R2: x1 = 0, m′ ≤ x2 < m′ + n′, h ≤ x3 < h + h′. It occurs when v1 = 0, 0 ≤ v2 < n′, 2h + h′ ≤ v3 < 2h + 2h′, because of the same reason for R1. Since t also occupies cell (n + m, v2, v3 − h′) in H1′(1, −1, 0)(0). Therefore v occupies cell (n + m + 1, v2, v3 − h′) in H1′(1, −1, −1)(0).. Path 3 starts with an s1-step and enter cell (1, 0, 0), followed by a. dimension routing to t in H1′(1, −1, 0)(0). Path 3 is completed by an s1-step to v in H1′(1, −1, −1)(0).. 3. R3: x1 = 0, m′ ≤ x2 < 2m′, 0 ≤ x3 < h. It is the same as the proof for R2, except that it occurs when v1 = 0, n′ ≤ v2 < m′ (if n′ < m′), h + h′ ≤ v3 < 2h + h′. 4. R4: x1 = 0, 2m′ + n′ ≤ x2 < 2m′ + n′, 0 ≤ x3 < h. It occurs when v1 = 0, m′ ≤ v2 < n′ + m′, h + h′ ≤ v3 < 2h + h′, because 33.

(46) of the same reason for R1. Since t also occupies cell (n, v2 − m′, v3 + h) in H1′(1, −1, 1)(0). Therefore v occupies cell (n + 1, v2 − m′, v3 + h) in H1′(1, −1, 0)(0).. Path 3 starts with an s1-step and enter cell (1, 0, 0), followed by a. dimension routing to t in H1′(1, −1, 1)(0). Path 3 is completed by an s1-step to v in H1′(1, −1, 0)(0).. Hence, path 3 is node-disjoint from paths 1 and 2, and it has length at most D + 2.. Since the lengths of paths 1 and 2 are equal to the distance from 0 to v, these two paths are shortest. Further, all shortest paths must start and end either with an s2-step or an s3-step (any combination allowed). Therefore a third disjoint path must start and end with an s1-step, i.e., the second node of the path is s1 and the penultimate node is t. Since our proposed third path uses dimension routing from s1 to t, it is shortest among the set of third disjoint paths given that the first and second paths are shortest. Hence the proposed routing is a weak minimum-3-routing.. Since the proofs of the two cases, v2 = 0 and v3 = 0, are analogous to v1 = 0, we only consider the conditions different from v1 = 0. b. v2 = 0. Suppose w is in the x2 = 0 plane. From the fact that w is 2-maximal, necessarily, n′ = 1 or m′ = 1. For n′ = 1, we only need to consider the condition that w is located in the following two regions R1 and R2: 1. R1: 0 ≤ x1 < n, x2 = 0, 2h + h′ ≤ x3 < 3h + h′. It occurs when 2m ≤ v1 < 2m + n, v2 = 0, 0 ≤ v3 < h for v also occupies cell (v1 − 2m, n′, v3 + 2h + h′) in H1′(0, 1, 1)(0). Thus we have that w occupies cell (v1 − 2m, 0, v3 + 2h + h′) in H1′(0). Since w also occupies cell (v1 − m, m′, v3 + h) in H1′(0, 0, −1)(0). Therefore, w′ occupies cell (v1 − m, m′ − 1, v3 + h) in H1′(0), and v occupies cell (v1 − m, m′ + 1, v3 + h) in H1′(0, 1, 34.

(47) 0)(0).. 2. R2: n ≤ x1 < n + m, x2 = 0, 2h ≤ x3 < 2h + h′. It occurs when 0 ≤ v1 < m, v2 = 0, 0 ≤ v3 < h′ for v also occupies cell (v1 + n, n′, v3 + 2h) in H1′(1, 0, 0)(0). Thus we have that w occupies cell (v1 + n, 0, v3 + 2h) in H1′(0). Since w also occupies cell (v1 + n + m, m′, v3 + h − h′) in H1′(0, 0, −1)(0). Therefore, w′ occupies cell (v1 + n + m, m′ − 1, v3 + h − h′) in H1′(0), and v occupies cell (v1 + n + m, m′ + 1, v3 + h − h′) in H1′(1, 0, −1)(0).. For m′ = 1, we only need to consider the condition that w is located in the following four regions R1, R2, R3 and R4: 1. R1: m ≤ x1 < m + n, x2 = 0, h + h′ ≤ x3 < 2h. (if h′ < h) It occurs when 0 ≤ v1 < n, v2 = 0, 2h + 2h′ ≤ v3 < 3h + 2h′ for v also occupies cell (v1 + m, m′, v3 − h − h′) in H1′(0, 0, −1)(0). Thus we have that w occupies cell (v1 + m, 0, v3 − h − h′) in H1′(0). Since w also occupies cell (v1 + 2m, m′, v3 − 2h − 2h′) in H1′(0, 0, −1)(0). Therefore v occupies cell (v1 + 2m, m′ + 1, v3 − 2h − 2h′) in H1′(0, 0, −2)(0). 2. R2: m ≤ x1 < m + n, x2 = 0, h ≤ x3 < h + h′. It occurs when 0 ≤ v1 < n, v2 = 0, 2h + h′ ≤ v3 < 2h + 2h′, because of the same reason for R1. Since w also occupies cell (v1, n′ + m′, v3 − h′) in H1′(0, 1, 0)(0). Therefore v occupies cell (v1, n′ + m′ + 1, v3 − h′) in H1′(0, 1, −1)(0).. 3. R3: m ≤ x1 < 2m, x2 = 0, 0 ≤ x3 < h. It is the same as the proof for R2, except that it occurs when n ≤ v1 < m (if n < m), v2 = 0, h + h′ ≤ v3 < 2h + h′.. 4. R4: 2m + n ≤ x1 < 2m + n, x2 = 0, 0 ≤ x3 < h. It occurs when m ≤ v1 < n + m, v2 = 0, h + h′ ≤ v3 < 2h + h′, because of the same reason for R1. Since w also occupies cell (v1 − m, n′, v3 + h) in H1′(0, 1, 1)(0). Therefore v occupies cell (v1 − m, n′ + 1, v3 + h) in H1′(0, 1,. 35.