弦環式網路之探討與研究

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(1)國立交通大學應用數學系碩士論文. 弦環式網路之探討與研究 The Study of Chordal Ring Networks. 研究生：陳建瑋指導老師：陳秋媛. 教授. 中華民國九十三年六月.

(2) 弦環式網路之探討與研究. The Study of Chordal Ring Networks 研究生：陳建瑋. Student: Komi Chienwei Chen. 指導老師：陳秋媛教授. Advisor: Dr. Chiuyuan Chen. 國立交通大學應用數學系碩. 士. 論. 文. A Thesis Submitted to Department of Applied Mathematics College of Science National Chiao Tung University In partial Fulfillment of Requirement For the Degree of Master In Applied Mathematics June 2004 Hsinchu, Taiwan, Republic of China 中華民國九十三年六月.

(3) 弦環式網路之探討與研究. 研究生：陳建瑋. 指導老師：陳秋媛教授. 國立交通大學應用數學系摘. 要. 「弦環式網路」是一種常被討論的區域網路架構 [1, 3, 8, 10, 11]。一個「無向的弦環式網路」是一個無向的三正則圖。在文獻[8, 10, 11]中，黃光明老師、陳尚寬學長、以及 Wright，將「無向的弦環式網路」推廣成「有向的弦環式網路」，並給出計算「有向的弦環式網路」的直徑的方法。在文獻[3]中，陳尚寬學長、黃光明老師、以及劉昱綺學姊又推廣「有向的弦環式網路」來提出另一種有向的網路的連法，稱為「混合的弦環式網路」。雖然「無向的弦環式網路」的直徑已被完整地研究、並且可以運用公式得出，但是截至目前為止，「有向的弦環式網路」的直徑、以及「混合的弦環式網路」的直徑卻還未被完全找出來。在這篇論文裡，我們首先推導「有向的弦環式網路」以及「混合的弦環式網路」的同構性質；我們接著得出某些特殊的「有向的弦環式網路」以及「混合的弦環式網路」的直徑，與之前文獻不同的是，我們並不需要先計算出對應的「雙環式網路」的直徑來得出這些直徑。關鍵詞：弦環式網路、有向的弦環式網路、混合的弦環式網路、雙環式網路、直徑、同構。. 中華民國九十三年六月 i.

(4) The Study of Chordal Ring Networks. Student : Komi Chienwei Chen. Advisor : Dr. Chiuyuan Chen. Department of Applied Mathematics National Chiao Tung University Hsinchu 300, Taiwan, R.O.C.. Abstract Chordal ring networks have been proposed as a popular architecture for local area networks [1, 3, 8, 10, 11]. An undirected chordal ring network is an undirected regular graph of degree 3. In [8, 10, 11], Hwang, Chen, and Wright proposed the directed version of the undirected chordal ring network and derived the diameter of a directed chordal ring network. Furthermore, in [3], Chen et al. proposed the mixed chordal ring network. While the diameter of an undirected chordal ring network has been well studied [1], the diameter of a directed chordal ring network and the diameter of a mixed chordal ring network are not known. In this thesis, we shall study the isomorphism property of chordal ring networks and we shall find out the diameter of some directed chordal ring networks and the diameter of some mixed chordal ring networks.. Keywords: Chordal ring network, directed chordal ring network, mixed chordal ring network, double-loop network, diameter, isomorphism.. ii.

(5) 誌. 謝. 首先，要特別感謝我的指導教授，陳秋媛老師，在研究所這兩年，不遺餘力的指導與幫助，不僅很照顧學生，而且總是在我最旁徨無助的時候及時給與協助。亦師亦友的相處，讓我在這兩年裡成長不少並且充滿信心的學習與研究。同時也要感謝系上組內的老師，黃光明老師、黃大原老師、傅恆霖老師及翁志文老師。修過你們的課，讓我知道組合數學裡又細分了很多不同的應用領域。「只要給我一個立足點，我將能移動地球。」這是希臘數學家阿基米德曾說過的一句話。對我而言，本論文的完成就好比地球一般的巨大，單憑一己之力，絕無法以蜉蟻之力來撼動。而幸運的是生活周遭遇到了一群關心我的師長、同學與及朋友們。在生論文的過程中，給了我無數的幫助及建議。特別要感謝同門師兄唐文祥，因為有你提供不少寶貴的意見讓我的論文才能順利地完成。同時也要感謝博士班學長郭君逸及張飛黃和我同研究室的同學，貴弘、致維、正傑及宏嘉、三樓研究室的抮君、昭芳、棨丰、啟賢、喻培、嘉文與及二樓研究室的學妹們。對我來說，生命中能和這些人相遇，真的是我的福氣。感謝我的女朋友佑寧長久默默地對我的支持與照顧，讓我能無後顧之憂的作研究。同時也要感謝一起在新竹唸書的錦文、坤宗、雅靜、崢佩、明泓、其儒及立業及在這兒工作的姿菁學姊。與及宿舍室友. 元，約漢及繼元。有你們的陪伴，. 一直都不是寂寞的。最後要感謝我的父母與家人，特別是住在台北的姊姊，若不是他們長久以來的支持，不可能有我今天的小小成果。在此獻上無限的感激，謝謝你們。. iii.

(6) Contents Abstract (in Chinese). i. Abstract (in English). ii. Acknowledgement. iii. Contents. iv. List of Figures. v. 1. Introduction. 1. 2. Previous results. 4. 3. Isomorphism. 7. 4. r DCR ( N , s , h ) and MCR ( N , s , h ) 18 The diameter of. References. 27. iv.

(7) List of Figures 1. The undirected chordal ring network U CR(16, 3). . . . . . . . . . . .. 1. 2. The directed chordal ring network DCR(16, 3, ~5). . . . . . . . . . . .. 2. 3. The mixed chordal ring network M CR(16, 3, 5). . . . . . . . . . . . .. 3. 4. Two examples of L-shapes. . . . . . . . . . . . . . . . . . . . . . . . .. 5. 5. An L-shape with parameters. . . . . . . . . . . . . . . . . . . . . . .. 5. 6. i is even, the links (i/2)∗ → (i/2 + s)∗ and (i/2)∗ → (i/2 + (s + h)/2)∗ . 14. 7. ~ into (a) The L-shape of DL(9, 5, 2). (b) Transforming DCR(18, 5, 17) DL(9, 5, 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. 8. i is odd, the links ((i − 1)/2)∗ → ((i − 1)/2 + s)∗ and ((i − 1)/2)∗ → ((i − 1)/2 + (h − s)/2)∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 15. 9. ~ into (a) The L-shape of DL(9, 5, 6). (b) Transforming DCR(18, 5, 17) DL(9, 5, 6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16. 10. M CR(16, 1, 9); it is isomorphic to Figure 3. . . . . . . . . . . . . . . 18. 11. Two equivalent L-shapes. . . . . . . . . . . . . . . . . . . . . . . . . . 19. 12. The nodes of DCR(18, 1, ~3) are divided into d18/(3 + 1)e groups. . . 22. 13. The diameter of M CR(10, 3, 5) is 5. . . . . . . . . . . . . . . . . . . . 26. v.

(8) 1. Introduction. Chordal ring networks were first proposed by Arden and Lee [1]. An undirected chordal ring network U CR(N, h) is an undirected graph with N nodes 0, 1, · · · , N −1 and 3N/2 edges of two types: i↔i+1. (mod N ) ∀ i = 0, 1, 2, · · · , N − 1,. i↔i+h. (mod N ) ∀ i = 1, 3, 5, · · · , N − 1,. where N is even, and h is odd. See Figure 1. 0 15. 1 2. 14. 13. 3. 12. 4. 5. 11 6. 10 7. 9. 8. Figure 1: The undirected chordal ring network U CR(16, 3).. Hwang and Wright [11] proposed the directed version of the undirected chordal − → ring network. A directed chordal ring network DCR(N, 1, h ) is a directed graph with N nodes 0, 1, · · · , N − 1 and 3N/2 links (i.e., directed edges) of two types: i→i+1. (mod N ) ∀ i = 0, 1, 2, · · · , N − 1,. i→i+h. (mod N ) ∀ i = 1, 3, 5, · · · , N − 1,. where N is even and h is odd.. 1.

(9) − → Hwang [8] generalized the directed chordal ring network DCR(N, 1, h ) to the − → − → directed chordal ring network DCR(N, s, h ). DCR(N, s, h ) is a directed graph with N nodes 0, 1, · · · , N − 1 and 3N/2 links of two types: i→i+s. (mod N ) ∀ i = 0, 1, 2, · · · , N − 1,. i→i+h. (mod N ) ∀ i = 1, 3, 5, · · · , N − 1,. where N is even, s is odd, and h is odd. See Figure 2. 0 1. 15 14. 2. 13. 3. 12. 4. 5. 11. 6. 10 7. 9 8. Figure 2: The directed chordal ring network DCR(16, 3, ~5).. Chen et al. [3] proposed another type of directed chordal ring networks and ← − ← − denoted it as DCR(N, s, h ). A directed chordal ring network DCR(N, s, h ) is a directed graph with N nodes 0, 1, · · · , N − 1 and 3N/2 links of two types: i→i+s i+h. (mod N ) ∀ i = 0, 1, 2, · · · , N − 1,. (mod N ) → i ∀ i = 1, 3, 5, · · · , N − 1,. − → where N is even, s is odd, and h is odd. Chen et al. [3] combined DCR(N, s, h ) ← − and DCR(N, s, h ) and proposed the mixed chordal ring network M CR(N, s, h). 2.

(10) More precisely, a mixed chordal ring network M CR(N, s, h) is a directed graph with N nodes 0, 1, · · · , N − 1 and 2N links types: i → i + s (mod N ) ∀ i = 0, 1, 2, · · · , N − 1, i→i+h. (mod N ) ∀ i = 1, 3, 5, · · · , N − 1,. i + h (mod N ) → i ∀ i = 1, 3, 5, · · · , N − 1, where N is even, s is odd, and h is odd. See Figure 3. 0 1. 15 14. 2. 13. 3. 12. 4. 5. 11. 6. 10 7. 9 8. Figure 3: The mixed chordal ring network M CR(16, 3, 5).. While the diameter of an undirected chordal ring network U CR(N, h) has been − → well studied [1], the diameter of a directed chordal ring network DCR(N, s, h ) and the diameter of a mixed chordal ring network M CR(N, s, h) are not known. In this thesis, we try to find the diameter of a chordal ring network. This thesis is organized as follows. In Section 2, we describe previous results of the chordal ring networks. In section 3, we discuss the isomorphism properties of chordal ring networks. In section 4, we derive the diameter of some directed chordal ring networks and the diameter of some mixed chordal ring networks. 3.

(11) 2. Previous results. In this section, we will briefly review previous results of chordal ring networks. Since most of these results depend on double-loop networks, we first define what is a double-loop network. Adouble-loop network DL(N, a, b) is a directed graph with N nodes 0, 1, · · · , N − 1 and 2N links: i → i + a (mod N ) ∀ i = 0, 1, 2, · · · , N − 1, i → i + b (mod N ) ∀ i = 0, 1, 2, · · · , N − 1. Fiol et al. [5] proved that DL(N, a, b) is strongly connected if and only if gcd(N, a, b) = 1. For surveys of the double-loop networks, please refer to [7, 9]. When DL(N, a, b) is strongly connected, then we can talk about a minimum distance diagram. This diagram gives a shortest path from node u to node v for any u, v. Since a double-loop network is node-symmetric, it suffices to give a shortest path from node 0 to any other node. Let 0 occupy cell (0,0). Then v occupies cell (i, j) if and only if ia + jb ≡ v (mod N ) and i + j is the minimum among all (i0 , j 0 ) satisfying the congruence, where ≡ means congruent modulo N . Namely, a shortest path from 0 to v is through taking i a-links and j b-links (in any order). Note that in a cell (i, j), i is the column index and j is the row index. A minimum distance diagram includes every node exactly once (in case of two shortest paths, the convention is to choose the cell with the smaller row index, i.e., the smaller j). Wong and Coppersmith [14] proved that the minimum distance diagram is always an L-shape (a rectangle is considered a degeneration). See Figure 4 for two examples. An L-shape is determined by four parameters l, h, p, n as shown in Figure 5. These four parameters are the lengths of four of the six segments on the boundary of the L-shape. For example, DL(9, 4, 1) in Figure 4 has l = 5, h = 3, p = 3, and n = 2. The diameter of a network is the maximum distance over all node-pairs; it is the maximum transmission delay between two stations. Arden and Lee [1] derived the 4.

(12) 2. 6. 3. 4. 5. 1. 5. 6. 7. 8. 0. 4. 0. 1. 2. 8. 3. 7. DL(9, 1, 6). DL(9, 4, 1). Figure 4: Two examples of L-shapes.. n. p. h l Figure 5: An L-shape with parameters. diameter of an undirected chordal ring network U CR(N, h) and proposed a routing algorithm. Without loss of generality, they assumed that h ≤ N/2. Theorem 1 [1] Let U CR(N, h) be an undirected chordal ring network and i = N e, 4 = d 2(h+1). When. N 2. h−3 2. i<. (mod h + 1). Then the diameter D of U CR(N, h) is given by and ⇒D =i+. h−1 ; 2. ⇒D =i+. h−3 ; 2. ⇒D =i+. h−1 ; 2. −i≤4≤h−i ⇒D =i+. h−3 ; 2. • 4=0 • 1≤4≤ • 4= •. h+5 2. h+1 2. h+3 2. −1. −i. • h−i+1≤4≤h When. i=. h−3 2. • 4=0. ⇒D =i+. h−1 . 2. and ⇒ D = h − 2;. 5.

(13) • 1≤4≤2. ⇒ D = h − 3;. • 3≤4≤h. ⇒ D = h − 2.. i≥. When. h−1 2. and. • 4=0. ⇒ D = 2i + 1;. • 4=1. ⇒ D = 2i − 1;. • 2≤4≤ •. h+5 2. h+3 2. ≤4≤h. ⇒ D = 2i; ⇒ D = 2i + 1.. − → Hwang and Wright [11] proposed the directed chordal ring network DCR(N, 1, h ). − → They observed that by combining two nodes in DCR(N, 1, h ) as a supernode, − → ). DCR(N, 1, h ) is reduced to the double-loop network DL( N2 , 1, 1+h 2 − → Hwang [8] generalized the directed chordal ring network DCR(N, 1, h ) to the − → − → directed chordal ring network DCR(N, s, h ). In [8], Hwang called DCR(N, s, h ) a 1.5 loop network. The 1.5 loop network is derived by allowing the full ring of − → DCR(N, 1, h ) to consist of several subrings instead of a hamiltonian circuit. Hwang proved that − → Lemma 2 [8] A necessary condition for DCR(N, s, h ) to be strongly connected is s is an odd integer. − → Hwang [8] observed that by combining two nodes in DCR(N, s, h ) as a supern− → ); he used ode, DCR(N, s, h ) is reduced to the double-loop network DL( N2 , s, s+h 2 this to prove → − Theorem 3 [8] The diameter of DCR(N, s, h ) = 1 + 2× the diameter of DL( N2 , s,. s+h ). 2. Hwang [8] also proved that 6.

(14) − → Theorem 4 [8] DCR(N, s, h ) is strongly connected if and only if gcd(N, s, h) = 1. → − Theorem 5 [8] DCR(N, s, h ) has a hamiltonian circuit if and only if its corre) does. sponding double-loop network DL( N2 , s, s+h 2 − → ← − Chen et al. [3] combined DCR(N, s, h ) and DCR(N, s, h ) and proposed the mixed chordal ring network M CR(N, s, h). They proved that the mixed chordal ring network also has the above two properties since Theorem 6 [3] M CR(N, s, h) is strongly connected if and only if gcd(N, s, h) = 1. ) Theorem 7 [3] M CR(N, s, h) has a hamiltonian circuit if and only if DL( N2 , s, s+h 2 or ) does. DL( N2 , s, s−h 2 Chen et al. [3] also proved 1. Theorem 8 [3] Let D be the diameter of M CR(N, s, h). Then D ≥ (2N ) 2 + o(N ). Chen et al. [3] observed that by combining two nodes in M CR(N, s, h) as a ); , s+h supernode, M CR(N, s, h) is reduced to the double-loop network DL( N2 , s−h 2 2 they used this to prove ) be , s+h Theorem 9 [3] Let D be the diameter of M CR(N, s, h). Let DL( N2 , s−h 2 2 the corresponding double-loop network of M CR(N, s, h) and assume that the L-shape of DL( N2 ,. 3. s−h s+h , 2 ) 2. has lengths l, h, p, n. Then D ≤ 2 max{l, h} − 1.. Isomorphism. Two directed graphs G1 and G2 are isomorphic if there is a bijection function f from V (G1 ) to V (G2 ) such that u → v is a link in E(G1 ) if and only if f (u) → f (v) is a link in E(G2 ). When G1 and G2 are isomorphic, we will write G1 ∼ = G2 . Note that unless otherwise specified, all the nodes in this thesis are considered to be taken modular N . That is, node i + 1 is the node i + 1 (mod N ) and node i + h is the node i + h (mod N ). We now prove 7.

(15) −−−−→ ← − Theorem 10 DCR(N, s, h ) ∼ = DCR(N, s, N − h). ← − Proof. By definition, DCR(N, s, h ) is a directed graph with N nodes 0, 1, · · · , N − 1 and 3N/2 links of two types: i→i+s i+h. (mod N ) ∀ i = 0, 1, 2, · · · , N − 1,. (mod N ) → i ∀ i = 1, 3, 5, · · · , N − 1,. −−−−→ where N is even, s is odd, and h is odd. By definition, DCR(N, s, N − h) is a directed graph with N nodes 0, 1, · · · , N − 1 and 3N/2 links of two types: i→i+s. (mod N ) ∀ i = 0, 1, 2, · · · , N − 1,. i→i+N −h. (mod N ) ∀ i = 1, 3, 5, · · · , N − 1,. where N is even, s is odd, and h is odd. Let f be a function from the nodes of ← − −−−−→ DCR(N, s, h ) to the nodes DCR(N, s, N − h) such that f (i) = i + 1. (mod N ) ∀ i = 0, 1, 2, · · · , N − 1.. ← − First consider the following type of links in DCR(N, s, h ): i → i + s (mod N ) ∀ i = 0, 1, 2, · · · , N − 1. Since f (i) = i + 1 (mod N ) and f (i + s) = i + 1 + s (mod N ), it is clear that −−−−→ f (i) → f (i + s) is a link in DCR(N, s, N − h). Now consider the following type of ← − links in DCR(N, s, h ): i+h. (mod N ) → i ∀ i = 1, 3, 5, · · · , N − 1.. Note that f (i + h) = i + h + 1 (mod N ) and f (i) = i + 1 (mod N ). Since i is odd and h is odd and N is even, i + h + 1 (mod N ) is odd. By the definition of −−−−→ DCR(N, s, N − h), the node i + h + 1 (mod N ) has a link to (i + h + 1 (mod N )) + N − h (mod N ), which is the node i + 1 (mod N ). Thus f (i + h) → f (i) is a −−−−→ link in DCR(N, s, N − h). From the above, we have proved that if u → v is a 8.

(16) ← − −−−−→ link in DCR(N, s, h ), then f (u) → f (v) is a link in DCR(N, s, N − h). Note −−−−→ ← − that DCR(N, s, N − h) has the same number of links as DCR(N, s, h ). Thus if −−−−→ ← − f (u) → f (v) is a link in DCR(N, s, N − h), then u → v is a link in DCR(N, s, h ). ← − −−−−→ Hence DCR(N, s, h ) ∼ = DCR(N, s, N − h). Similarly, we have − → ←−−−− Theorem 11 DCR(N, s, h ) ∼ = DCR(N, s, N − h). We now prove Theorem 12 M CR(N, s, h) ∼ = M CR(N, s, N − h). − → Proof. By definition, M CR(N, s, h) is the combination of DCR(N, s, h ) and ← − −−−−→ DCR(N, s, h ). Also, M CR(N, s, N − h) is the combination of DCR(N, s, N − h) ←−−−− → − ←−−−− and DCR(N, s, N − h). By Theorem 11, DCR(N, s, h ) ∼ = DCR(N, s, N − h). By −−−−→ ← − Theorem 10, DCR(N, s, h ) ∼ = M CR(N, = DCR(N, s, N − h). Thus M CR(N, s, h) ∼ s, N − h).. Theorem 13 Suppose gcd(N, s) = 1. Then − → − → DCR(N, s, h ) ∼ = DCR(N, 1, h1 ), where h1 is the unique integer in {1, 2, · · · , N − 1} satisfying h1 s ≡ h. (mod N ).. Proof. Since gcd(N, s) = 1, we have {i × s (mod N ) : i = 0, 1, 2, · · · , N − 1} = {0, 1, 2, · · · , N − 1}. → − Consider the nodes s and s + h in DCR(N, s, h ). Suppose (3.1). s+h=k×s. (mod N ) 9.

(17) for some integer k in {0, 1, , · · · , N − 1}. Let h1 = k − 1. (mod N ).. Since s 6= s + h, we have k 6= 1 and h1 6= 0. Since h1 s ≡ (k − 1) × s (mod N ), by (3.1), we have h1 s ≡ h (mod N ). From the above, h1 is the unique integer in {1, 2, · · · , N − 1} such that h1 s ≡ h. (mod N ).. − → Let f be a function from the nodes of DCR(N, s, h ) to the nodes DCR(N, 1, − → h 1 ) such that f (i × s) = i ∀ i = 0, 1, 2, · · · , N − 1. − → First consider the following type of links in DCR(N, s, h ): i → i + s (mod N ) ∀ i = 0, 1, 2, · · · , N − 1. Suppose i = m×s (mod N ) for some integer m in {0, 1, , · · · , N −1}. Then f (i) = m and f (i + s) = f (ms + s) = f ((m + 1)s) = m + 1. Since m → m + 1 is a link in − → → − DCR(N, 1, h 1 ), it is clear that f (i) → f (i + s) is a link in DCR(N, 1, h 1 ). − → Now consider the following type of links in DCR(N, s, h ): i → i + h (mod N ) ∀ i = 1, 3, 5, · · · , N − 1. Let i be an odd integer in {1, 3, 5, · · · , N − 1}. Suppose i=m×s. (mod N ). for some integer m in {0, 1, , · · · , N − 1}. Since i is odd, m is odd. Consider the set − → of nodes {0, 1, 2, · · · , N − 1} of DCR(N, s, h ). Since {0, 1, 2, · · · , N − 1} = {i × s (mod N ) : i = 0, 1, 2, · · · , N − 1} − → and i + h is a node in DCR(N, s, h ), we have i + h ≡ q × s (mod N ) 10.

(18) for some integer q in {0, 1, 2, · · · , N − 1}. Then f (i) = m and f (i + h) = q. Since i = m × s (mod N ) and i + h ≡ q × s (mod N ), we have (q − m) × s ≡ h (mod N ). Since (q − m) × s ≡ h (mod N ) and h1 s ≡ h (mod N ), we have (q − m) × s ≡ h1 s. (mod N ).. Thus ms + h1 s ≡ qs. (mod N ).. Since gcd(N, s) = 1, m + h1 ≡ q. (mod N ).. − → Since m is odd, there is a link m → m + h1 in DCR(N, 1, h 1 ); i.e., m → q is a link − → → − in DCR(N, 1, h 1 ). That is, f (i) → f (i + h) is a link in DCR(N, 1, h 1 ). − → ¿From the above, we have proved that if u → v is a link in DCR(N, s, h ), then − → − → f (u) → f (v) is a link in DCR(N, 1, h 1 ). Note that DCR(N, 1, h 1 ) has the same − → − → number of links as DCR(N, s, h ). Thus if f (u) → f (v) is a link in DCR(N, 1, h 1 ), − → − → → − then u → v is a link in DCR(N, s, h ). Hence DCR(N, s, h ) ∼ = DCR(N, 1, h 1 ). Similarly, we have Theorem 14 Suppose gcd(N, s) = 1. Then ← − − → DCR(N, s, h ) ∼ = DCR(N, 1, h1 ), where h1 is the unique integer in {1, 2, · · · , N − 1} satisfying h1 s ≡ −h. (mod N ).. 11.

(19) −−−−→ ← − Proof. By Theorem 10, DCR(N, s, h ) ∼ = DCR(N, s, N − h). By Theorem 13, −−−−→ − → DCR(N, s, N − h) ∼ = DCR(N, 1, h1 ), where h1 is the unique integer in {1, 2, · · · , N − 1} satisfying h1 s ≡ N − h. (mod N ).. Thus we have this theorem. Furthermore, we have Theorem 15 Suppose gcd(N, s) = 1. Then M CR(N, s, h) ∼ = M CR(N, 1, h1 ), where h1 is the unique integer in {1, 2, · · · , N − 1} satisfying h1 s ≡ h. (mod N ).. − → ← − Proof. M CR(N, s, h) is the combination of DCR(N, s, h ) and DCR(N, s, h ). By Theorem 13, − → − → DCR(N, s, h ) ∼ = DCR(N, 1, h1 ), where h1 is the unique integer in {1, 2, · · · , N − 1} satisfying h1 s ≡ h. (mod N ).. and by Theorem 14, ← − − → DCR(N, s, h ) ∼ = DCR(N, 1, h2 ), where h2 is the unique integer in {1, 2, · · · , N − 1} satisfying h2 s ≡ −h. (mod N ).. Thus h2 s ≡ −h (mod N ); i.e., −h2 s ≡ h (mod N ). Hence h1 ≡ −h2. (mod N ).. 12.

(20) ←−−−− − → By Theorem 11, DCR(N, 1, h2 ) ∼ = DCR(N, 1, N − h2 ). Since h1 ≡ −h2 (mod N ), we have ← − ←−−−− ←−−−− ← − DCR(N, s, h ) ∼ = DCR(N, 1, h1 ). = DCR(N, 1, N + h1 ) ∼ = DCR(N, 1, N − h2 ) ∼ We have this theorem. Before going further, we describe how to transform the directed chordal ring − → ). For each node network DCR(N, s, h ) into the double-loop network DL( N2 , s, s+h 2 − → i in DCR(N, s, h ), there is a link from i to i + s. Since s is odd, when i is even, i + s is odd. For each i in {0, 2, 4, · · · , N − 2}, merge the pair of nodes i and i + s as a supernode and denote it by (i/2)∗ . Let the set of supernodes {(i/2)∗ : i = 0, 2, 4, · · · , N − 2} be the set of nodes of the double-loop network. See Figure 6 for an illustration. The set of links of the double-loop network is derived as follows. For each node (i/2)∗ in the double-loop network, since − → i + s → i + 2s is a link in DCR(N, s, h ) and − → i + s → i + s + h is a link in DCR(N, s, h ), in the double-loop network, (i/2)∗ → (i/2 + s)∗ is a link and (i/2)∗ → (i/2 + (s + h)/2)∗ is a link. − → ). From the above, DCR(N, s, h ) is transformed into the double-loop network DL( N2 , s, s+h 2 See Figure 7 for an example.. 13.

(21) (i / 2)*. i. i+s. means a supernode.. i + 3s. i 2s. i + 2s + h. i+s+h. (i / 2 + ( s + h) / 2)*. (i / 2 + s )*. Figure 6: i is even, the links (i/2)∗ → (i/2 + s)∗ and (i/2)∗ → (i/2 + (s + h)/2)∗ .. 8. 16. 3. 6. 12. 17. 4. 8. 13. 2. 7. 3. 4. 9. 14. 1. 6. 11. 0. 5. 1. 0. 5. 10. 15. 2. 7. (a). (b). ~ into Figure 7: (a) The L-shape of DL(9, 5, 2). (b) Transforming DCR(18, 5, 17) DL(9, 5, 2).. We now describe how to transform the directed chordal ring network DCR(N, s, − → − → ). For each node i in DCR(N, s, h ), h ) into the double-loop network DL( N2 , s, h−s 2 there is a link from i to i + s. Since s is odd, when i is odd, i + s is even. For each i in {1, 3, 5, · · · , N − 1}, merge the pair of nodes i and i + s as a supernode and denote it by ((i − 1)/2)∗ . Let the set of supernodes {((i − 1)/2)∗ : i = 1, 3, 5, · · · , N − 1} be the set of nodes of the double-loop network. See Figure 8 for an illustration. The set of links of the double-loop network is derived as follows. For each node 14.

(22) ((i − 1)/2)∗ in the double-loop network, since − → i + s → i + 2s is a link in DCR(N, s, h ) and → − i → i + h is a link in DCR(N, s, h ), in the double-loop network, ((i − 1)/2)∗ → ((i − 1)/2 + s)∗ is a link and ((i − 1)/2)∗ → ((i − 1)/2 + (h − s)/2)∗ is a link. − → From the above, DCR(N, s, h ) is transformed into the double-loop network DL( N2 , s,. h−s ). 2. See Figure 9 for an example. ((i 1) / 2)* i. is. means a supernode. ish. ih. i 2s. ((i 1) / 2 (h s ) / 2)*. i 3s. ((i 1) / 2 s )*. Figure 8: i is odd, the links ((i − 1)/2)∗ → ((i − 1)/2 + s)∗ and ((i − 1)/2)∗ → ((i − 1)/2 + (h − s)/2)∗ . ← − −−−−→ −−−−→ Furthermore, since DCR(N, s, h ) ∼ = DCR(N, s, N − h) and DCR(N, s, N − h) ) ) (when i is even) and into DL( N2 , s, −s−h can be transformed into DL( N2 , s, s−h 2 2 ← − (when i is odd), DCR(N, s, h ) can be transformed into the double-loop networks ) and DL( N2 , s, DL( N2 , s, s−h 2. −s−h ). 2. As for M CR(N, s, h), we describe how to transform M CR(N, s, h) into the ). For each node i in M CR(N, s, h) where i , s+h double-loop network DL( N2 , s−h 2 2 15.

(23) 3. 8. 4. 7. 12. 17. 4. 9. 14. 6. 2. 7. 13. 0. 5. 10. 15. 2. 0. 5. 1. 1. 6. 11. 16. 3. 8. (a). (b). ~ into Figure 9: (a) The L-shape of DL(9, 5, 6). (b) Transforming DCR(18, 5, 17) DL(9, 5, 6).. is odd, there is a link from i to i + h. Since h is odd, i + h is even. For each i in {1, 3, 5, · · · , N − 1}, merge the pair of nodes i and i + h as a supernode and denote it by ((i − 1)/2)∗ . Let the set of supernodes {((i − 1)/2)∗ : i = 1, 3, 5, · · · , N − 1} be the set of nodes of the double-loop network. The set of links of the double-loop network is derived as follows. For each node ((i − 1)/2)∗ in the double-loop network, since i → i + s is a link in M CR(N, s, h) and i + h → i + h + s is a link in M CR(N, s, h), in the double-loop network, ((i − 1)/2)∗ → ((i − 1)/2 + (s − h)/2)∗ is a link and ((i − 1)/2)∗ → ((i − 1)/2 + (s + h)/2)∗ is a link. From the above, M CR(N, s, h) is transformed into the double-loop network DL( N2 , s−h s+h , 2 ). 2. 16.

(24) − → Theorem 16 Given a directed chordal ring network DCR(N, s, h ), if gcd(k, N ) = 1, then − → − → DCR(N, s, h ) ∼ = DCR(N, ks, k h ). → − ). Proof. From previous discussion, DCR(N, s, h ) corresponds to DL( N2 , s, s+h 2 Note that if gcd(k, N ) = 1, then DL(N, a, b) ∼ = DL(N, ka, kb); see [6]. Since )). Since )∼ gcd(k, N ) = 1, gcd(k, N/2) = 1. Thus DL( N2 , s, s+h = DL( N2 , ks, k( s+h 2 2 ) corresponds to ) and DL( N2 , ks, ks+kh )) is exactly DL( N2 , ks, ks+kh DL( N2 , ks, k( s+h 2 2 2 − → DCR(N, ks, kh), we have this theorem. Similarly, we have ← − Theorem 17 Given a directed chordal ring network DCR(N, s, h ), if gcd(k, N ) = 1, then ← − ← − DCR(N, s, h ) ∼ = DCR(N, ks, kh). Theorem 18 Given a mixed chordal ring network M CR(N, s, h), if gcd(k, N ) = 1, then M CR(N, s, h) ∼ = M CR(N, ks, kh). − → ← − Proof. Since M CR(N, s, h) is the combination of DCR(N, s, h ) and DCR(N, s, h ). By Theorem16 and Theorem17, we have M CR(N, s, h) ∼ = M CR(N, ks, kh).. Corollary 19 M CR(N, s, h) ∼ = M CR(N, −s, −h). Proof. Take k = −1. Then gcd(k, N ) = 1. By Theorem 18, M CR(N, s, h) ∼ = M CR(N, −s, −h). Corollary 20 M CR(N, s, h) ∼ = M CR(N, −s, h). 17.

(25) Proof. By Theorem 12 and Corollary 19, M CR(N, s, h) ∼ = M CR(N, s, N − h) ∼ = M CR(N, −s, −N + h). Since M CR(N, −s, −N + h) ∼ = M CR(N, −s, h), we have this corollary. Let’s look at an example. By Corollary 20, M CR(16, 3, 5) ∼ = M CR(16, 13, 5). Since gcd(16, 13) = 1, by Theorem 15, we have M CR(16, 13, 5) ∼ = M CR(16, 1, 9). Thus the mixed chordal ring networks in Figure 3 and Figure 10 are isomorphic. 0 1. 15 14. 2. 13. 3. 12. 4. 5. 11. 6. 10 7. 9 8. Figure 10: M CR(16, 1, 9); it is isomorphic to Figure 3.. 4. The diameter of DCR(N, s, ~h) and M CR(N, s, h). Note that the diameter of a double-loop network DL(N, a, b) can be computed in O(log N ) time using the Cheng-Hwang algorithm [4]. Therefore, by Theorem 3 the − → diameter of DCR(N, s, h ) can be derived in O(log N ) time. However, unless we perform the Cheng-Hwang algorithm, the diameter of DCR(N, s, h) is not known.. 18.

(26) − → In this section, we will derive of the diameter of some DCR(N, s, h ) directly. We will also derive the diameter of some M CR(N, s, h) directly. → − In the previous section, we have shown how to transform DCR(N, s, h ) into DL( N2 , s,. s+h ). 2. Recall that Wong and Coppersmith [14] proved that the minimum. distance diagram of a double-loop network is an L-shape. Let d(k) denote the number of cells (i, j) in an L-shape of a double-loop network with i + j = k. Hwang and Xu [12] defined two double-loop networks to be equivalent if they have the same d(k) for every k. Note that two equivalent double-loop networks have the same diameter. In [13], R¨odseth proved that DL(N, a, b) is equivalent to DL(N, N − a, b − a). In [2], Chen and Hwang proved that DL(N, N − a, b − a) is equivalent to DL(N, a, a − b) and thus DL(N, a, b) is equivalent to DL(N, a, a − b). For example, DL(9, 1, 7) is equivalent to DL(9, 1, −6), which is DL(9, 1, 3). See Figure 11.. 5. 6. 6. 7. 8. 7. 8. 3. 4. 5. 0. 1. 0. 1. 2. 2. 3. 4. DL(9,1,7). DL(9,1,3). Figure 11: Two equivalent L-shapes.. We have the following theorem. − → Theorem 21 Let D1 be the diameter of DCR(N, s, h ) and D2 be the diameter of ← − DCR (N, s, h ). Then D1 = D2 . → − ) Proof. The corresponding double-loop network of DCR(N, s, h ) is DL( N2 , s, s+h 2 ← − ). and the corresponding double-loop network of DCR(N, s, h ) is DL( N2 , s, s−h 2 ) is equivalent to DL( N2 , s, s − Note that DL( N2 , s, s+h 2 s+h ) 2. s+h ). 2. Since DL( N2 , s, s −. ). Thus ) is equivalent to DL( N2 , s, s−h ), DL( N2 , s, s+h is exactly DL( N2 , s, s−h 2 2 2 19.

(27) ) have the same diameter. By Theorem 3, we can ) and DL( N2 , s, s−h DL( N2 , s, s+h 2 2 have D1 = D2 . Let’s see an application of the above theorems. It can be seen from Figure 11 − → that the diameter of DL(9, 1, 7) is 4. By Theorem 3, the diameter of DCR(18, 1, 13) ← − is 9. By Theorem 21, the diameter of DCR(18, 1, 13) is also 9. By Theorem 11, the − → diameter of DCR(18, 1, 5 ) is also 9. − → Theorem 22 Let D be the diameter of DCR(N, s, h ). If s = h, then D = N − 1. → − Proof. When s = h, the corresponding double-loop network of DCR(N, s, h ) is DL(N/2, s, s), whose diameter is N/2−1. Thus by Theorem 3, D = 1+2(N/2−1) = N − 1. − → Theorem 23 Let D be the diameter of DCR(N, s, h ). If s + h = N , then D = N − 1. − → Proof. When s+h = N , the corresponding double-loop network of DCR(N, s, h ) is DL(N/2, s, 0), whose diameter is N/2−1. Thus by Theorem 3, D = 1+2(N/2−1) = N − 1. ← − Corollary 24 Let D be the diameter of DCR(N, s, h ). If s = h or s + h = N , then D = N − 1. Proof. This corollary follows from Theorem 21, Theorem 22, and Theorem 23. → − In the following, we try to derive the diameter of DCR(N, 1, h ). Recall that h − → is odd. Let D be the diameter of DCR(N, 1, h ) and let d(u, v) be the length of the shortest path from u to v. Let Di = max{d(i, v) : v ∈ {0, 1, · · · , N − 1}} for i = 0, 1, · · · , N − 1. We have following properties. 20.

(28) Lemma 25 D = D0 . Proof. In a directed chordal ring network, all even numbered nodes are symmetric, and all odd numbered nodes are symmetric, too. Thus D = max{D0 , D1 }. Suppose D1 > D0 . Let i be a node such that d(1, i) = D1 . Then d(1, i) > d(0, i). Note that node 0 has only one link 0 → 1 going out from it. Therefore the shortest path from node 0 to node i consists of the link 0 → 1 and a shortest path from node 1 to node i. Thus d(0, i) = 1 + d(1, i); this contradicts with the assumption that d(1, i) > d(0, i). Hence D1 ≤ D0 and therefore D = D0 . − → We divide the N nodes of DCR(N, 1, h ) into dN/(h + 1)e groups, each group contains h + 1 nodes (except possibly the last group). For i = 1, 2, · · · , dN/(h + 1)e − 1, the i-th group contains nodes {(i − 1)(h + 1) + 1, (i − 1)(h + 1) + 2, · · · , i(h + 1)}. The last group (i.e., the dN/(h + 1)e-th group) contains nodes {(dN/(h + 1)e − 1)(h + 1) + 1, (dN/(h + 1)e − 1)(h + 1) + 2, · · · , N − 1, 0}. For convenience, we will say that (i − 1)(h + 1) + 1 is the first node of the i-th group. See Figure 12 for an illustration. − → Lemma 26 Let x be a node of DCR(N, 1, h ) such that x 6= 0. Let x0 be the first node of the group containing x (i.e., the first node of the dx/(h+1)e-th group). Then there exists a shortest path P from node 0 to x that passes through x0 . Proof. Let P be an arbitrary shortest path from 0 to x. If P passes through x0 , then we are done. In the following, assume that P does not pass through x0 . Suppose P contains i 1-links and j h-links. Clearly, i ≥ j. Let P 0 be a path from 0 to x derived by rearranging the links in P so that every 1-link follows immediately an h-link unless there is no more h-links. (For example, if P contains five 1-links 21.

(29) N-1. 1. 0. 4(h+1). x. h+1. 3(h+1). 2(h+1). Figure 12: The nodes of DCR(18, 1, ~3) are divided into d18/(3 + 1)e groups.. and three h-links, then the links in P 0 are: a 1-link, an h-link, a 1-link, an h-link, a 1-link, an h-link, a 1-link, and a 1-link.) Note that P 0 is also a path from 0 to x. Furthermore, P 0 is also shortest since it has the same number of links as P . If j ≥ dx/(h + 1)e, then clearly P 0 passes through x0 . If j < dx/(h + 1)e, then after P 0 passes through the j-th group, all the remaining links in P 0 are 1-links; thus P 0 will also pass through x0 . We have this lemma. The following lemma is obvious and we omit its proof. Lemma 27 Let x0 be the first node of the i-th group. Then d(0, x0 ) = 2i − 1. Lemma 28 Let X = {x : d(0, x) = D0 }. Then all the elements of X belong to the last two groups. Proof. Suppose this lemma is not true and there is an x ∈ X such that x does not belong to the last two groups. Choose t such that x + t(h + 1) is in the last two groups. Then d(0, x) ≥ d(0, x + t(h + 1)). Let x0 be the first node of the 22.

(30) group containing x and let (x + t(h + 1))0 be the first node of the group containing x + t(h + 1). Set y = x + t(h + 1) and set y 0 = (x + t(h + 1))0 for easy writing. By Lemma 26, there is a shortest path from 0 to x that passes through x0 . Thus d(0, x) = d(0, x0 ) + d(x0 , x). Also by Lemma 26, there is a shortest path from 0 to y that passes through y 0 . Thus d(0, y) = d(0, y 0 ) + d(y 0 , y). Since x does not belong to the last two groups and y belongs to the last two groups, by Lemma 27, d(0, y 0 ) ≥ d(0, x0 ) + 2. Note that d(y 0 , y) = d(x0 , x). Thus d(0, y) ≥ d(0, x) + 2, i.e., d(0, x + t(h + 1)) ≥ d(0, x) + 2. This contradicts with the assumption that d(0, x) ≥ d(0, x + t(h + 1)).. Lemma 29 If 3 ≤ h ≤. N 2. N − 1) + h. − 1 and h + 1 | N , then D = 2( h+1. Proof. Let X = {x : d(0, x) = D0 }. By Lemma 28, all the elements of X belong to the last two groups. Since h + 1 | N , each group contains exactly h + 1 nodes. Let {y, y + 1, y + 2, · · · , y + h} be the set of nodes in the previous group of the last group and let {x, x + 1, x + 2, · · · , x + h} be the set of nodes in the last group. Note that the node x + h is node 0 and the node x + h − 1 is node N − 1. By Lemma 27, we have d(0, x) > d(0, y). Since h + 1 | N , we have d(0, x + 1) > d(0, y + 1), d(0, x + 2) > d(0, y + 2), · · · , d(0, x + h − 1) > d(0, y + h − 1). Moreover, d(0, x) > d(0, y + h). Since h + 1 | N , we have d(0, x) < d(0, x + 1) < d(0, x + 2) < · · · < d(0, x + h − 1). From the above, D0 = d(0, N − 1). By Lemma N − 1) + h, we have this lemma. 25, D = d(0, N − 1). Since d(0, N − 1) = 2( h+1. → − Theorem 30 Let D be the diameter of DCR(N, 1, h ). Then ( N −1 if h = 1 or h = N − 1, D= N 2( h+1 − 1) + h if 3 ≤ h ≤ N2 − 1 and h + 1 | N . 23.

(31) Proof. The case that h = 1 follows from Theorem 22; the case h = N − 1 follows from Theorem 23. The case that 3 ≤ h ≤. N 2. − 1 and h + 1 | N follows from Lemma. 29. → − Let D be the diameter of DCR(N, s, h ). By Theorem 21, D is also the diameter ← − → − of DCR(N, s, h ). Since M CR(N, s, h) is derived by combining DCR(N, s, h ) and ← − − → DCR(N, s, h ), the diameter of DCR(N, s, h ) is an upper bound for the diameter of M CR(N, s, h). One might suspect that the diameter of M CR(N, s, h) is also D. − → Unfortunately, this is not true. For example, the diameter of DCR(18, 1, 5 ) is 9 and the diameter of M CR(18, 1, 5) is 5. → − Recall that Hwang [8] proved that the diameter of DCR(N, s, h ) = 1 + 2× ). In the following, we give an example to show that the diameter of DL( N2 , s, s+h 2 this is not true for a mixed chordal ring network. The corresponding double-loop ). Consider M CR(18, 1, 5); its corre, s+h network of M CR(N, s, h) is DL( N2 , s−h 2 2 sponding double-loop network is DL(9, 7, 3). The diameter of DL(9, 7, 3) is 4, but the diameter of M CR(18, 1, 5) is 5, which is not equal to 1 + 2 × 4. In the remaining part of this thesis, we shall derive the diameter of some mixed chordal ring networks. Theorem 31 Let D be the diameter of M CR(N, s, h). If s = h or s + h = N , then D = N − 1. Proof. First consider the case that s = h. Let d(u, v) denote the length of the shortest path from u to v. Since gcd(N, s, h) = 1, we have gcd(N, s) = 1. Therefore M CR(N, s, s) has a hamiltonian circuit 0, s, 2s, 3s, · · · , (N −1)s and hence d(u, v) ≤ N − 1 for every u and v. Thus D ≤ N − 1. On the other hand, the shortest path from s to 0 is s, 2s, 3s, · · · , (N − 1)s, 0, which is of length N − 1. Thus D ≥ N − 1 and therefore D = N − 1. Now consider the case that s+h = N . From the above discussion, the diameter of M CR(N, h, h) is N −1. When s+h = N , M CR(N, −s, h) is exactly M CR(N, h, h). 24.

(32) By Corollary 20, M CR(N, −s, h) ∼ = M CR(N, s, h). Thus when s + h = N , the diameter of M CR(N, s, h) is N − 1. Theorem 32 Let D be the diameter of M CR(N, s, h). If h = N/2, then D = N/2. Proof. Let d(u, v) be the length of the shortest path from u to v. Let Di = max{d(i, v) : v ∈ {0, 1, · · · , N − 1}} for i = 0, 1, · · · , N − 1. In a mixed chordal ring network, all even numbered nodes are symmetric, and all odd numbered nodes are symmetric, too.. Thus. D = max{D0 , D1 }. First consider D0 . For every even node i, since i → i − h and i − h → i, we can view the two nodes i and i−h as a supernode. Thus there are total N/2 supernodes: {(i(h + s), i(h + s) − h) : i = 0, 1, · · · , N/2 − 1}. See Figure 13 for an illustration. Consider the i-th supernode (i(h + s), i(h + s) − h) and the two s-links going out from this supernode: i(h + s) → i(h + s) + s and i(h + s) − h → i(h + s) − h + s. Note that node i(h + s) + s is node (i + 1)(s + h) − h. Moreover, node i(h + s) − h + s is node (i + 1)(s + h) − 2h; since h = N/2, node (i+1)(s+h)−2h is node (i+1)(s+h). The two nodes (i+1)(s+h) and (i+1)(s+h)−h are in the (i + 1)-th supernode. Thus both of the two s-links going out from the i-th supernode go to the (i + 1)-th supernode. Now consider the distance from node 0 to the two nodes in the i-th supernode (i(h + s), i(h + s) − h). Then ½ i + 1 if i is odd d(0, i(h + s)) = i if i is even and. ½ d(0, i(h + s) − h) =. i if i is odd . i + 1 if i is even. Since h is odd and h = N/2, it is impossible that 2 |. N . 2. Hence 2 -. N 2. and. D0 = max{d(0, (N/2 − 1)(h + s) − h), d(0, (N/2 − 1)(h + s))} = (N/2 − 1) + 1 = N/2. 25.

(33) 0. 5. 8. 3. 6. 1. 4. 9. 2. 7. 1. 6. 9. 4. 7. 2. 5. 0. 3. 8. Figure 13: The diameter of M CR(10, 3, 5) is 5.. Now consider D1 . For every odd node i, since i → i + h and i + h → i, we can view the two nodes i and i+h as a supernode. Thus there are total N/2 supernodes: {(1 + i(h + s), 1 + i(h + s) + h) : i = 0, 1, · · · , N/2 − 1}. See Figure 13 for an illustration. Consider the i-th supernode (1 + i(h + s), 1 + i(h + s)+h) and the two s-links going out from this supernode: 1+i(h+s) → 1+i(h+s)+s and 1 + i(h + s) + h → 1 + i(h + s) + h + s. Note that node 1 + i(h + s) + s is node 1 + (i + 1)(s + h) − h. Moreover, node 1 + i(h + s) + h + s is node 1 + (i + 1)(s + h). Since h = N/2, node 1 + (i + 1)(s + h) − h is node 1 + (i + 1)(s + h) + h. The two nodes 1 + (i + 1)(s + h) and 1 + (i + 1)(s + h) + h are in the (i + 1)-th supernode. Thus both of the two s-links going out from the i-th supernode go to the (i + 1)-th supernode. Now consider the distance from node 1 to the two nodes in the i-th supernode (1 + i(h + s), 1 + i(h + s) − h). Then ½ d(1, 1 + i(h + s)) = and. i + 1 if i is odd i if i is even. ½ d(1, 1 + i(h + s) + h) =. i if i is odd . i + 1 if i is even. Since h is odd and h = N/2, it is impossible that 2 |. N . 2. Hence 2 -. N 2. and D1 =. max{d(1, 1 + (N/2 − 1)(h + s) + h), d(1, 1 + (N/2 − 1)(h + s)} = (N/2 − 1) + 1 = N/2. 26.

(34) From the above, we have D = max{D0 , D1 } = N/2.. References [1] B. W. Arden and H. Lee, Analysis of chordal ring networks, IEEE Trans. Comput. 30 (1981) 291-295. [2] C. Y. Chen and F. K. Hwang, Equivalent Nondegenerate L-Shapes of DoubleLoop Neworks, Networks. 36(2) (2000), 118-125. [3] S. K. Chen, F. K. Hwang and Y. C. Liu, Some combinatorial properties of mixed chordal rings, J. Interconnection Networks 4 (2003), 3-16. [4] Y. Cheng and F. K. Hwang, Diameters of weighted double loop networks, J. Algorithms 9 (1988), 401-410. [5] M. A. Fiol, M. Valero, J. L. A. Yebra, I. Alegre, and T. Lang, Optimization of double-loop structures for local networks, in Proc. XIX Int. Symp. MIMI’82, Paris, France (1982), 37-41. [6] M. A. Fiol, J. L. A. Yebra, I. Alegre, and M. Valero, A discrete optimization problem in local networks and data alignment, IEEE Trans. Comput. C-36 (1987), 702-713. [7] F. K. Hwang, A survey on double-loop networks, in Reliability of Computer and Communication Networks, Eds: F. Roberts, F. K. Hwang and C. Monma, AMS series (1991), 143-151. [8] F. K. Hwang, The 1.5-Loop Network, unpublished manuscript. [9] F. K. Hwang, A complementary survey on double-loop networks, Theoret. Comput. Sci. A 263 (2001), 211-229.. 27.

(35) [10] F. K. Hwang and S. K. Chen, The 1.5-loop network and the mixed 1.5-loop network, SIROCCO (2000), 297-306. [11] F. K. Hwang and P. E. Wright, Survival reliability of some double-loop netwroks and chordal rings, IEEE Trans. Comput. 44 (1995) 1468-1471. [12] F. K. Hwang and Y. H. Xu, Double loop networks with minimum delay, Disc. Math. 66 (1987), 109-118. [13] O. J. R¨odseth,Weighted multi-connected loop networks, Discr Math.. 148. (1996), 161-173. [14] C. K. Wong and D. Coppersmith, A combinatorial problem related to multimodule memory organizations, J. Assoc. Comput. Mach. 21 (1974), 392-402.. 28.

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