Fault-Free Ring Embedding in Faulty Wrapped Butterfly Graphs

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(1)FAULT-FREE RING EMBEDDING IN FAULTY WRAPPED BUTTERFLY GRAPHS ∗ Chang-Hsiung Tsai, Tyne Liang and Lih-Hsing Hsu Department of Computer and Information Science National Chiao Tung University, Hsinchu, Taiwan, R.O.C. Email: [email protected] ABSTRACT In this paper, we study cycle embedding in a faulty wrapped butterfly BFn with at most two faults in vertices and/or edges. Let F be a subset of V (BFn ) ∪ E(BFn ) with |F | ≤ 2. Let fv denote |F ∩ V (BFn )|. In this paper, we prove that BFn − F contains a cycle of length n × 2n − 2fv . Moreover, BFn − F contains a cycle of length n × 2n − fv if n is an odd integer. In other words, BFn − F contains a hamiltonian cycle if n is an odd integer. 1 INTRODUCTION Performance of the distributed system is significantly determined by the choice of the network topology. The hypercube (binary n-cube) is one of the most popular interconnection networks. It has been used to design various commercial multiprocessor machines. One basic drawback with hypercubes is that the degree of nodes increases with the number of nodes. Hence it is not suitable to apply hypercubes to the area layout from the viewpoint of VLSI implementation. Among all networks of fixed degrees, wrapped butterfly network is one of the most promising networks due to its nice topological properties. On the other hand, cycle (ring) contains several attractive properties such as simplicity, extensibility, and feasible implementation. Hence embedding a cycle into wrapped butterfly network has received many researchers’ efforts in recent years [1, 3, 5, 6, 8]. To embed a cycle into a faulty butterfly network, it is desirable to isolate those faulted components from the rest ones so that a maximal-length cycle can be still embedded. Assume that F ⊂ V (BFn ) ∪ E(BFn ) be the fault set with |F | ≤ 2. In [6], Vadapalli and Srimani verified that BFn − F contains a cycle of length n × 2n − 2 if F is a set with only one vertex and that BFn − F contains a cycle of length n × 2n − 4 if F is a set with two vertices. In [3], Hwang and Chen proved that there still exists a cycle of length n × 2n in a BFn − F where F is a subset of E(BFn ). In other words, BFn − F remains hamiltonian with at most two edges faults. In the previous study of cycle embedding into wrapped butterfly, faults are limited into ei∗ This work was supported in part by the National Science Council of the Republic of China under Contract NSC 89-2115-M-009-020.. ther node faults or edge faults. However some faults on both nodes and edges may occur. Therefore we want to improve the results of [3, 6]. We use fv to denote |F ∩ V (BFn )|. In this paper, we prove that BFn − F contains a cycle of length n × 2n − 2fv . Moreover, BFn − F contains a cycle of length n × 2n − fv if n is an odd integer. In other words, BFn − F contains a hamiltonian cycle if n is an odd integer. In the following section, we discuss some properties of the wrapped butterfly graphs. In section 3, we first present a short proof that BFn − F remains hamiltonian if F is a subset of E(BFn ). Then we prove that BFn − F contains a cycle of length n × 2n − 2 if F is a set with one vertex and one edge. Finally, we prove that BFn − F contains a cycle of length n × 2n − fv if n is an odd integer. 2 WRAPPED BUTTERFLY AND ITS PROPERTIES A graph G = (V, E) consists of a finite set V and a subset E of {(u, v) | u = v, (u, v) is an unordered pair of elements of V }. We call V = V (G) the vertex set of G and E = E(G) the edge set of G. Let F = V1 E1 for E1 ⊂ E and V1 ⊂ V . Weuse G − F to denote the graph G = (V − V1 , (E − E1 ) ((V − V1 ) × (V − V1 ))). The wrapped butterfly (butterfly for short) BFn is a graph with n × 2n vertices such that each vertex is labeled by

(2) a0 a1 . . . an−1 , i with 0 ≤ i ≤ n − 1 and aj ∈ {0, 1} for all 0 ≤ j ≤ n − 1. We say the vertex

(3) a0 a1 . . . an−1 , i is at level i. Edges of BFn are described as follows. Node

(4) a0 a1 . . . ai . . . an−1 , i is adjacent to node

(5) a0 a1 . . . ai . . . an−1 , (i + 1) mod n by a straight edge and ¯i . . . an−1 , (i + 1) mod n by a adjacent to node

(6) a0 a1 . . . a cross edge. Lemma 1 [4] For any integer k with 0 ≤ k < n, the mapping σk from V (BFn ) into V (BFn ) defined by σk (

(7) a0 a1 . . . an−1 , l ) =

(8) ak ak+1 . . . an−1 a0 a1 . . . ak−1 , (l − k) mod n is an automorphism of BFn . Similarly, we can easily obtain the following lemma. Lemma 2 For any integer i with 0 ≤ i < n, the mapping ϕi from V (BFn ) into V (BFn ) defined by.

(9) ϕi (

(10) a0 a1 . . . an−1 , l ) =

(11) a0 a1 . . . a ¯i ai+1 . . . an−1 , l is an automorphism of BFn . Thus, we have the following corollary. Corollary 1 BFn is vertex transitive. In [5], Vadapalli et al. proposed a family of degree four Cayley graphs, Gn . Later, Chen and Lau [2] point out that Gn is isomorphic to BFn . Thus, we can combine all the results of Gn and BFn . Each vertex of Gn is represented by a circular permutation of n different symbols in lexicographic order, where the n symbols are presented in either uncomplemented or complemented form. Let dk , 0 ≤ k ≤ n − 1, denote the kth symbol in the set of n symbols. We use the English alphabets: thus for n = 3, d0 = a, d1 = b, and d2 = c. We use tk to denote either dk or d¯k . Therefore, for n distinct symbols, there are exactly n different cyclic permutations of the symbols in lexicographic order. Moreover, each symbol can be presented in either uncomplemented or complemented form. So the vertex set of Gn has a cardinality of n × 2n . If a0 a1 . . . an−1 denotes the label of an arbitrary vertex and a0 = tk for some integer k, then for all i and 0 ≤ i ≤ n − 1, we have ai = tl where l = k + i (mod n). The edges of Gn are defined by the following four generators in the graph: g(tk tk+1 . . . tn−1 t0 . . . tk−2 tk−1 ) = tk+1 . . . tn−1 t0 . . . tk−1 tk , f (tk tk+1 . . . tn−1 t0 . . . tk−2 tk−1 ) = tk+1 . . . tn−1 t0 . . . tk−1 t¯k , g −1 (tk tk+1 . . . tn−1 t0 . . . tk−2 tk−1 ) = tk−1 tk . . . tn−1 t0 . . . tk−2 , and f −1 (tk tk+1 . . . tn−1 t0 . . . tk−2 tk−1 ) = t¯k−1 tk . . . tn−1 t0 . . . tk−2 . In [7], Wei et al. point out the isomorphism maps the vertex

(12) a0 a1 . . . an−1 , k of BFn into the vertex tk . . . tn−1 t0 . . . tk−1 of Gn , where ti = di if and only if ai = 0, or ti = d¯i if and only if ai = 1. Therefore, throughout this paper, the nodes of the butterfly graph will be labeled in the form of

(13) a0 a1 . . . an−1 , k rather than tk . . . tn−1 t0 . . . tk−1 . Therefore, the four generators g, g −1 , f and f −1 can be rewritten as follows: g(

(14) a0 a1 . . . an−1 , k ) =

(15) a0 a1 . . . an−1 , k + 1 , f (

(16) a0 a1 . . . an−1 , k ) ¯k ak+1 . . . an−1 , k + 1 , =

(17) a0 a1 . . . ak−1 a g −1 (

(18) a0 a1 . . . an−1 , k ) =

(19) a0 a1 . . . an−1 , k − 1 , and f −1 (

(20) a0 a1 . . . an−1 , k ) =

(21) a0 a1 . . . ak−2 a ¯k−1 ak . . . an−1 , k − 1 . Hence the g-edges, (u, g(u)) or u, g −1 (u) for some u ∈ V (BFn ), correspond to the straight edges and the f -edges, (u, f (u)) or u, f −1 (u) for some u ∈ V (BFn ), correspond to the cross edges of BFn .. Lemma 3 f −1 (g(u)) = g −1 (f (u)) for any node u in BFn . Let u be any vertex of BFn . Obviously, g n (u) = u. Moreover,

(22) u, g(u), g 2 (u), . . . , g n (u) = u forms a simple cycle of length n, denoted by Cgu . We call such cycle of BFn a g-cycle at u. It is easy to see that Cgv = Cgu if and only if v ∈ Cgu . Thus all g-cycles form a partition of the straight edges of BFn . There is no g-edge joining vertices of two different g-cycles. Any f -edge joins vertices of two different g-cycles. Obviously, (u, f (u)) joins vertices of Cgu f (u). and Cg. . The following lemma can be proved easily.. Lemma 4 g(u), g −1 (f (u)) is an f -edge joining vertices f (u) and Cg . Moreover, the of Cgu −1 path

(23) u, f (u), g (f (u)) , g(u), u forms a cycle of length 4. Any Cgu contains exactly one vertex at each level. In particular, Cgu contains exactly one vertex at level 0, say (a a ...a. ).

(24) a0 a1 . . . an−1 , 0 . We use Cg 0 1 n−1 as the name for Cgu . Now, we form a new graph BFnG with all the g-cycles of BFn as vertices, two different g-cycles are joined with an edge if and only if there exists an f -edge joining them. The vertex of BFnG corresponding to Cgu is denoted by C¯gu . The following theorem is proved in [5] [6]. Lemma 5 BFnG is isomorphic to the n-dimensional hypercube. Moreover, the set of vertices adjacent to the vertex corresponding to (a a ...a ) Cg 0 1 n−1 is the set of vertices corresponding to the g(¯ a a ...a ) (a a ¯ ...a ) (a a ...¯ a ) cycles in {Cg 0 1 n−1 , Cg 0 1 n−1 , . . . Cg 0 1 n−1 }. Let h = (C¯gu , C¯gv ) be any edge of BFnG . We use X(h) to denote the set of edges in BFn joining vertices of Cgu and Cgv . Using standard counting technique, we have the following two corollaries. Corollary 2 Let h = (C¯gu , C¯gv ) be any edge of BFnG . Then |X(h)| = 2. Moreover, the vertices of edges in X(h) induces a 4-cycle in BFn . f (u). Corollary 3 There is a unique g-cycle, namely Cg , such f (u) that edges of BFn joining vertices between Cgu and Cg are exactly (u, f (u)) and g(u), f −1 (g(u)) . According to Corollaries 2 and 3, any edge h = (C¯gu , C¯gv ) in BFnG induces a unique 4-cycle in BFn , with two f -edges and two g-edges. We use Xf (Cgu , Cgv ) to denote the set of f -edges in this 4-cycle, and Xg (Cgu , Cgv ) to denote the set of g-edges in this cycle..

(25) Lemma 6 Assume that T be any subtree of BFnG . Let CgT denote the graph generated by the edge set    E(Cgu ) ∪ Xf (Cgu , Cgv ). g(u). Corollary 5 There is a unique f -cycle, namely Cf. , such. Cfu. g(u). that edges of BFn joining vertices between and Cf −1 are exactly (u, g(u)), f (u), g (f (u)) , (˜ u , g(˜ u)), and −1 f (˜ u), g (f (˜ u)) .. ¯ u ,C ¯ v )∈E(T ) (C g g. ¯ u ∈V (T ) C g. −. . Xg (Cgu , Cgv ).. ¯ u ,C ¯ v )∈E(T ) (C g g. Then CgT is a cycle of BFn of length n × |V (T )|. Let u =

(26) a0 a1 . . . an−1 , k be any vertex of BFn . We use u ˜ to denote the node

(27) ¯ a0 a ¯1 . . . a ¯n−1 , k . Obviously, f n (u) = u ˜ and f 2n (u) = u. Moreover,

(28) u, f (u), f 2 (u), . . . , f 2n (u) = u forms a simple cycle of length 2n, denoted by Cfu . It is easy to see that all f -cycles form a partition of the cross edges of BFn . There is no f edge joining vertices of two different f -cycles. Any g-edge joins vertices of two different f -cycles. The g-edge (u, g(u)) g(u) joins vertices of Cfu and Cf . The following lemma can be proved easily. Lemma 7 u, g(˜ u)), (f (˜ u), g −1 (f (˜ u))) are also g(f (u), g −1 (f (u))), (˜ g(u) u edges joining vertices of Cf and Cf . Moreover, the paths

(29) u, f (u), g −1 (f (u)) , g(u), u , and

(30) ˜ u, f (˜ u), g −1 (f (˜ u)) , g(˜ u), u˜ , form two 4-cycles in BFn . Any Cfu contains exactly two vertex at each level. Suppose that u is one of the vertex in Cfu at level i. Obviously, the other vertex in Cfu at level i is u˜. Thus, Cfu contains exactly one vertex at level 0, say

(31) a0 a1 . . . an−1 , 0 with (a a ...a ) an−1 = 0. We use Cf 0 1 n−2 as the name for Cgu . Now, F we form a new graph BFn with all the f -cycles of BFn as vertices, two different f -cycles are joined with an edge if and only if there exists a g-edge joining them. The vertex of BFnF corresponding to Cfu is denoted by C¯fu . The following theorem is proved in [5] [6].. According to Corollaries 4 and 5, any edge h = (C¯fu , C¯fv ) induces two 4-cycles in BFn . Let α be an assignment of (C¯fu , C¯fv ) ∈ E(BFnF ) with one of the 4-cycles it induced. We use Yfα (Cfu , Cfv ) to denote the set of f -edges induced by α(h) and Ygα (Cfu , Cfv ) to denote the set of g-edges induced by α(h). Hence |Yfα (Cfu , Cfv )| = |Ygα (Cfu , Cfv )| = 2. Lemma 9 Assume that T is any subset of BFnF . Let CfT,α denote the graph generated by the edge set    E(Cfu ) ∪ Ygα (Cfu , Cfv ) ¯ u ∈V (T ) C f. ¯ u ,C ¯ v )∈E(T ) (C f f. . −. Yfα (Cfu , Cfv ).. ¯ u ,C ¯ v )∈E(T ) (C f f. Then CfT,α is a cycle of BFn of length 2n × |V (T )|. In the following, we introduce three basic cycles B1 , B2 , and B3 . The cycle B1 is constructed as follows: Let a1 =

(32) 00 . . . 0 , 1 . Let P1 be the path a1 , g(a1 ), . . ., g n−2 (a1 ) =.

(33) n. Obviously, a2. a2 .. =.

(34) 00 . . . 0 , n − 1 , f (a2 ).

(35). =. n.

(36) 00 . . . 0 1, 0 = a3 , and f (a3 ) =

(37) 1 00 . . . 0 1, 1 = a4 . Let.

(38).

(39) n−1. n−2. P2 be the path a4 , g(a4 ), . . ., g n−1 (a4 ) = a5 . Obviously, . . . 0 1, 0 and f (a5 ) =

(40) 1 00

(41) . . . 0 , n − 1 = a6 . a5 =

(42) 1 00

(43) n−2. n−1. Let P3 be the path a6 , g −1 (a6 ), . . ., g −(n−1) (a6 ) = a7 . Ob. . . 0 , 0 and f (a7 ) = a1 . Then B1 is viously, a7 =

(44) 1 00

(45) n−1. Lemma 8 BFnF is isomorphic to the (n − 1)-dimensional folded hypercube. Moreover, the set of vertices ad(a a ...a ) jacent to the vertex corresponding to Cf 0 1 n−2 is the set of vertices corresponding to the f -cycles in (¯ a a ...a ) (a a ¯ ...a ) (a a ...¯ a ) ∪ {Cf 0 1 n−2 , Cf 0 1 n−2 , . . . Cf 0 1 n−2 } (¯ a0 a ¯1 ...¯ an−2 ). {Cf. }..

(46) a1 → P1 → a2 , a3 , a4 → P2 → a5 , a6 → P3 → a7 , a1 . Let W1 = V (Cga1 ) ∪ V (Cga3 ) ∪ V (Cga5 ) ∪ V (Cga7 ) and ¯ 1 = {C¯ a1 , C¯ a3 , C¯ a5 , C¯ a7 }. W g g g g The cycle B2 is constructed as follows: Let b1 =

(47) 00 . . . 0 , 1 . Let Q1 be the path b1 , g(b1 ), . . ., g n−2 (b1 ) =.

(48) n. b2 . Obviously, b2 =

(49) 00 . . . 0 , n − 1 and f −1 (b2 ) =.

(50) n.

(51) 00 . . . 0 10, n − 2 = b3 . Let Q2 be the path b3 , g −1 (b3 ),.

(52). Let h = (C¯fu , C¯fv ) be any edge of BFnF . We use Y (h) to denote the set of edges of BFn joining vertices of Cfu and Cfv . Using standard counting technique, we have the following two corollaries.. . . . 0 10, 1 . . ., g −(n−3) (b3 ) = b4 . Obviously, b4 =

(53) 00.

(54). Corollary 4 Let h = (C¯fu , C¯fv ) be any edge of BFnF . Then |Y (h)| = 4. Moreover, the vertices of edges in Y (h) induce two 4-cycles in BFn .. path b5 , g(b5 ), . . ., g n−1 (b5 ) = b6 . Obviously, b6 =

(55) 1 00

(56) . . . 0 10, n−1 and f −1 (b6 ) =

(57) 1 00 . . . 0 , n−2 = b7 ..

(58). n−2. n−2. . . . 0 10, 0 = b5 . Let Q3 be the and f −1 (b4 ) =

(59) 1 00

(60) n−3. n−3. n−1.

(61) Let Q4 be the path b7 , g(b7 ), g 2 (b7 ) = b8 . Then b8 =

(62) 1 00

(63) . . . 0 , 0 and f (b8 ) =

(64) 00 . . . 0 , 1 = b1 . Then B2 is.

(65) n. n−1.

(66) b1 → Q1 → b2 , b3 → Q2 → b4 , b5 → Q3 → b6 , b7 → Q4 → b8 , b1 . Let W2 = V (Cgb1 ) ∪ V (Cgb3 ) ∪ V (Cgb5 ) ∪ ¯ 2 = {C¯gb1 , C¯gb3 , C¯gb5 , C¯gb7 }. V (Cgb7 ) and W Let 2 ≤ j ≤ d − 1. The cycle B3 is constructed as follows: Let c1 =

(67) 00 . . . 0 , 1 . Let R1 be the.

(68) n. path c1 , g(c1 ), . . ., g j−2 (c1 ) = c2 . Obviously, c2 = . . . 0 1 00 . . . 0 , j =

(69) 00 . . . 0 , j − 1 and f (c2 ) =

(70) 00.

(71).

(72).

(73) n. j−1. n−j −j. c3 . Let R2 be the path c3 , g −1 (c3 ), . . ., g (c3 ) = . . . 0 1 00 . . . 0 , 0 and f (c4 ) = c4 . Obviously, c4 =

(74) 00.

(75).

(76) j−1.

(77) 1 00

(78) . . . 0 1 00 . . . 0 , 1 = c5 ..

(79) j−2. n−j. Let R3 be the path c5 ,. n−j. g −1 (c5 ), . . ., g −(n−j+1) (c5 ) = c6 . Obviously, c6 =

(80) 1 00

(81) . . . 0 1 00 . . . 0 , j and f −1 (c6 ) =

(82) 1 00 . . . 0 , j − 1 =.

(83).

(84) j−2. n−j. n−1. c7 . Let R4 be the path c7 , g −1 (c7 ), . . ., g −(n−j+1) (c7 ) = c8 . . . . 0 , 0 and f (c8 ) =

(85) 00 . . . 0 , 1 = c1 . Then c8 =

(86) 1 00.

(87).

(88) n−1. n. Then B3 is

(89) c1 → R1 → c2 , c3 → R2 → c4 , c5 → R3 → c6 , c7 → R4 → c8 , c1 . Then, the length of B3 is 2n + 4. Let W3 = V (Cgc1 ) ∪ V (Cgc3 ) ∪ V (Cgc5 ) ∪ V (Cgc7 ) ¯ 3 = {C¯gc1 , C¯gc3 , C¯gc5 , C¯gc7 }. and W When n = 3, it is observed that b3 = b4 and c1 = c2 . All the vertices of Bi is a proper subset of Wi for every 1 ≤ i ≤ 3. Moreover, the length of Bi is 3n for i = 1, 2. 3. CYCLE EMBEDDING IN A FAULTY WRAPPED BUTTERFLY In this section, we assume that F ⊂ V (BFn ) E(BFn ) with |F | ≤ 2. In the following lemmas, we just state the results and omit the proofs. Lemma 10 For any integer n with n ≥ 3, BFn −F is hamiltonian if F ⊂ E(BFn ) and |F | = 2. Lemma 11 Assume that n ≥ 3. Then BFn − F contains a cycle of length n × 2n − 2 where F consists of a vertex and an edge in BFn . Lemma 12 For any odd integer n with n ≥ 3, BFn − F is hamiltonian where F consists of a vertex and an edge in BFn . Lemma 13 For any odd integer n with n ≥ 3, BFn − F is hamiltonian where F ⊂ V (BFn ) and |F | = 2. Since BFn is hamiltonian for all n ≥ 3, by lemmas 10, 11, 12, 13, and Vadapalli et. al. [6], we have the following theorem.. Theorem 1 Assume that n ≥ 3, F ⊂ V (BFn ) E(BFn ), and |F | ≤ 2. Then BFn − F contains a cycle of length n × 2n − 2|F ∩ V (BFn )|. Moreover, BFn − F contains a hamiltonian cycle if n is an odd integer. References [1] D. Barth and A. Raspaud, Two edge-disjoint hamiltonian cycles in the butterfly graph, Info. Process Lett. 51(1994) 175-179. [2] G. Chen and F.C.M. Lau, Comments on a new family of cayely graph interconnection networks of constant degree four, IEEE Trans. Parallel and Distributed Systems, 8(1997) 1299-1300. [3] S.C. Hwang and G.H. Chen, Cycles in butterfly graphs, Networks 35, 2(2000) 1-11. [4] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes (Morgan Kaufmann, Los Altos, CA, 1992). [5] P. Vadapalli and P.K. Srimani, A new family of cayley graph interconnection networks of constant degree four, IEEE Trans. Parallel and Distributed Systems 7, 1(1996). [6] P. Vadapalli and P.K. Srimani, Fault tolerant ring embedding in tetravalent cayley network graphs, Journal of Circuits, Systems, and Computers 6, 5(1996) 527536 [7] D.S.L. Wei, F.P. Muga, and K. Naik, Isomorphism of degree four cayley graph and wrapped butterfly and their optimal permutation routing algorithm, IEEE Trans. Parallel and Distributed Systems 10, 11(1999) 1290-1298. [8] S.A. Wong, Hamilton cycles and paths in butterfly graphs, Networks 26(1995) 145-150..

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