A note on the alternating group network - Semantic Scholar
J Supercomput DOI 10.1007/s11227-010-0434-y
A note on the alternating group network Eddie Cheng · Ke Qiu · Zhizhang Shen
© Springer Science+Business Media, LLC 2010
Abstract The class of alternating group networks was introduced in the late 1990’s as an alternative to the alternating group graphs as interconnection networks. Recently, additional properties for the alternating group networks have been published. In particular, Zhou et al., J. Supercomput (2009), doi:10.1007/s11227-009-0304-7, was published very recently in this journal. We show that this so-called new interconnection topology is in fact isomorphic to the (n, n − 2)-star, a member of the well-known (n, k)-stars, 1 ≤ k ≤ n − 1, a class of popular networks proposed earlier for which a large amount of work have already been done. Specifically, the problem in Zhou et al., J. Supercomput (2009), doi:10.1007/s11227-009-0304-7, was addressed in Lin and Duh, Inf. Sci. 178(3), 788–801, 2008, when k = n − 2. Keywords Alternating group network · (n, k)-star graph · Isomorphism 1 ANn is isomorphic to (n, n − 2)-star The alternating group network [6] was introduced as an alternative to the alternating group graphs [7]. Supposing n ≥ 3, the alternating group network AN n has the E. Cheng Dept. of Mathematics and Statistics, Oakland University, Rochester, MI 48309-4401, USA e-mail: [email protected] K. Qiu () Department of Computer Science, Brock University, St. Catharines, Ontario L2S 3A1, Canada e-mail: [email protected] Z. Shen Dept. of Computer Science and Technology, Plymouth State University, Plymouth, NH 03264-1595, USA e-mail: [email protected]
E. Cheng et al.
vertex set of even permutations from {1, 2, . . . , n}, two vertices [a1 , a2 , a3 , . . . , an ] and [b1 , b2 , b3 , . . . , bn ] are adjacent if one of the following three conditions is satisfied. The first is that there exists an i ∈ {4, 5, . . . , n} such that a1 = b2 , a2 = b1 , a3 = bi , ai = b3 and aj = bj for j ∈ {4, 5, . . . , n} \ {i}. The second is a1 = b2 , a2 = b3 , a3 = b1 and aj = bj for 4 ≤ j ≤ n. The third is a1 = b3 , a2 = b1 , a3 = b2 and aj = bj for 4 ≤ j ≤ n. Since its proposal, the alternating group network has attracted some attention and research has been done in [2, 3]. In [2], the network is studied in terms of node-tonode distance and optimal routing, while the Whitney numbers of the second kind, also known as the surface areas, are studied in [3]. Most recently, vertex-disjoint paths are constructed for this network in this very journal [9]. In this short note, we would like to point out that this so-called new interconnection topology is in fact isomorphic to the well-known (n, k)-star, proposed earlier [4, 5] to address deficiency (most notably the scalability) in the star network [1]. Because the alternating group network is a special case of the (n, k)-star graph, many results are already known and they need not be studied separately. For example, the same problem addressed in [9] was studied in [8] when k = n − 2. The (n, k)-star has been studied extensively in areas such as broadcasting and sorting, distance formula and shortest distance routing, fault tolerant routings, superconnectedness, vulnerability, Hamiltonianicity and various generalizations. Suppose n ≥ 3 and 2 ≤ k ≤ n − 1. The vertex set of an (n, k)-star graph is the k-permutations from {1, 2, . . . , n}, and two vertices [a1 , a2 , a3 , . . . , ak ] and [b1 , b2 , b3 , . . . , bk ] are adjacent if one of the following two conditions is satisfied. The first is there exists an i ∈ {2, 3, . . . , k} such that a1 = bi , ai = b1 and aj = bj for j ∈ {1, 2, . . . , k} \ {1, i} (that is, one is obtained from the other by switching the symbols in the first and the ith position). The second is a1 = b1 and aj = bj for 2 ≤ j ≤ k (that is, one is obtained from the other by replacing the symbol in the first position by a symbol not in the permutation). Theorem 1 Let n ≥ 3. Then AN n is isomorphic to (n, n − 2)-star. Proof Consider a vertex π = [a1 , a2 , . . . , an−2 ] in (n, n − 2)-star. We will extend this to a full even permutation π E on {1, 2, . . . , n}. Let {y, z} = {1, 2, . . . , n} \ Then exactly one of [y, z, a1 , a2 , . . . , {a1 , a2 , . . . , an−2 }. an−2 ] and [z, y, a1 , a2 , . . . , an−2 ] is even. Let π E be the even permutation. We claim that φ : V (Sn,n−2 ) −→ V (AN n ), defined by φ(π) = π E , is an isomorphism. Consider π = [a1 , a2 , . . . , an−2 ] and π E = [y, z, a1 , a2 , . . . , an−2 ]. Let α = [b1 , b2 , . . . , bn ] be obtained from π by switching the symbols in the first and the ith position. Then α E = [z, y, b1 , b2 , . . . , bn ] as we want α E to be even. This is precisely Condition 1 for AN n . Now suppose we replace a1 by y in π to obtain α = [y, a2 , . . . , an ]. Then α E is either [z, a1 , y, a2 , . . . , an ] or [a1 , z, y, a2 , . . . , an ]. Since we need to choose the even one, α E = [z, a1 , y, a2 , . . . , an ]. But this is Condition 2 for AN n . Similarly, if we replace a1 by z in π to obtain α, then π E and α E are related as in Condition 3 for AN n .
A note on the alternating group network
References 1. Akers SB, Krishnamurthy B (1989) A group theoretic model for symmetric interconnection networks. IEEE Trans Comput 38(4):555–566 2. Chen B, Xiao W, Parhami B (2006) Internode distance and optimal routing in a class of alternating group networks. IEEE Trans Comput 55(12):1645–1648 3. Cheng E, Qiu K, Shen ZZ (2009) On the surface area of the alternating group networks. In: Proc 21st IASTED parallel and distributed computing and systems (PDCS 2009), Nov, Cambridge, MA, pp 85– 90 4. Chiang WK, Chen RJ (1995) The (n, k)-star graph: a generalized star graph. Inf Process Lett 56:259– 264 5. Chiang WK, Chen RJ (1998) Topological properties of the (n, k)-star graphs. Int J Found Comput Sci 9:235–248 6. Ji Y (1999) A new class of Cayley networks based on the alternating groups. Appl Math A, J Chin Univ 14:235–239 (in Chinese) 7. Jwo JS, Lakshmivarahan S, Dhall SK (1993) A new class of interconnection networks based on the alternating group. Networks 23:315–326 8. Lin TC, Duh DR (2008) Constructing vertex-disjoint paths in (n, k)-star graphs. Inf Sci 178(3):788– 801 9. Zhou SM, Xiao WJ, Parhami B (2009) Construction of vertex-disjoint paths in alternating group networks. J Supercomput. doi:10.1007/s11227-009-0304-7
A note on the alternating group network Eddie Cheng · Ke Qiu · Zhizhang Shen
© Springer Science+Business Media, LLC 2010
Abstract The class of alternating group networks was introduced in the late 1990’s as an alternative to the alternating group graphs as interconnection networks. Recently, additional properties for the alternating group networks have been published. In particular, Zhou et al., J. Supercomput (2009), doi:10.1007/s11227-009-0304-7, was published very recently in this journal. We show that this so-called new interconnection topology is in fact isomorphic to the (n, n − 2)-star, a member of the well-known (n, k)-stars, 1 ≤ k ≤ n − 1, a class of popular networks proposed earlier for which a large amount of work have already been done. Specifically, the problem in Zhou et al., J. Supercomput (2009), doi:10.1007/s11227-009-0304-7, was addressed in Lin and Duh, Inf. Sci. 178(3), 788–801, 2008, when k = n − 2. Keywords Alternating group network · (n, k)-star graph · Isomorphism 1 ANn is isomorphic to (n, n − 2)-star The alternating group network [6] was introduced as an alternative to the alternating group graphs [7]. Supposing n ≥ 3, the alternating group network AN n has the E. Cheng Dept. of Mathematics and Statistics, Oakland University, Rochester, MI 48309-4401, USA e-mail: [email protected] K. Qiu () Department of Computer Science, Brock University, St. Catharines, Ontario L2S 3A1, Canada e-mail: [email protected] Z. Shen Dept. of Computer Science and Technology, Plymouth State University, Plymouth, NH 03264-1595, USA e-mail: [email protected]
E. Cheng et al.
vertex set of even permutations from {1, 2, . . . , n}, two vertices [a1 , a2 , a3 , . . . , an ] and [b1 , b2 , b3 , . . . , bn ] are adjacent if one of the following three conditions is satisfied. The first is that there exists an i ∈ {4, 5, . . . , n} such that a1 = b2 , a2 = b1 , a3 = bi , ai = b3 and aj = bj for j ∈ {4, 5, . . . , n} \ {i}. The second is a1 = b2 , a2 = b3 , a3 = b1 and aj = bj for 4 ≤ j ≤ n. The third is a1 = b3 , a2 = b1 , a3 = b2 and aj = bj for 4 ≤ j ≤ n. Since its proposal, the alternating group network has attracted some attention and research has been done in [2, 3]. In [2], the network is studied in terms of node-tonode distance and optimal routing, while the Whitney numbers of the second kind, also known as the surface areas, are studied in [3]. Most recently, vertex-disjoint paths are constructed for this network in this very journal [9]. In this short note, we would like to point out that this so-called new interconnection topology is in fact isomorphic to the well-known (n, k)-star, proposed earlier [4, 5] to address deficiency (most notably the scalability) in the star network [1]. Because the alternating group network is a special case of the (n, k)-star graph, many results are already known and they need not be studied separately. For example, the same problem addressed in [9] was studied in [8] when k = n − 2. The (n, k)-star has been studied extensively in areas such as broadcasting and sorting, distance formula and shortest distance routing, fault tolerant routings, superconnectedness, vulnerability, Hamiltonianicity and various generalizations. Suppose n ≥ 3 and 2 ≤ k ≤ n − 1. The vertex set of an (n, k)-star graph is the k-permutations from {1, 2, . . . , n}, and two vertices [a1 , a2 , a3 , . . . , ak ] and [b1 , b2 , b3 , . . . , bk ] are adjacent if one of the following two conditions is satisfied. The first is there exists an i ∈ {2, 3, . . . , k} such that a1 = bi , ai = b1 and aj = bj for j ∈ {1, 2, . . . , k} \ {1, i} (that is, one is obtained from the other by switching the symbols in the first and the ith position). The second is a1 = b1 and aj = bj for 2 ≤ j ≤ k (that is, one is obtained from the other by replacing the symbol in the first position by a symbol not in the permutation). Theorem 1 Let n ≥ 3. Then AN n is isomorphic to (n, n − 2)-star. Proof Consider a vertex π = [a1 , a2 , . . . , an−2 ] in (n, n − 2)-star. We will extend this to a full even permutation π E on {1, 2, . . . , n}. Let {y, z} = {1, 2, . . . , n} \ Then exactly one of [y, z, a1 , a2 , . . . , {a1 , a2 , . . . , an−2 }. an−2 ] and [z, y, a1 , a2 , . . . , an−2 ] is even. Let π E be the even permutation. We claim that φ : V (Sn,n−2 ) −→ V (AN n ), defined by φ(π) = π E , is an isomorphism. Consider π = [a1 , a2 , . . . , an−2 ] and π E = [y, z, a1 , a2 , . . . , an−2 ]. Let α = [b1 , b2 , . . . , bn ] be obtained from π by switching the symbols in the first and the ith position. Then α E = [z, y, b1 , b2 , . . . , bn ] as we want α E to be even. This is precisely Condition 1 for AN n . Now suppose we replace a1 by y in π to obtain α = [y, a2 , . . . , an ]. Then α E is either [z, a1 , y, a2 , . . . , an ] or [a1 , z, y, a2 , . . . , an ]. Since we need to choose the even one, α E = [z, a1 , y, a2 , . . . , an ]. But this is Condition 2 for AN n . Similarly, if we replace a1 by z in π to obtain α, then π E and α E are related as in Condition 3 for AN n .
A note on the alternating group network
References 1. Akers SB, Krishnamurthy B (1989) A group theoretic model for symmetric interconnection networks. IEEE Trans Comput 38(4):555–566 2. Chen B, Xiao W, Parhami B (2006) Internode distance and optimal routing in a class of alternating group networks. IEEE Trans Comput 55(12):1645–1648 3. Cheng E, Qiu K, Shen ZZ (2009) On the surface area of the alternating group networks. In: Proc 21st IASTED parallel and distributed computing and systems (PDCS 2009), Nov, Cambridge, MA, pp 85– 90 4. Chiang WK, Chen RJ (1995) The (n, k)-star graph: a generalized star graph. Inf Process Lett 56:259– 264 5. Chiang WK, Chen RJ (1998) Topological properties of the (n, k)-star graphs. Int J Found Comput Sci 9:235–248 6. Ji Y (1999) A new class of Cayley networks based on the alternating groups. Appl Math A, J Chin Univ 14:235–239 (in Chinese) 7. Jwo JS, Lakshmivarahan S, Dhall SK (1993) A new class of interconnection networks based on the alternating group. Networks 23:315–326 8. Lin TC, Duh DR (2008) Constructing vertex-disjoint paths in (n, k)-star graphs. Inf Sci 178(3):788– 801 9. Zhou SM, Xiao WJ, Parhami B (2009) Construction of vertex-disjoint paths in alternating group networks. J Supercomput. doi:10.1007/s11227-009-0304-7
