Mutually Independent Hamiltonian Paths in Star Networks

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1 Mutually Independent Hamiltonian Paths in Star Networks Cheng-Kuan Lin a, Hua-Min Huang a, Lih-Hsing Hsu b,, and Sheng Bau c, a Department of Mathematics National Central University, Chung-Li, Taiwan b Department of Computer Science National Chiao Tung University, Hsinchu, Taiwan c School of Mathematics, University of Natal Pietermaritzburg, South Africa Abstract Let P = v,v,,v k and P = u,u,,u k be any two hamiltonian paths of G. We say that P and P are independent if u = v, v k = u k, and v i u i for <i<k. We say a set of hamiltonian paths P,P,,P s of G are mutually independent if any two different hamiltonian paths are independent. We prove that there exist (n ) mutually independent hamiltonian paths in S n between any two nodes from different bipartite sets if n. Moreover, this result is optimal. Keywords: hamiltonian, hamiltonian laceable. Introduction For definitions and notations we follow []. G =(V,E) isagraph if V is a finite set and E is a subset of {(u, v) (u, v) is an unordered pair of V }. We say that V is the node set and E is the edge set. For a node u, N(u) denotes the neighborhood of u which is the set {v (u, v) E}. For any node x of V,denotethedegree of x by deg G (x) = N(x). Two nodes u and v are adjacent if (u, v) E. A path is represented by v,v,,v k where v i v j for i, j {,,,k} and i j and (v i,v i+ ) E. Denote by Q(i) the i-th node v i of path Q = v,v,,v k. We also write the path v,v,,v k as v,q,v i,v i+,,v j,q,v t,,v k,whereq is the path v,v,,v i and Q is the path Corresponding auther: lhhsu@cc.nctu.edu.tw The work of this author was supported in parts by the University of Natal URF fund and the National Chiao Tung University of Taiwan. The great hospitality of Professor Lih-Hsing Hsu and the Department of Computer Science and Department of Applied Mathematics of National Chiao Tung University is gratefully acknowledged. 96

2 v j,v j+,,v t.weused(u, v) todenotethedistance between u and v, i.e., the length of the shortest path joining u and v. ApathP is a hamiltonian path if V (P )=V (G). Let P = v,v,,v k and P = u,u,,u k be two paths of G. WesaythatP and P are independent if u = v, v k = u k, and v i u i for <i<k. We say a set of hamiltonian paths P,P,,P s of G are mutually independent if any two different paths in the set are independent. An interconnection network connects the processors of the parallel computer. Its architecture can be represented as a graph in which the nodes corresponds to processors and the edges to connections. Hence we use graphs and networks interchangeably. There are many mutually conflicting requirements in designing the topology for computer networks. The n-cube is one of the most popular topologies [7]. The star network S n wasproposedin[]as an attractive alternative to the n-cube topology for interconnecting processors in parallel computers. Since its introduction, the network received considerable attention. Some interesting properties of star networks are studied in [,,, 6, 8]. For example, it is known that the star network is a bipartite graph and there is a hamiltonian path joining nodes from different partite sets. In particular, Fragopoulou and Akl [, ] study the embedding of (n ) directed edge-disjoint spanning trees on the star network S n. These spanning trees are used in communication algorithms for star networks. In this paper, we prove that there exist (n ) mutually independent hamiltonian paths of S n between any two nodes from different bipartite sets. Obviously, these mutually independent paths can be used for communication algorithms for star networks. In the following section, we give the definition of the star networks and some of the previous work that are used in this paper. In section, we prove that there exist (n ) mutually independent hamiltonian paths of S n between any two nodes from different bipartite sets. Moreover, we prove that the value (n ) is tight. Basic properties of the star networks Let n be a positive integer. We use n to denote the set {,,,n}. The n-dimensional star network, denoted by S n, is a graph with the node set V (S n )={u u u n u i n and u i u j for i j}. The edges are specified as follows: u u...u i...u n is adjacent to v v...v i...v n through an edge in dimension i with i n if v j = u j for j / {,i}, v = u i and v i = u. By definition, S n is an (n )-regular graph with n! nodes. Moreover, it is it is node transitive and edge transitive. We use bold face letters to denote nodes in S n. Hence u, u,, u n is a sequence of nodes in S n. It is known that the connectivity of S n is n. We use e to denote the element...n. It is known that S n is a bipartite graph with one partite set containing those nodes corresponding to odd permutations and the other partite set containing those nodes 97

3 corresponding to even permutations. We will use white nodes to represent nodes for even permutations and black nodes to represent nodes for odd permutations. Let u = u u...u n be any node of the star network S n.wesaythatu i is the i-th coordinate of u, denoted by (u) i, for i n. By the definition of S n, there is exactly one neighbor v of u such that u and v are adjacent through an edge in the i-th dimension, i n. For this reason, we use (u) i to denote the unique i-neighbor of u. Obviously, ((u) i ) i = u. For i n, let S i n denote the subgraph of S n induced by those nodes u with (u) n = i. Obviously, S n can be decomposed into n subgraphs S i n, i n, andeachs i n is isomorphic to S n. This furnishes a recursive definition (construction) for star networks. For i j n, weuse E i,j to denote the set of edges between S i n and S j n. Lemma E i,j =(n )! for any i j n. Moreover, there are (n )! edges joining black nodes of S i n to white nodes of S j n. Proof. Let i j. For each of (n )! permutations of n {i, j} thereisexactlyone transposition that represents an edge between a node of Sn i and a fixed node of Sn. j Hence E i,j =(n )!. Exactly half of these correspond to transpositions of an odd permutation in Sn i to an even permutation in Sn. j Lemma Let u and v be two distinct nodes of S n with (u) n =(v) n such that d(u, v). Then((u) n ) n ((v) n ) n. Theorem [8] S n is hamiltonian laceable if and only if n. Theorem Assume that n. Let {a,a,,a r } be a subset of n for some r n. For any white node u of S a n and any black node v of Sn ar, there exists a path P as x,p, y, x,p, y,, x r,p r, y r joining u and v such that x = u, y r = v, andp i is a hamiltonian path of S a i n joining x i to y i for i r. Moreover, P is a hamiltonian path of the subgraph induced by r i=s a i n joining u to v. Proof. We set that x as u and set that y r as v. Since S a n is isomorphic to S n,by Theorem, this theorem holds for r =. Assume that r. By Lemma, we choose (y i, x i+ ) E a i,a i+ with y i is a black node and x i+ is a white node for i r. Obviously, x i y i in S a i n. Since S a i n is isomorphic to S n, by Theorem, there is a hamiltonian path P i of S a i n joining x i to y i. Obviously, x,p, y, x,p, y,, x r,p r, y r is the required path. Lemma Let w be any black node and u, v be any two distinct white nodes of S n with n. Then there exists a hamiltonian path in S n {w} joining u to v. Proof. We prove this lemma by induction. For n =, we can set w = because S is node transitive. The corresponding hamiltonian path between u and v in S {w} are listed below. 98

4 ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()() Assume that n. Since S n is node transitive and edge transitive, we may assume that u = e and (w) n = n. Case. v Sn, i i n. Let {a,a,,a n } = n with a n =(v) n. Since Sn n is isomorphic to S n, by Theorem, there exists a hamiltonian path P of Sn n joining u to a black node s with (s) n = n. Since Sn n is isomorphic to S n, by induction, there exists a hamiltonian path Q of Sn n {w} joining the white node (s) n to a white node t with (t) n = a.sinces a i n is isomorphic to S n for i n, by Theorem, there exists a hamiltonian path R of the subgraph induced by n i= Sa i n joining the black node (t) n to v. 99

5 Then u,p,s, (s) n,q,t, (t) n,r,v forms the desired path. Case. v Sn n. Let {a,a,,a n } = n {n } with a = n. Since Sn i is isomorphic to S n with i {a,a,,a n }, by Theorem, there exists a hamiltonian path R of the subgraph induced by n i= Sa i and (s) n v. Since S n n path Q of Sn n desired path. n joining u to a black node s with (s) n = n is isomorphic to S n, by induction, there exists a hamiltonian {w} joining the white node (s) n to v. Then u,r,s, (s) n,q,v forms the Case. v S n n. Note that N(v) (v) n. We can choose x as a neighbor of v with (x) n n. Since S n n is isomorphic to S n, there exists a hamiltonian path P of S n n {x} joining u to v. Without loss generality, we can write P as u,p, s, u. Sinced(s, x), by Lemma, ((s) n ) n ((x) n ) n. Obviously, x and s both are black nodes. Assume that (s) n = n. Let {a,a,,a n } = n with a n =(x) n. Since Sn n areisomorphictos n, by induction, there exists a hamiltonian path Q of Sn n {w} joining the white node (s) n to a white node t with (t) n = a. By Theorem, there exists a hamiltonian path R of the subgraph induced by r i=s a i n joining the black node (t) n to the white node (x) n.then u,p, s, (s) n,q,t,(t) n,r,(x) n, b, v forms the desired path. Assume that (s) n n. Let {a,a,,a n } = n with a n = (y) n and a n =(s) n. Since S a n n is isomorphic to S n, by Theorem, there exists a hamiltonian path Q of S a n n joining the white node (s) n to a black node t with (t) n = n. Since Sn n is isomorphic to S n, by induction, there exists a hamiltonian path H of Sn n {w} joining the white node (t) n to a white node z with (z) n = a. By Theorem, there exists a hamiltonian path R of the subgraph induced by r i=s a i n joining the black node (z) n to the white node (x) n.then u,p, s, (s) n,q,t,(t) n,h,z,(z) n,r,(x) n, x, v form the desireds path. Theorem Let u be any white node of S n with n and {a,a,,a n } be an n subset of n. Then there exist hamiltonian paths P,P,,P n of S n joining u to some black node z i such that () (z i ) =(P i (n!)) = a i for i n, and () {P (i),p (i),,p n (i)} = n for i n!. Proof. Since S n is node transitive, we may assume that u = e. We prove the theorem holds on S by exhibiting three required hamiltonian paths in S as follows: {a,a,a } = {,, } ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() {a,a,a } = {,, } ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() {a,a,a } = {,, } ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() 00

6 Assume that the theorem holds for S k, k < n. Without loss of generality, we suppose that a <a < <a n.wesetthata i = a i for i n, a n = a n,and a n = a n. Obviously, (e) n is a black node in Sn.Wesetthat i + if i n b i = i n + if n i n i +n + if n i n By induction hypothesis, there exist hamiltonian paths H,H,,H n of S n n joining e to some black node v i such that () (v i ) =(H i ((n )!)) = i + for i n, (v n ) =(H n ((n )!)) =, (v n ) =(H n ((n )!)) =,and () {H (i),h (i),,h n (i)} = n for i (n )!. Since {v Sn i+ v is a black node with (v) = a i} = (n )!, we choose a black node z i S b i+n n with (z i ) = a i for i n. Since Sn i is isomorphic to S n for i n, by Theorem, there exists a path W i as x i,t, i y, i x i,t, i y, i, x i n,tn, i yn i joining the white node (v i ) n to the black node z i such that x i =(v i ) n, yn i = z i,andtj i is a hamiltonian path of S b i+j n joining x i j to yj i for j n, and i n. Again, there exists a path W n as x n,t n, y n, x n,t n, y n,, x n n,tn n, yn n joining the white node (v n ) n to the black node z n such that x n =(v n ) n, yn n = z n,and T n j is a hamiltonian path of S b i+n n joining x n j to y n j for j n. We set that P i as e,h i, v i, (v i ) n,w i, z i for i n. By Theorem, there exists a path W as x,q, y, x,q, y,, x n,q n, y n joining the black node (e) n to the white node ((e) n ) n such that x =(e) n, y n =((e) n ) n, and Q i is a hamiltonian path of Sn i joining x i to y i for i n. By Lemma, {v Sn v n is a black node with (v) = a n } = (n )!, we choose a black node z n Sn with (z n ) = a n and z n (e) n. Since Sn n is isomorphic to S n,by Lemma, there exists a hamiltonian path R of Sn n {e} joining the black node (e) n to z n.wesetthatp n as e,w,((e) n ) n, (e) n,r,z n. Then {P,P,...,P n } forms the desired paths. We illustrate the proof of Theorem with n = as follows: Obviously, a,a, a,anda. By induction hypothesis, there exist hamiltonian paths H,H,H of joining e to some black node v i such that () (v ) =(H ()) =,(v ) =(H ()) =,(v ) =(H ()) =,and () {H (i),h (i),h (i)} = for i. Then we choose a black node z in S with (z ) = a,ablacknodez in S with (z ) = a,ablacknodez in S with (z ) = a, and a black node z in S with (z ) = a. By Theorem, there exists a path W as x,t, y, x,t, y, x,t, y, x,t i, y joining 6 0

7 the white node (v ) to the black node z such that x =(v ), y = z,andtj is a hamiltonian path of S b j n joining x j to yj for j, and a path W. Furthermore, a path W. By Theorem, there exists a path W as x,q, y, x,q, y, x,q, y, x,q, y joining the black node (e) to the white node ((e) ) such that x =(e), y =((e) ),andq i is a hamiltonian path of S i joining x i to y i for i. Since is isomorphic to S,by Lemma, there exists a hamiltonian path R of {e} joining the black node (e) to z. We set that P as e,w,((e) ), (e),r,z. Thus {P,P,P,P } forms the desired paths. See Figure for illustration. e H v ( v ) W P z e H v ( v ) W P z e H v ( v ) W P z P -{e} e W ( e) (( e) ) ( e) R z Figure : Illustration for Theorem on. Mutually independent hamiltonian paths Theorem Assume that n. Then there exist (n ) mutually independent hamiltonian paths of S n between any two nodes u and v from different bipartite sets. Moreover the value (n ) is tight. Proof. Since star networks are node transitive, we may assume that u = e. We prove this theorem by induction. For n, the required hamiltonian paths are list below: 7 0

8 v = () ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() v = () ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() v = () ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() v = () ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() v = () ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() v = () ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() v = () ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() v = () ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() v = () ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() v = () ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() v = () ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() v = () ()()()()()()()()()()()()()()()()()()()()()()()() ()()()()()()()()()()()()()()()()()()()()()()()() Assume that the theorem holds for S k, k<n.sinces n is node transitive and edge transitive, we assume that u = e Sn n and v Sn n. We set b i = i for i n and b i = i n + for n i n. By Theorem, there exist hamiltonian paths Q,Q,,Q n of Sn n joining e and some black node s i such that () (s i ) = b i for i n, and () {Q (i),q (i),,q n (i)} = n for i (n )!. Again, there exist hamiltonian paths R,R,,R n of Sn n joining v and some white node y i such that () (t i ) = b i+n for i n, and () {R (i),r (i),,r n (i)} = n for i (n )!. By Theorem, there exists a path W i as x i,t i, y i, x i,t i, y i,, x i n,t i n, y i n joining the white node x i to the black node y n n such that x i =(s i ) n, y i n =(t i ) n,andt i j is a hamiltonian path of S b i+j n joining x i j to yj i for j n, and i n. Hence {P,P,,P n } form the desired paths. See Figure for illustration the case n =. Suppose that u = e and v =(e) n. Let P,P,,P s be a set of mutually independent hamiltonian path between u and v. Obviously, {P i () i s} {(u) i i n}. Thus s n. Thus, the value (n ) is tight. References [] S. B. Akers and B. Krishnamurthy, A group-theoretic model for symmetric interconnection networks, IEEE Trans. Comput., Vol. 8, pp. 66, 989. [] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North Holland, New York, (980). 8 0

9 P P P e Q s ( s ) W ( ) t R v e Q s ( s ) W ( ) t R v e Q s ( s ) W ( ) t R v t t t Figure : Illustration for mutually independent hamiltonian paths on. [] A. Bouabdallah, M. C. Heydemann, J. Opatmy, and D. Sotteau, Embedding complete binary trees into star and pancake graphs, Theoory Comput. Syst., pp. 79 0, 998. [] P. Fragopoulou and S.G. Akl, Optimal communication algorithms on the star graphs using spanning tree constructions, Journal of Parallel and Distributed Computing, Vol., 7, 99. [] P. Fragopoulou and S.G. Akl, Edge-disjoint spanning trees on the star networks with applications to fault tolerance, IEEE Transactions on Computers, Vol., 7 8, 996. [6] L. Gargano, U. Vaccaro, and A. Vozella, Fault tolerant routing in the star and pancake interconnection networks, Information Processing Letters, pp. 0, 99. [7] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays Trees Hypercubes, Morgan Kaufmann, San Mateo, CA 99. [8] S.Y. Hsieh, G.H. Chen, and C.W. Ho, Hamiltonian-laceability of star graphs, Networks, vol. 6, no., pp. -,

Theoretical Computer Science. Fault-free longest paths in star networks with conditional link faults

Theoretical Computer Science. Fault-free longest paths in star networks with conditional link faults Theoretical Computer Science 410 (2009) 766 775 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs Fault-free longest paths in star networks