Embedding cycles in IEH graphs

(1)

ELSEVIER Information Processing Letters 64 ( 1997) 23-27

Informqtion

;r$=rw

Embedding cycles in IEH graphs

Hung-Yi Chang, Rong-Jaye Chen *

Department of Computer Science and Information Engineeri?g, National Chiao Tung Universify, 1001 TA Hsueh Road, Hsinchu, 30050, Taiwan, ROC

Received I April 1997; revised 1 July 1997 Communicated by S.G. Akl

Abstract

We embed cycles into IEH graphs. First, IEH graphs are proved to be Hamiltonian except when they are of size 2” - 1 for all n > 2. Next, we show that for an IEH graph of size N, an arbitrary cycle of even length N, where 3 < Ne < N is found. We also find an arbitrary cycle of odd length NO where 2 < NO < N if and only if a node of this graph has at least one forward 2-Inter-Cube (IC) edges. These results help describe the whole cycle structure in IEH graphs. @ 1997 Elsevier Science B.V.

Keywords: Hypercubes; Embedding; Hamiltonian cycles; Incrementally Extensible Hypercubes; Interconnection networks

1. Introduction

Hypercube graphs are one class of the most popular

topologies for implementing massively parallel ma- chines. It has many advantages: regularity, symmetry, low diameter, optimal fault tolerance, and so on [ 71. However, the hypercube has one major drawback: that it is not incrementally extensible. The number of nodes for hypercubes must be a power of two, and thus con- siderably limit the choice of the number of nodes in the graphs. A few papers have so far been written to improve this drawback [2,5,8,9] but still have prob- lems as described briefly in the following. Bhuyan and Agrawal [ 21 proposed Generalized Hypercubes that have two drawbacks: the network reduces to a com- plete graph when the number of nodes is prime and changes significantly when a new node is added. Kat- seff [5] proposed Incomplete Hyperwbes that suffer

l Corresponding author.

from the problem of fault tolerance: failure of a single node will cause the entire network to be disconnected. Sen [ 81 proposed Super-cubes that become more ir- regular as the size of the networks grows. Recently, Sur and Srimani [ 93 have proposed a new generalization class of hypercube graphs, Incrementally Exten-

sible Hypercube (IEH) graphs. This topology can be

defined for an arbitrary number of nodes and still pre- serves several advantages such as optimal fault tolerance, a low diameter, a simple routing algorithm, and almost regularity.

Many papers have addressed the problem of finding cycles in various network topologies [ 1,3,4,6,7]. However, finding cycles in IEH graphs has never been studied. In this paper, we focus on IEH graphs and obtain the following results. First, IEH graphs are proved to be Hamiltonian except when they are of size 2n - 1 for all n > 2. Next, we show that for an IEH graph of size N, an arbitrary cycle of even length Ne where 3 < N, < N is found. We also find an arbitrary cy-

(2)

24 H.-L Chung, R.-J. Chen/lnfiwmation Processing Letter.7 64 (1997) 23-27

,, ,, ,,, ,,.... ,. ,, ”

Fig. I. IEH( I I ) graph.

cle of odd length N, where 2 < N, < N if and only if a node of this graph has at least one forward 2-

Inter-Cube (IC) edges. These results help describe

the whole cycle structure in IEH graphs.

The rest of this paper is organized as follows. In Section 2, we introduce basic terminology for hypercube graphs and IEH graphs. In Section 3, we show IEH graphs are Hamiltonian when their sizes are not equal to 2n - I for all n 2 2. In Section 4, we describe the cycle structure of IEH graphs. Finally, we give a conclusion in Section 5.

2. Preliminaries

A hypercube H, is a graph G( YE), where V is the set of 2n nodes which are labeled as binary numbers of length n; E is the set of edges that connects two nodes if and only if they differ in exact one bit in their labels. An IEH graph, a generalized hypercube graph, is composed of several hypercubes of different sizes. These hypercubes are connected with Inter-Cube (IC) edges. Let IEH( N) be an IEH graph of N nodes. The graph is constructed by the following algorithm [ 93. Algorithm CONSTR

Express N as a binary number (c,, . . , cl ,co)2 where c, = 1. For each ci, with Ci # 0, construct a hypercube Hi. The edges constructed in this step are called regular edges.

For all H,, label each node with a dedicated binary number 11.. . lObi_,. . . bo where the number of leading 1s is n - i and bi_1 . . . bo is the label of this node in the regular hypercube of dimension i. Find the minimum i such that ci = 1, set G; = Hi, and set j = i.

i=i+l.

While i < n if Ci # 0 then

Connect the node 11 . . . 1 b,ib,i_1 . . . bo in G.i to the following i - j nodes in Hi:

n-i i-j-l

IT-‘ . . . lOll...lbjbj_l . ..bo. n-i i-J-1

TTT?Obl...b,ib,i_l . . . bo,

. . . .

n-i i-.j- I --

11 . ..lOll . ..Objbj_l . ..bo.

Set j = i and let Gi be the composed graph obtained in this step. /* Gi is the graph which is composed of the Hks for k < i. */

endif i=i+l. endwhile

Thus obtain the IEH( N) graph, G,.

In Algorithm CONSTR, we observe two useful properties. First, Gi is the IEH( ch ~2~) graph. Second, two nodes which are joined by IC edges differ in one or two bits of their labels. For illustration, Fig. 1 shows the IEH( I 1) graph. Note that solid lines represent regular edges and dot lines represent IC edges.

For convenience of discussion, we divide IC edges into two classes: I-ZC edges and 2-IC edges. A I-IC edge connects nodes which differ in exactly one bit in their labels; and a 2-IC edge connects nodes which differ in exactly two bits. Let (u, v) be an IC edge with u in Hi and u in Hj for i # j. We call (u, U) a forward IC edge of u if i < j, and a backward one otherwise. Fig. 1 shows that (1100,1110) is a forward I-IC edge of node 1110 and that (0000, 1100) is a backward 2-IC edge of node 0000. Note that a node u which has forward 2-IC edges, connecting to some nodes in Hk for k > i, has exactly one for- ward l-IC edge to a dedicated node in the same hypercube.

(3)

H.-E Chang, R.-J. Chedlnformation Processing Letters 64 (1997) 23-27

Fig. 2. IEH(2” - 1) graphs are not Hamiltonian for all n 3 2.

3. Hamiltonian cycles in IEH graphs

In this section, we show IEH( N) graphs are Hamil- tonian for all N # 2” - 1 and n 2 2. To prove this theorem, we need the following lemmas.

Lemma 1 (Saad-Schultz [7]). A hypercube dues not admit odd cycles.

Lemma 2 (Saad-Schultz [7]). A cycle of length 1 can be mapped into H,, when 1 is even between 4 and 2”.

Lemma 3. IEH( 2” - 1) graphs are not Hamiltonian for all n 3 2.

Proof. By Algorithm CONSTR, let 2”- I =(1,1,...,1)2

where ( 1 , 1, . . . , 1)2 is the binary representation of 2” - I. We construct the IEH( 2” - 1) graph which is a composite graph of H,,_I, H+2, . ., HO. Since all His exist for 0 < i < n - 1, any IC edge connects two nodes when they are different in exactly one bit. Consider a node u in H; (see Fig. 2). Observe that u has II - i - 1 forward IC edges, i regular hypercube edges, and one backward IC edge if and only if the last i significant bits of its label are not all 1. Thus, the IEH( 2” - 1) graph is a subgraph of H, induced bytheverticesV(H,)\{(l,l,...,l)}.Bythisand

Lemma I, the IEH(2” - I ) graph is not Hamilto- nian. 0

Fig. 3 shows the IEH(7) graph is not Hamiltonian since 7 = 2” - 1. Except for the case of IEH( N) where N = 2” - I and n > 2, we prove in the following theorem that IEH graphs are Hamiltonian.

Fig. 3. IEH(7) is not Hamiltonian.

Theorem 4. IEH( N) graphs are Hamiltanianfor all N # 2” - 1 and n 3 2.

Proof. For simplicity, we define that Ptz is a Hamil- tonian path from u to u in a graph G. We consider two situations: (I) N is even and (2) N is odd.

Cusel:Niseven,N>3.LetN=(c,,...,c,,co)2,

with c, = 1. Because N is even, HO does not exist. Let Hj and Hi be two adjacent hypercubes. Without loss of generality, let j < i. Let Uj and l1.i be adja- cent nodes in H,i. Hence, there exists a Hamiltonian path from U.i to Uj in Hj by Lemma 2. By Algorithm CONSTR, L’,, is connected to one node of H, by a forward I-IC edge. For the same reason, U, also has a neighbor node in H;. Let these two neighbor nodes of 0.j and l~,i in Hi be ui and Ui, respectively. Note that u, and u; will differ in the same bit that u,; and L’.~ do. Since there exists a Hamiltonian path from ui to U, in H;, we get a cycle

u./ 4,“: ii --L’~--C,-P~~~,-Ui--llj.

Now, we prove Gk is Hamiltonian by induction on n, where n is the number of hypercubes of Gk. The base cases for n = I and 2 can be easily verified by Lemma 1 and the previous argument. Assume Gk is Hamiltonian and consists of 1 hypercubes. Let G, be the graph composed of Gk and H, where x > k. Let u and v be adjacent nodes in Gk and let u, and u,, respectively, denote their neighbor nodes in H,. We then obtain a Hamiltonian cycle

u-p,:: --“--L’x-P,~*;, --ux-u

in G,, which consist of I+ 1 hypercubes. This proves the induction and we have that IEH( N) is Hamilto- nian.

Case 2: N is odd where N f 2” - 1 and n > 2. Since N # 2” - I, there exists a cJ with c, = 0, for j # 0,~ By Algorithm CONSTR, HO has at least

(4)

26 H.-Y Chang, R.-J. Chen/Informuiion Processing Letters 44 (1997) 23-27

Fig. 4. IEH(N) graph is Hamiltonian for odd N and N # 2” - I for all n 2 2.

two forward IC edges connecting to some nodes in H; with ci = 1 and ck = 0 for j 6 k < i < n. Let v be the neighbor node of Ho in

Hi

connected by the I-IC edge and u be one of the neighbor nodes in Hi connected by the 2-IC edges. Note that u and v are adjacent. Thus we have a cycle from u to v in

V( IEH( N) ) \ Ho by the previous case. By adding two

edges, (HO, u) and (Ho, u) and deleting (u, v), we find that IEH(N) graphs are Hamiltonian since we have a cycle

as Fig. 4 shows. Cl

4. Cycles in IEH graphs

In this section, we describe the cycle structure of IEH graphs. In the following theorem, we show that there exists a cycle of even length Ne in an IEH( N) graph for 2 < N, < N.

Theorem 5. Given an arbitrary even number Ne with 2 < Ne < N, there exists a cycle of length Ne in an

IEH( N) graph.

Proof. Let N = (c,, . . . .co)~. with c, = 1, and N, = (4,. . . , do)2. Since N > N,, there exists some j such

thatci=d;=l,forn+l >i>jandcj=l #

dj = 0. Let Ne = N,i + Ne2 where Nei = Cy=,+, di2’

and Ne2 = c{i,’ di2’. Since ci = di for n + 1 > i > j, we find a cycle of length Net in the IEH( Nei ) graph, G,, composed of H,, H, - 1, . . ., Hi + 1,

in the same way as we do in Case 1 of Theorem 4. By Lemma 2, we find a cycle C of length Ne2 in

Hj since 2.i > Ne2. Let u,/ and Uj be two neighbor

nodes in C. Note that both Uj and v,i have neighbor nodes in H, by l-IC edges. Let u, and v, be these two nodes in H,,. Thus the following cycle of length N, exists:

U,--pU%, -v,,-vj-PUG; _{I I -u,j--u,, .} 0

In hypercubes, there exist no odd cycles. However, it is interesting to note that there exist odd cycles in IEH( N) graphs for certain special integers N. The following theorem shows that for an odd integer N,, where 2 < N, < N, there exists a cycle of length N, in an IEH( N) graph if and only if there exists one node in Hj which has at least one forward 2-IC edge.

Theorem 6. Given an arbitrary odd number No with 2 < No < N, there exists a cycle of length No in the

IEH( N) graph if and only if there exists some node u

in Hj that has at least one forward 2-IC edge, Proof. Assume that all nodes in the IEH( N) graph have no 2-IC edges. It is obvious that the IEH( N) graph is a subgraph of H, for some n and 2” > N. By Lemma 1, there exist no odd cycles in these IEH( N) graphs.

For the converse part, recall that the two adjacent graphs Gj and Hi are connected by IC edges for i >

j in step 3 of Algorithm CONSTR. Thus, if some

node n in H.i has forward 2-IC edges, then all nodes in Hk have 2-IC edges for all k < j. Without loss

(5)

Table I

H.-l! Chang, R.-J. Chen/Informarion Processing Letters 64 (1997) 23-27 21

If the graph is Hamiltonian If the graph contains even cycles If the graph contains odd cycles

IEH( N) Yes. except when N = 2” - I Yes Yes, when it has 2-IC edges

lH(N) Yes, only N is even Yes No

of generality, let

HO

have forward 2-IC edges, let u be the neighbor node by a forward l-IC edge of

HO,

and let u be the neighbor node by a forward 2-IC edge of

HO.

By applying Theorem 5, we find a cycle C’ of length N,, - 1 through the edge (u, 0). By adding

HO

and its IC edges and deleting the edge (u, u), we find a cycle of length

No

in the IEH( N) graph. Cl

5. Conclusion

In this paper, we describe the whole cycle structure in IEH graphs. The main results are summarized in the second row of Table 1. The properties of Incomplete Hypercubes (IHs) are listed in the third row [ 51 to be compared with the above results in the second row. It is obvious that IEH graphs are superior to IHs in embedding cycles. References III I21 131 141 151 161 171 181 191

V. Auletta, A.A. Rescigno and V. Scarano, Embedding graphs onto the supercube, IEEE Trans. Comput. 44 (4) ( 1995) 593-597.

L. Bhuyan and D.P Agrawal, Generalized hypercube and hyperbus structure for a computer network, IEEE Trans. Comput. 33 (3) ( 1984) 323-333.

K. Day and A. Tripathi, Embedding of cycles in arrangement graphs, IEEE Trans. Comput. 42 (8) (1993) 1002-1006. A. Kanevsky and C. Feng, On the embedding of cycles in pancake graphs, Parallel Comput. 21 ( 1995) 923-936. HP Katseff, Incomplete Hypercubes, IEEE Trans. Comput. 37 (5) (1988) 604-607.

S. Latifi and N. Bagherzadeh, On the clustered-star graph and its properties, Comput. System Sci. Engineering I I (3) (1996) 145-159.

Y. Saad and M.H. Schultz, Topological properties of hypercubes, IEEE Trans. Comput. 37 (7) (1988) 867-872. A. Sen, Supercube: An optimal fault tolerant network architecture, Acta Inform. 26 ( 1989) 741-748.

S. Sur and P.K. Srimani, IEH graphs: A novel generalization of hypercube graphs, Acta Inform. 32 ( 1995) 597-609.