Information Processing Letters

Information Processing Letters 61 (1997) 157-160

The diameter of the cube-connected cycles Ivan FriS a,*, Ivan Have1 byL, Petr Liebl c

a Department of Mathematics, Statistics and Computing Science, University of New England, Armidale. NSW 2351, Australia b Mathematical Institute. Academy of Sciences of the Czech Republic, i&n& 25, II5 67 Prague I. Czech Republic

c Na Petiindch 78, 162 00 Prague 6, Czech Republic

Received 1 August 1996 Communicated by S.G. Akl

Abstract

Cube-connected cycles, or CCC, are graphs with properties which make them possible candidates for switching patterns of multiprocessor computers. In this paper, the diameter of CCC is calculated. In fact, the same calculation works for somewhat more general graphs than just CCC. @ 1997 Elsevier Science B.V. @ 1997 Elsevier Science B.V.

Keywords: Combinatorial problems; Interconnection networks: Cube-connected cycles; Graph diameter: Hamiltonian path; Hypercube

1. Introduction and notation

One of the problems in designing a network of pro-

cessors is to find an underlying structure which is practical and versatile at the same time. Many dif- ferent processor organizations are being used or eval- uated. One of the popular designs is the Hypercube (HC) which is suitable for a large number of prob- lems occurring in practice. HC has some nice prop- erties, namely small diameter, high bisection width, and a simple routing algorithm. Its major weakness is the number of edges (connections) per vertex (pro- cessor) which could make this design impractical for very large systems. HC is not a scalable design. [7] suggested a remedy, namely to use the cube-connected cycles (CCC) network, a network organization which

* Corresponding author. Email: ivan@cs.une.edu.au. ’ The author acknowledges the support by the Grant Agency

of the Czech Republic under Grant 201/94/1064. Email: haveli@mbox.cesnct.cz.

is similar to HC but has three edges per vertex regard- less of the network size. The motivation for this paper was our finding different asymptotic or approximate values for the diameter of CCC (see [5,6,8]). * In this paper we derive the actual value.

The construction which creates CCC from a single HC and several copies of the cycle graph can be easily generalized by using an arbitrary graph G instead of the cycle graph. This generalization is considered here, as the calculation of the diameter works in exactly the same way for the more general case. A further generalization in which HC is also replaced, this time by another “highly symmetrical” graph, is possible and looks interesting. This latter generalization has been deferred to another paper.

Our graphs will be finite and undirected, without loops and multiple edges. For a graph G, V(G) de- notes its vertex set and E(G) its edge set. A sequence

2 After submitting the manuscript we Lamed that the problem and a proof sketch were also considered in [ 91.

0020-0190/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PII SOO20-0190(97)00013-6

158 I. FriS et al./Information Processing Letters 61 (1997) 157-160

of vertices ua, ui , . . . , u, (I b 0) will be called a walk (or a walk in Gfrom ug to u,) of length r if [ ui, Ui+l] E

E(G), 0 < i < r. We shall also say that the walk contains the vertices ui, (0 f i < r). If all vertices of a walk (except possibly the first and the last) are distinct, the walk is called a path.

As usual, the length of the shortest path in G from u to u is called the distance between u and u and is

denoted dc( u, u) . The diameterdiam( G) of the graph G is defined by diam( G) = max,,#V(G) dG( u, u). For a finite set S, IS/ will denote the number of elements

of S. If Si and S2 are sets, then A( SI , SZ ) denotes the symmetric difference of Si and &, namely the set of elements which are in one of the two sets but not in both. The set of all subsets of a set V is denoted by 2”, and x used between two sets denotes their Cartesian product.

The graph of the cycle on n 2 3 vertices will be denoted by C,.

2. Cube-connected graphs

Definition 1. For an arbitrary graph G, the graph QG

is defined as follows:

V(QG) = V(G) x 2V(G),

E(QG> =B(QG) uC(QG)v

where

B(QG> ={[(hW,(6Wl:

[u’,u”l E E(G), M C V(G)},

C(QG) ={[(~,M’),(uAf”)l: u E V(G),

M’,M” G V(G), A(M’,M”) = {u}},

We shall call the edges from B( QG) basic, and those from C( QG) cubical. If IV(G) 1 = n, then QG con- sists of 2” copies of G connected mutually by n. 2”-’ cubical edges.

QG is connected iff G is connected. QG is (k + 1) - regular iff G is k-regular.

As an example consider G = C,, for n > 3. The graph QC, is the graph of cube-connected cycles. In accordance with other authors we denote it by CCC,,.

Definition 2. Let G be a connected graph, u, u E V(G) and M C V(G). We denote by hd$(u,u) the length of the shortest walk in G from u to u con- taining every vertex of M. Instead of hdEcG’ (u, u) we write simply hdo(u, u). We define the Humil- tonian diameter hdiam(G) of G by hdiam( G) =

m&,uEV(G) hdG(w u>.

If MI C_ M2 G V(G) then hdt’(u,u) <

hd,M’(u,u). In particular, for any A4 C V(G),

dc(u,u) < hdg(u,u) < hdo(u,u) < hdiam(G).

Remark 3. Graph characteristics similar to hdiam(G) have already been studied. In [2,3] a Hamiltonian walk in a connected graph G is defined to be any closed sequence of edges of minimal length which contains every vertex of G. The length of a Hamiltonian walk in G can, in our notation, be ex- pressed as hdo( u, u) - a value which is independent

of u E V(G). Hamiltonian walks are called Pseudo-

Hamiltonian circuits in [ 11, cf. also [ 41. In [ 10,111, an open Hamiltonian walk in a connected graph G is defined to be any open sequence of edges of minimal length which contains every vertex of G. Using our notation, the length of an open Hamiltonian walk in G equals min,+“hdG(u,u).

Theorem 4. Let G be a connected graph. Then

diam(QG) = IV(G)( +hdiam(G).

The theorem follows from Lemma 5. The proof of Lemma 5 is constructive, and could be easily made into a routing algorithm for QG. (The proof is rather long, so it was placed in Section 3.)

Lemma 5. Let G be a connected graph, and u, u E

V(G), L,M C V(G). Then

dQG((KL),(%M))

= IA(L,M)l+ hd;(L,M)(U,u).

Proof of Theorem 4. The inequality diam(QG) < lV( G) 1 + hdiam( G) follows from Lemma 5 as hd$L’M)(U, u) < hdiam(G). To show that equality is attained, take the vertices u, u of G (not necessar- ily distinct) such that hdo(z& u) = hdiam( G), and considerdQG((U,@),(u,V(G))). 0

1. Fri.? ef d/Information Processing Letters 61 (1997) 157-160 159

Lemma 6. anypathinQGfrom(u,L) to(u,M)isatleastr+s and then we construct a walk from (u, L) to (u, M) of length r + s. hdiam( C3 ) = 3,

hdiam(C,,)=n+j_n/2J -2 forn34. Let

Proof. Let us show for n > 3 that for any two vertices u # UOfC,,

p: (WL),..., (x’,K’), (.X”,K”,, . . .) (U.M)

be a path in QG, of length 1. Notice that

hdc,,(u,u) =n+dc,,(u,u) -2.

In general, there is more than one walk in C, from u

to u of length n + dc,, (u, u) - 2. For example: start at U, follow the edge away from u and take n - 1 steps, then turn back and reach u in dc,s (u, u) - 1 steps.

To show that there is no shorter walk, assume that w is a walk in C, from u to u containing all vertices. Consider the multigraph H induced on the vertices of C, by w. In H, u and u have odd, and all the other vertices even, degree, which means that the parity of the number of edges in H between two neighboring vertices changes only in u and in u. So, on one of the two paths from u to u in C,, each edge is used by w an odd number of times, while on the other path, an

even number of times. Any two edges removed from C, would disconnect it, so, all the edges of C,,, with at most one exception, must be used by w at least once. Consequently, H has at least n - 2 + dc,, (u, u) edges.

U,..., x’,x”,. . . ,u (1)

is not necessarily a walk in G yet. Consider an edge

e = [(x’,K’), (x”,K”)] of p. If e is basic, i.e. if x’ # x” then K’ = K” and [ax’, x”] is an edge of G. If e is cubical then x’ = X” and two identical vertices appear in ( 1) . Observe that since p is a path, there are no two consecutive cubical edges in it, hence there are at most two consecutive vertices equal to each other in ( 1) . By omitting one vertex from each pair of re- peating vertices, ( 1) becomes a walk, call it w’; let I’ be its length. For every x E A( L, M), there is at least one cubical edge [(x, K’), (x, K”)] in p, otherwise

p could not reach (u, M) starting from (u, L). Hence I’ < I - s. Since w’ contains all vertices of A( L, M),

1’ 3 hd$L’M’ (u, u) = r. Thus the length of an arbi- trary path from (u, L) to (u, M) is at least r + s.

Next we construct a walk

Clearly, hdc,, (u, U) = IZ for any vertex U, and for IZ > 4, n < II + [n/21 - 2. So, to establish the required equality, use max,,L, de,, (u, u) = \n/2]. IJ

The following theorem gives the value of the diam-

yo,...>yr+s

in QG from yo = (u, L) to yrfs = (u, M).

eter of cube-connected cycles.

Theorem 7.

diam( CCC,) = 6,

diam( CCC,) = 2n + [n/21 - 2 for n 3 4.

Proof. Combine Lemma 6 and Theorem 4. 0

3. Proof of Lemma 5

By the definition of hd$L’M’ (u, u) there is a walk w: X0 = c&Xl,..., x, = u in G from u to u which contains all the vertices of A( L, M). Choose the set of indices J = {jt,. . . , j,}, 0 6 j,, 6 r for 1 <

m 6 s SO that A(L, M) = {xj,, . . . ,xjy} - in general, this can be done in more than one way. We can now definethesequence(2).Form=O,...,r+sputy,,,= (Xi,, L,,) where the sequence io, . . . , ir+,T of indices and the sequence L,,-,, . . . , L,+, of subsets of V(G) are defined inductively as follows:

is = 0, Lo = L

and for 1 < m < r + s

Assume that G is connected, U, u E V(G), L, M C

V(G) . Put Y = hd;(L,M’ (u,u) and s = IA(L,M)l. We will prove the desired equality dec( (u, L) , (u, M) ) = Y + s in two steps. First, we show that the length of

. . l,, = ~,,-I 9 L = A(L-I, {xi,,_,}) if&-l E J,

i,, = in--l + 1, L,, = L,_l otherwise.

Note that ia,. . . , ir+s is a nondecreasing sequence of indices between 0 and r; it is increasing with the

(2)

160 I. FriS et al./Information Processing Letters 61 (1997) 157-160

exception of one pair of duplicate values for each el- ement of J. In (2) we get a basic edge [ y,+~, ym] when in,-1 < i,, and a cubical edge when i,_l = i,. This construction guarantees that (2) is a walk in QG

from (u, L) to (u, M) whose length is r + s. 0

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