Undirected Loop Networks

Sheng Chen and Xing-De Jia

Department of Mathematics, Southwest Texas State University, San Marcos, Texas 78666

Let M, d , and k be positive integers. We say that M is feasible with respect to d and k if there exists a set A = (0, 21, +a2. . ., + a k } such that the Cayley graph associated with Z / ( M ) and A has diameter less than or equal to d . Such a Cayley graph is a model for undirected loop communication networks. Denote M(d, k) the maximal feasible number M with respect to d and k. In the paper, an explicit formula for M(d, 2) is obtained. Also, when k 2 3, a lower bound for M(d, k) is established. 0 7993 by John Wiley & Sons, Inc.

1. INTRODUCTION

Let M be a positive integer. Let A = (0, ? 1 = + a , , +a2 , . . . , +-ak} be a set of integers. The Cayley graph associated with Zl(M) and A is a graph whose vertices are the elements in the cyclic group Zl (M) and that the vertex i is adjacent to i + aj fo r j = 1, 2 , . . . , k . Let M , d , and k be positive integers. We say that M is feasible with respect to d and k if there exists a set A = (0, 2 1, ' a , , . . . , +ak} such that the Cayley graph G ( M , A ) associated with Zl(M) and A has diameter less than or equal to d . Denote M(d, k ) as the maximum positive integer M such that there exists a set A = (0, a l , +az, . , . , + a k } of integers satisfying that every integer is congruent to a sum of d not necessarily dis- tinct elements from A modulo M.

The Cayley graphs defined above have applications to local area communication networks and in the con- struction of massively parallel processors (see, e.g., Bermond et al. [ l , 21 and Hsu and Shapiro IS, 91). In fact, they are also known in the literature as undirected loop communication networks (see [ l , lo]). In this pa- per, we study the extremal function M(d, k ) . In Section

*The authors were told that this theorem was also described by some other authors in some form.

2 , we prove an explicit formula for M(d, 2 ) . In Section 3 , we give a lower bound of M(d, k ) in terms of d for any k 2 3. In Section 4, we list some computer-gener- ated data and discuss some problems.

2. EXPLICIT FORMULA FOR M(d, 2)

Let dA denote the set of all sums of d not necessarily distinct elements from A.

Theorem l.* For any d 2 1,

M(d, 2 ) = 2d2 + 2d + 1.

Proof. It is obvious that M(d, 2 ) I 2d2 + 2d + 1. Let a = 2d + 1, A = (0, 21, * a } . We show that M ( d , A) 2 2d2 + 2d + 1. Let n be any integer. We may assume without loss of generality that 0 5 n I 2d2 + 2d + 1. We consider two cases that may occur:

CASE I. There exists an i: 0 5 i 5 d such that

ia - (d - i ) d n I ia + (d - i ) .

Then, n = ia + r , where - ( d - i ) I r I (d - i ) . Noting that i + Irl I i + (d - i ) = d , we see that n E dA.

NETWORKS, Vol. 23 (1993) 257-260 0 1993 by John Wiley 8 Sons, Inc. CCC 0028-30451931040257-04

257

258 CHEN AND JIA

CASE 11. There exists an i : 0 I i i d such that

ia + (d - i) + 1 I n

5 (i + 1)a - (d - (i + 1 ) ) - 1.

Suppose that

n = i a + ( d - i ) + 1 + r ,

where

0 5 r 5 {(i + l)a - ( d - (i + 1 ) ) - 1 )

- {ia + (d - i) + 1 ) = 2i.

Therefore.

n = i d + ( d - i ) + l + r = 2 i d + d + I + r = 2dz + 2d + 1 - ( d - i)(2d + 1) + ( r - 1 ) - ( d - i ) ( -a ) + (r - i)(mod2& + 2d + 1).

Noting that - i 5 r - i I i, we see that (d - i ) + ) r - i( 5 d . Hence, n E d A .

Therefore, M ( d , 2) 2 2d2 + 2d + 1. This com- pletes the proof of the theorem.

3. LOWER BOUND FOR M(d, k ) WHEN k 2 3

Let n be a positive integer. For any positive integer n, define

gn (x)

x - 2kn,

and

= !Ik,

if (2k - 1)n < x I (2k + 1)n i f - n s x s n if - (2k + l ) n s x < -(2k - 1)n

if (2k - 1)n < x 5 (2k + 1)n i f - n l x l n if - (2k + I)n 5 x < - (2k - 1)n

Lemma 1. For any positive integer n and any real number x.

IgnCx>l+ Ig,(x - n)l= n. Proof This is straightforward from the definition.

Theorem 2. Let d and k be any positive integers such that d 2 k 2 3. Let n = [ (d - k + 3) / kJ . Then

M ( d , k ) z 2 n y (4n)' = 2 ( 4 7 d k + 0 ( d h - I ) . i = O

Proof. We may assume without loss of generality that d = k n + k - 3. Let A = (0, +alp + a 2 , * - - , +ak} , where a, = (4n)'-1 and let M = 2na,. For any positive integer x (x -= M ) , let x1 be the integer in ( - 2n , 2n] such that xI = x (mod4n). For any 2 I i I k, let xi be the integer in ( - 2 n , 2n] such that

xi = (x - xjaj ) (mod 4n) . a;

Then, x = xlal + x2a2 + * + X k a k , with 0 5 xk 5 2n and -2n < xi 5 2n for all i = I , 2 , * * , k - I .

We claim that either x E d A or x - M E d A . Note that gzn(xi ) = xi for i = 1 , 2 , . . . , k and that

So, it suffices to show that

or

Suppose that is not the case. Then, we have

and

UNDIRECTED LOOP NETWORKS 259

which is impossible when k L 4.

Therefore, If k = 3, then all above equalities must hold.

m 0 1

17

16

15

14

9 11 10

A =(0.*l . i3 , f8)

Fig. 1.

ifx, 2 0 - 1 ifx,<O' i" EZn(Xj - 2n) =

TABLE 1. M(d, 3) for 2 s d 5 10, where A = (0, el, *a, *b)

2 21 3 55 4 117 5 203 6 333 7 515 8 737 9 1027

10 1237

as 2n + x2 1 0. This shows that (3) and (4) cannot hold at the same time; hence, either x or x - M is in dA, which completes the proof of Theorem 2.

4. EXAMPLES AND PROBLEMS

Table I lists M ( d , 3) for all 2 i d 5 10 and all corre- sponding sets A = (0, +1 , k u , kb} , except for the multiples of an n invertible element.

Figure 1 shows the network associated with M(2, 3) where a = 3, b = 8.

Problem 1. Is it true that M(d, k) is odd for all d and k ( k 1 2)? Note that, by Theorem 1, M ( d , 2) is odd and, from Table I, M ( d , 3) is odd for all 2 5 d 5 10. The problem seems easy, but we could not prove or disprove it.

260 CHEN AND JIA

TABLE II. d(M, 3) and corresponding M's

d ( M , 3) M

2 3 4 5

6

7

8

9

10

4-2 I 22-50, 55 51-54, 56-102, 105, 1 1 1 , 117 103, 104, 106-110, 112-116, 118-185, 187, 190,

191. 197, 203 186, 188, 189, 192-196, 198-202. 204-293,

295-302, 304-307, 313, 314, 317, 323, 333 294, 303, 308-312, 315, 316, 318-322, 324-332,

334-436, 438-465, 468-471, 479, 480, 495, 505, 515

437, 466, 467, 472-478, 481-494, 496-504. 506-514,516-668,670-674,676,677,680-684, 687, 689, 694, 695, 703, 713, 717, 727, 737

669, 675, 678, 679, 685, 686, 688. 690-693, 696-702,704-712.714-716,718-726.728-736, 738-926,928-938,941-943,947-950,953,955, 957, 958, 960-963, 974, 979, 989. 999, 1003, 1013, 1027

927, 939, 940, 944-946, 951, 952, 954, 956, 959, 964-973, 975-978, 980-988, 990-998, 1000- 1002, 1004- 101 2, 101 4- 1026, 1028- 1097, 1099, 1101, 1103, 1104, 1110-11l2, 1116, 1121, 1125-1127, 1129, 1131, 1135, 1141, 1143, 1145-1 150, 1152, 1157, 1160, 1162-1 165. 1169, 1181, 1211, 1227, 1237

Problem 2. It is obvious that, for any k ,

( 5 ) 2k k !

M ( d , k ) I - dL + O(dX-' ).

A nonobvious upper bound is very desirable. Given two positive integers M and k , let d(M, k )

denote the least positive integer d such that M is feasi- ble with respect to d and k. In other works, d(M, k ) is the diameter of the Cayley graph G(M, A), where A = (0, +al = k 1, k u 2 , . . . , Lak} for which G(M. A) has minimum diameter. It follows from ( 5 ) and Theo- rem 2 that

k -

where the upper bound holds for an infinite family of positive integers M and the lower bound holds for all

positive integers M. It is interesting to improve the above estimates. Table I1 lists d ( M , 3) for M I 1000.

Problem 3. Note that, when k is fixed, d(M, k ) is not a monotone increasing function of M. For example, we have d(54, 3) = 4 > 3 = 4 5 5 , 3). Is it true that d ( M , k ) 5 d ( M ' , k) + 1 for any M < M'? When k = 2, Tzvieli [lo] verified that this is true for M < M' 5 8 - lo6 and she conjectured that this true for all M < M'. Table I1 shows that d(M, 3) 5 d(M' , 3) + 1 for all M < M' 5 1000. We like to remark that the answer to a similar question in the directed case is negative [ 6 ] . A weaker version of this question is as follows: Does there exist a constant c depending only on k such that, for M < M',

d ( M , k ) 5 d(M', k ) + c ?

The authors would like to thank Professor D. Frank Hsu for his helpful conversation with one of the authors about the subject and thank Professor Ricardo Torrej6n for his help in computing the data that appeared in this paper.

REFERENCES

J.-C. Bermond, F. Comellas, and D. F. Hsu, Distrib- uted loop computer networks, a survey, J. Parallel and Distrih. Comput. to appear. J.-C. Bermond, G. Illiades, and C. Peyrat, An optimiza- tion problem in distributed loop computer network. Ann. N . Y . Acad. Sci. (1989). J.-C. Bermond and D. Tzvieli, Minimal diameter double-loop networks: Part 11: Dense optimal families. Networks, 21 (1991) 1-10. D. V. Chudnovsky, G. V. Chudnovsky. and M. M. Denneau, Regular graphs with small diameter as mod- els for interconnection networks. 3rd Int. Conf. on Supercomputing, Boston, May ( 1988) 232-239. D.-Z. Du, D. F. Hsu, Q. Li, and J. Xu, A combinatorial problem related to distributed loop networks. Net- works, 20 (1990) 173-180. D. F. Hsu and J. Shapiro, Bounds for the minimal number of transmission delays in double loop net- works. J. Comhinatorics, Information and System Sci- ences, 16, (1991) 55-62. D. F. Hsu and J . Shapiro, A census of tight one-optimal double loop networks, Graph Theory, Combinatorics. Algorithms and Applications (ed. by J. Alavi et al.),

D. Tzvieli, Minimal diameter double-loop networks: Part I: Large infinite optimal families. Networks, 21

SIAM (1991) 254-265.

(1991) 387-416.