Computers Elect. Engng Vol. 18, No. 5, pp. 411-420, 1992 0045-7906/92 $5.00+0.00 Printed in Great Britain. All rights reserved Copyright © 1992 Persamon Press Ltd

TECHNICAL NOTE

A MEASURE OF THE PERFORMANCE OF BINOMIAL INTERCONNECTION NETWORKS

)

SANJAY KHANNA t AT&T Bell Laboratories, Room 3M-621, Crawfords Comer Road, Holmdel, NJ 07733-1988, U.S.A.

(Received 20 April 1990; accepted in final revised form 13 February 1991; received for publication 8 October 1991)

Abstract--An important measure of the performance of an interconncction network is its capacity. The capacity of an ('~) binomial network is discussed as a measure of the average number of inputs that can be connected to the outputs. It is shown that for input subset of size less than or equal to d + 2, all inputs can be connected to outputs. For input subset of size greater than d + 2, the network is considered in terms of its prime decomposition class (pdc). We present an empirical formula for the capacity of (7) networks based on the number of configurations and capacity contribution of different factors in the pdc. It is shown that an (7) binomial network can provide capacity close to that of the complete crossbar switch while avoiding the latter's expensive hardware cost of contact switches. Next, we suggest a method to maximize the capacity of binomial network when selecting inputs. We illustrate how concepts of constant weight codes and residue classes can be applied to construct a type of (7) network called Optimally Designed Binomial (ODB) Network. We present an algorithm which selects inputs corresponding to different residue classes and uses the left cyclic shift feature to optimize the capacity of the network.

1. I N T R O D U C T I O N

Switching networks provide interconnecting routes in a distributed system from a set of network inputs to a set of network outputs through a set of contact switches, called crosspoints. These networks are commonly used in a large class of telecommunication and memory/processor/ peripheral systems [1]. The associated switching network in a multiprocessor or distributed system significantly affects the overall performance. Thus, the analysis and design of switching networks is an area of considerable research interest. A complete crossbar switch with N inputs and N outputs can provide N simultaneous connections, but has a prohibitive O ( N ~) hardware cost o f contact switches. We consider a class of sparse interconncction networks for distributed systems called binomial networks. The pattern of crosspoints between inputs and outputs in a binomial network follows the binomial distribution [2]. An (~') binomial network has (~') inputs connected to m outputs with d crosspoints per input. Here d is called the degree of the network. As an illustration a (~) network is shown in Fig. 1. One important measure of the performance of the network is its capacity, i.e. what is the average number of the input subset o f size k, k ~< m, that can be connected to the outputs at a time. Binomial networks find application in concentration and distribution [2]. Garey, Hwang and Richards [4] discuss the capacity of partial concentrations using a graph-the- oretic approach. N a k a m u r a and Masson [6] determine a lower bound on the number of crosspoints in sparse crossbar concentrators. Masson and Morris [3] analyze the capacity of binomial networks of degree 2. In Section 3, we discuss the capacity of the binomial network of any specified degree d, in terms of the number of configurations and capacity contribution of single and multiple factors. In Section 4, we illustrate how ideas from coding theory can be used to maximize the capacity of binomial networks when selecting inputs. We now review the concept o f prime decomposit ion class [3] of a binomial network.

A crossbar network can be expressed as a relation R between the input set and the output set. For i ~ input set and o • output set, o ~ R( i ) implies that there exists a crosspoint between input i and output o. Consider any I, a subset o f the input set and O, a subset o f the output set. A decomposit ion of I is an association of the subsets o f a partitioning of L denoted as ( I , , / 2 . . . . Iq), with the subsets of an equal sized partitioning of O, denoted as (O,, 02 . . . . . Oq), where for

tThis manuscript was written while the author was at Syracuse University, Syracuse, NY 13244, U.S.A.

411

412 Technical Note

Fig. I. A (~) binomial

1 2 3 4 5 6 7 8 9 10 I 1 I I I I I I I , - - - , - - - , - - - , - - - , - - - , - - - , - - - , - - - , . . . . a

I I I I I 1 I I I , - - - , - - - , - - - , - - - , - - - , - - - , - - - , - - - , . . . . b

I I I I I I I I I , - - - , - - - , - - - , - - - , - - - , - - - , - - - , - - - , . . . . c I I I I 1 I I I I , - - - , - - - , - - - , - - - , - - - , - - - , - - - , - - - , - - - , - d I I I I I I I I I I , - - - , - - - , - - - , - - - , - - - , - - - , - - - , - - - , - - - , - o

network. Number of inputs = (I) = 10. Number of outputs = 5. Degree = cross- points per input = 2.

all 1 <~j ~< q, / j and Oj are not empty, and R(/j) = O/. This decomposition is denoted as (I~/O~ . . . . . Iq/Oq) and each/r iO/is called a component of the decomposition. A component of a decomposition is a prime component if it cannot be further decomposed. A prime decomposition contains only prime components. Also classes of decompositions can be formed based on the sizes of input and output subsets. The prime decomposition class is denoted as pdc (1I, 1/I O,I . . . . . I tql/I Oqt) and each I/ : l /IOjl is called a factor of the pdc.

Example 1: Let us consider an input subset of the (5) network (Fig. 1) represented as I = 1, 2, 3, 10. A decomposition of I is {1/(a, b)}, {(2, 3, 10)/(c, d, e)}. This decomposition is also a prime decomposition since neither of the two components can be further decomposed. Here I is a member of pdc (1/2, 3/3), where 1/2 and 3/3 are factors of the prime decomposition class.

Two (~') networks can be combined with the help of m links to provide a connection between (~') inputs and (~') outputs. Letting N = (~'), the hardware cost of crosspoints for the two stage binomial network is 2.d.N. The complete crossbar switch, on the other hand, requires N 2 crosspoint switches to connect N inputs with N outputs. Thus, the relative hardware saving with the two stage binomial network compared to the complete crossbar switch is given by

N : - 2 N d N - 2 d (~')-2d N 2 N (~')

2. P R E L I M I N A R I E S

Theorem I. Given an G') network, for y <~ m, x/y is a potential factor of a IXlC only in the following range

F(y - 1 ) / (d - 1)] ~< x ~< (~)

where x and y are the size of the input and output subsets. PROOF: It is easy to see that the maximum number of inputs that can be connected to y outputs,

given that the degree of the input node equals d, is (,~). Thus, x ~< (,~). Also the network configuration shows that maximum number of outputs, y, that can be

connected to x inputs, is possible when any two input lines X 1 and X2 have crosspoints in common on exactly one output line.

Thus y ~<d + ( x - l ) x ( d - l ) o r y ~ < l + x x ( d - 1 ) or x >t (y - l)/(d- 1) or [ '(y - 1 ) / ( d - 1)7 ~< x, since number of inputs is an integer. Q.E.D.

The potential factors for the (s3) network are shown in Fig. 2. Thus, 1/3, 2/4, 2/5, 3/4 . . . . . 3/7, etc. are all potential factors.

Corollary I. Given an (~') network and an input subset I of size k less than or equal to d + 2, it is always possible to connect all inputs in I to k outputs. Here we say that the capacity measure of the subnetwork is d + 2.

Proof: For an G') network, we know that for x/y to be a potential factor the upper bound on x is given by, x ~< (~). To be able to connect all k inputs in an input subset to k outputs, we need

Technical Note 413

I N P U T S X

1 2 3 4 5 6 7

Y 3 x

0 U 4 x x x

T P 5 x x x x x x U

T 6 x x x x x s

7 x x ! x x ~

8 9 1 0 1 1

X X X

X X ~ X

X X X X

120 21 35 36 56

- - - X

- - - X X . . . . .

e x x x x x x ~ x . . . . . x x - - - x x . . . . . x

Fig. 2. Potential factors of (~) network. An 'X' in column X and row Y indicates that X/Y is a factor of the pdc.

to find the greatest value of x (say x,) such that for all factors of the form x,/y for the (~') network, x, is less than or equal to y. This will happen when we select the least y( = x~) such that, (~) > y. Since (d~")= d + 1 and (d~-e) > d + 2 , we have x , = d + 2 . Q.E.D.

Definition 1. Let iv(x, d) refer to the minimum value of z such that (~)/> x. We call iv(x, d) the inverse combinatorial of x with respect to d. As an example, iv (5, 2) is 4.

3. C A P A C I T Y C A L C U L A T I O N S F O R B I N O M I A L N E T W O R K S

Having determined the capacity measure for input subsets of size less than or equal to d + 2, we now consider the case when the number of inputs, k, is in the following range:

d + 2 < k < . m .

It may be pointed out that the maximum possible capacity for an (~') network is m, since we have exactly m outputs and therefore only m inputs can be connected at a time. To perform capacity calculation for k in the above range, we take into consideration the number of configurations of all possible factors in the network. Two types of factors, called single and multiple factors, are possible, which we discuss next.

3.1. Single and multiple factors

3.1.1. Single factors. Single factors are of the type (x/y), where x is equal to the number of inputs, and y is in the following range:

iv (x, d) ~< y <~ mini1 + ( d - 1) x x, m].

The capacity contribution of a factor x/y is given by the minimum of the number of inputs and outputs, i.e. rain(x, y). As an example, for the (]) network with input size o fk = 7, the single factors are as follows:

Factor Capacity contribution x/y min(x, y) 7/5 5 7/6 6 7/7 7 7/8 7.

The capacity contribution follows from the fact that only the minimum of the number of inputs and the number of outputs can be connected together.

414 Technical Note

3.1.2. Multiple factors. Multiple factors are of type (x 1/y 1 . . . . . xp/yp). The capacity contri- bution of each multiple factor is given by,

P

min(xj, yj). j = l

To illustrate the concept of multiple factors, let us again take the example of the (]) network with input size k = 7:

Multiple factor Capacity contribution (1/3, 6/5) 1 + 5 = 6 (3/4, 4/4) 3 + 4 = 7.

It must be noted here that the sum of the number of inputs in a multiple factor should be k and that the number of outputs should not exceed m. Thus (3/5, 4/4) is not a valid multiple factor.

For an (~') network with a given input size k, the following factors of multiplicity 2 are possible:

[x/yl, (k-x)/y2] where l<~x~<k div 2, iv(x, d)~< yl ~< l + ( d - 1 ) x x, and iv(k-x,d)<~y2<, min[ l+(d-1) x ( k - x ) , m - y l ] .

The above bounds on inputs and outputs in the multiple factors follow from the observation that the total number of inputs is k, that both y 1 and y2 are in the range specified by Theorem l, and that y 1 + y2 can be at most equal to m. The multiple factors for which y2 lies outside the range specified by the above inequality are excluded.

3.2. Number of configurations

We will now present expressions for the number of configurations of single and multiple factors of a prime decomposition class.

3.2.1. Single factors. Single factors have the form k/yl , where k is the number of inputs. For a (7) network the following is an upper bound on the number of configurations for this factor:

A single factor x/yl is sNd to cover a single factor x/y2 i f y l >y2. A single factor x/y is also said to cover a multiple factor (x 1/yl . . . . . xr/yr) if:

( x l + . . . + x r ) = x and ( y l + . . . + y r ) ~ y .

As an example, for the (1) network the single factor 7/8 covers the factors 7/7, 7/6, 7/5, (1/3, 6/5) and (3/4, 4/4). For single factors which cover other factors the number of configurations is reduced to account for the covered factors. The point is illustrated in the next subsection with the help of an example.

3.2.2. Multiple factors. For a given multiple factor of an (~') network

(x 1/y 1, x2/y2 . . . . . xr/yr) w h e r e d + 2 < ( x l + x 2 + . . . + x r ) = k <~m a n d y l + y 2 + . . . +yr=Y<~m,

the following is an upper bound on the number of configurations for the factor:

t)(,

Technical Note 415

which for factors of multiplicity 2 reduces to:

¢:,i (I lll) (¢:ilI As an example, for the multiple factor (1/3, 6/5) of the (~) network the total number of

configurations with Y = 3 + 5 = 8 are as follows:

A multiple factor ( x l / y l . . . . . x i / y i . . . . . x r / y r ) is said to cover another multiple factor ( a l / b l . . . . . a j /b j , . . . . a p / b p ) , where x l + . . . + x r = a l + . . . + ap, if there exist one or more factor component(s) of the form x i / y i in the first multiple factor each of which cover one or more factor component(s) of the form a j / b j in the latter multiple factor (in accordance with the definition of cover for single factors), and the remaining factor components, if any, are the same between the two multiple factors. As an example, the multiple factor (1/2, 7/6) of the (s) network covers the multiple factors (1/2, 7/5) and (1/2, 1/2, 6/4) for input subsets of size 8. For multiple factors which cover other multiple factors, the number of configurations is reduced to account for the covered factors.

3.3. Capac i t y calculat ions

To calculate the capacity measure for a given (~') network, we first identify all single and multiple factors for the specified input size. We calculate the capacity contribution and the number of configurations for all the factors as explained above. To illustrate, let us consider the (~) network with input size, k, of 7. We note here that there are (~6) subsets of size 7 that we must cover with the six factors in Fig. 3. We only need to calculate the number of configurations for factors whose capacity contribution is less than 7 (k, in general). The remaining configurations will automatically have capacity contribution of 7. For the purpose of enumerating the number of configurations of the factor 7/6, let us limit ourselves for the time being to a specific choice of six of the eight outputs.

FACTOR CAPACITY ~ OF CONFIGURATIONS CONTRI BUT, ON

(") (ci ) ( ' } ( ] / 3 , 6 /5) 6 . . . = 11760 3 8

(3/4, 4/4) 7

6720

7/6 6 [ C ( i ) ) - ( ( ! ) ) . { : } ] . ( : } = 2150400

7/7 7

7/8 7

CC~iFIG~TI(~IS WITH CAPACITY CORI~IBUTIC~I LESS THAN 7 = 2168880

TOT~ I~NSEZ OF ~ Y S 0F SELECTING 7 II~UTS 0UT 0F 56 = (5677

Ac,Tyo. C : ) : [ 6720'*5 + 2150400*6 + 11760'*6 + ( ( ' ~ 7 ) " 2168880 ) * 7 ]

= 6 .99

Fig. 3. Capacity calculation for (~) binomial network with input subset, k, of size 7.

416 Technical Note

The number of subsets of seven inputs with crosspoints only to these specific six outputs is given by:

We need to reduce the number of these configurations to account for the covered factor 7/5. The number of subsets of 7 inputs with crosspoints to only five of these six outputs is given by:

Given six specific outputs, the number of choices of seven inputs covered by 7/6 is 77,520-720 = 76,$00, Next we observe that given eight outputs in the (I) network there are (~) choices of six of these eight outputs. Thus, the number of configurations of the factor 7/6 is, (I). 76,800 = 2,150,400.

The number of configurations for other factors is shown in Fig. 3. We take a weighted average of the capacity contribution of all configurations to give us a prol~bilistic measure of the capacity. We notice from Fig. 3 that for a (I) network the single factors have a much higher number of configurations compared to the multiple factors. This is generally true for a (~') network. Thus, a fairly accurate estimation of the capacity of an (~') network can be obtained by taking into account the reduced capacities due to single factors only. Generalizing the capacity calculations for the (I) network discussed above, we can obtain a closed form for the capacity of (~') networks. The capacity of the network for a given input subset size k can be expressed as follows:

c ( k ) =

~=~,~ _ d

k

(/i)) + k '

The first term in the above expression represents the configurations with capacity y, where iv(k, d)~<y ~< k - 1, whereas, the second term represents the configurations with capacity k. The above capacity expression simplifies to the following form:

[(/!)) d .y • .(k - y ) - k

c ( k ) = k

Technica l N o t e 417

Next, we observe that:

is much larger than:

Thus, we can approximate the above expression to obtain an estimate of the capacity of (~') network:

(/!)) (:/ ~_~:~k, ~ • "(k - y )

c(k) ~, k

We verify the accuracy of the empirical formula for the capacity of (~') networks, expressed in equation (1), by comparing the results obtained from our equation with the expected capacities calculated in [3]. The capacities are calculated in Table 1 for an (~') network, with input subsets of size k. Here E represents the expected capacity calculated in [3] and C represents the capacity obtained from equation (1).

As can be seen from the above computations, the closed form of the capacity of an (~') network, expressed in equation (1), provides better than 95% accuracy.

The binomial network has the potential to provide capacity very close to that of a complete crossbar switch while avoiding the prohibitive hardware cost associated with the latter. We illustrate this point for an (~') network in Table 2. Here CC expresses the capacity of the complete crossbar switch for input subsets of size k and C represents the capacity of the (~') binomial network, as determined by equation (1).

It can be seen from Table 2 that binomial networks with high degree, d, provide capacities close to that of the complete crossbar switch.

4. M A X I M I Z I N G T H E C A P A C I T Y OF (7) N E T W O R K S

Certain applications provide us with the flexibility to connect an unspecified set of inputs to outputs. For instance, a concentration network [5] is a switching network which is capable of

Table I. Accuracy of capacity calculations for (T) binomial networks

m k E C % Accuracy

Table 2. Comparison of capacity of (~') network and the complete crossbar switch

m d k C CC 6 5 4.881 4.881 !00.00 6 6 4.970 4.970 ! 00.00 8 3 6 5.9996 6.0 7 5 4.990 4.990 lO0.O0 8 3 7 6.9907 7.0 7 6 5.908 5.919 99.81 8 3 8 7.8675 8.0 7 7 6.569 6.613 99.34 9 3 7 6.9986 7.0 8 5 4.996 4.996 100.00 9 3 8 7.9808 8.0 8 6 5.962 5.969 99.88 9 3 9 8.8149 9.0 8 7 6.819 6.854 99.50 I0 2 8 7.8996 8.0 8 8 7.405 7.479 99.00 10 2 9 8.6950 9.0 9 5 4.998 4.998 100.00 10 2 10 9.2446 10.0 9 6 5.983 5.987 99.94 I 0 3 9 8.9682 9.0 9 7 6.916 6.939 99.66 10 3 10 9.7627 i0.0 9 8 7.709 7.777 99.12 I0 4 9 8.9984 9.0 9 9 8.239 8.353 98.61 10 4 10 9.9479 I0.0

418 Technical Note

POSITIONAL VALUE

6 5 4 3 2 1 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 1 1 1 1 0 0

0 0 1 0 1 1 1

1 1 0 1 0 0 1

1 0 1 1 0 1 0

1 1 0 0 1 1 0

Fig. 4. Generators for constant weight codes of weight, d = 4 and length, m = 7.

connecting, within the capacity limits of the network, a specific idle output terminal to an unspecified, idle input terminal.

Some ideas from coding theory can be used to maximize the capacity of (~') networks when selecting inputs. Binary codes of length m in which each code has d number of l 's are called constant weight codes of weight d. The number of such codewords, n, is equal to (~'). Since the weight and length of the codeword is fixed, the complete list can be obtained by making m - 1 cyclic shifts on a set of codewords called generators. For a codeword of weight d and length m there are g = n/m generators. As an example, for d = 4 and m = 7,

the number of codewords, n = (7) = 35, and the number of generators, g = 35•7 = 5.

The 5 generators are listed in Fig. 4.

4.1. Residue classes

Consider a constant weight codeword, Y, of length m and weight d.

Y = (Y,,-i . . . . . Yl, Y0)

Let P(Y) = (~Tj0~j x yj) mod m.

Let Rj denote the set of codewords for which P(Y) = i, 0 ~< i ~< m - 1. These R;s constitute the residue classes of the code. As an illustration, for the codeword Y = 0 1 1 1 0 0 1, P(Y) is calculated as follows:

P ( Y ) = ( 0 × l + l × 0 + 2 x 0 + 3 × l + 4 x l + 5 × l + 6 x 0 ) mod 7 = 5 .

Here Y belongs to the residue class Rs. As a convention, the generators belongto residue class R0. Moreover, each cyclic shift to the left changes the residue class of the codeword from Ri to R ( i + d)mod(m) •

I N P U T S 1 6 11 16 21 26 31 35

t i t & i &

! i I IF i ! [ -1-T i IIW

. i , t ~ h e h . , ~. el, i .,lh~

:[ [ ~ -"I [ Z: I [.,7,[ ] F.I ] [ I,

I = = = { I I I 2 [ ] { l ~ ] [ ] [ ] .

,.,.Ik,,I.,,,Jk ~ ]

i::::=:.-]]l[]IIl]] . . . . . . . . . .

R4 RI R5 1:12 R6 R3

Fig. 5. Optimally d e d ~ e d binomial (47) nq

I O

I

2

3

4

. . . . 6

network.

T e c h n i c a l N o t e 4 1 9

I N P U T S

1 6 11 16 21 26 31

I I I I . . . . [ . . . . x . . . . x . . . . x . . . . o . . . . I - 0 0 I I I I . . . . I . . . . I . . . . x . . . . x . . . . x . . . . o - 1 U I I I o . . . . I . . . . I . . . . I . . . . x . . . . x . . . . x- 2 T I I I x . . . . o . . . . I . . . . I . . . . l . . . . x . . . . x- 3 P I I I x . . . . x . . . . o . . . . I . . . . I . . . . I . . . . x- 4 U I I I

x . . . . x . . . . x . . . . o . . . . l . . . . . . . . I - S T

I I I I . . . . x . . . . x . . . . x . . . . o . . . . I . . . . I - 6 S

Fig. 6. An illustration of the algorithm to optimize the capacity of a (47) network. An "x " in the network represents a crosspoint, and an "o" represents a closed crosspoint, i.e. one in which a connection is made between the input and the corresponding output. For an input subset size of 7, selecting inputs I, 6, 11, 16, 21, 26 and 31 to be connected to the outputs, ensures that the network will achieve its maximum

c a p a c i t y o f 7 .

An (~') network can be constructed using a constant weight code of weight d and length m. Each of the output lines in the network is assigned a number between 0 and m - 1. For a 1 at position p in the codeword, there is a crosspoint on output line p in the network. The first g input lines are constructed to correspond to the g generators belonging to the residue class R0. Each subsequent set of g input lines correspond to codewords obtained by a left cyclic shift of the codewords associated with the previous set of g lines. We refer to such a network as an Optimally Designed Binomial (ODB) network. The illustration for a (7) ODB network is given in Fig. 5. Having designed an (~') network in this fashion, we can use the following algorithm to obtain the maximum possible capacity.

4.2. Algorithm to maximize the capacity of an ODB network

Given an ODB Network, let k ~< m be the number of input-output connections that we wish to establish. We select the first input from one of the g input lines corresponding to the residue class R0. The remaining k - 1 inputs are selected, one from each of the succeeding residue classes. Specifically, input lines whose codewords correspond to those obtained by successive left cycle shifts of the codeword associated with the first input line are selected. Thus, if the first input line selected in an ODB network is i (where i ~ g), then the other input lines for an input subset size k > 1 are i + j x g, where 1 ~<j ~< k - 1.

Example: For the (7) network shown in Fig. 5, with input size, k = 7, if the first selected input line is 1, then the other input lines would be 6, 11, !6, 21, 26 and 31 belonging to residue classes R4, RI, Rs, R2, R 6 and Rs, respectively (Fig. 6).

Claim: It is always possible to connect k inputs (k ~< m), selected using the above algorithm, to k outputs (i.e capacity = k).

Proof: The following crosspoint selection gives capacity k. Connect the first input using the crosspoint connecting to the lowest order output line (say p). Connect subsequent input lines to output lines numbered (p + j ) mod m, 1 ~<j ~ k - I. Such a connection is always able to connect all k inputs to k outputs due to the fact that input lines were selected to correspond to the left cyclic shift of the codeword of the first line. This is further illustrated for the (I) network in Fig. 6.

5. C O N C L U S I O N

In this paper, we have discussed an approach to calculate the capacity of (~') binomial networks. We have presented empirical expressions to calculate the number and type of various factors in a prime decomposition class of the binomial network. These configurations along with the capacity contribution of each configuration give us a probabilistic measure of the capacity of the binomial network. As demonstrated by our calculations, a binomial network provides the necessary

420 Technical Note

in te rconnec t ion in a d i s t r ibu ted archi tec ture with capac i ty very close to tha t o f a comple te c rossbar switch, but wi thou t its expensive archi tecture .

We have in t roduced a class o f ne tworks , called Op t ima l ly Des igned Binomia l ne tworks , i nco rpo ra t i ng concepts f rom coding theory and have presented an a lgor i thm to maximize the capac i ty o f the ne twork when selecting inputs . The discussion in this p a p e r has significance in the pe r fo rmance o f in te rconnec t ion ne tworks for d i s t r ibu ted archi tectures .

Acknowledgement--The author would like to thank Dr Arif Ghafoor for some useful discussions on this topic.

R E F E R E N C E S

1. c. D. Thompson, Generalized connection networks for parallel intercommunication. IEEE Trans. Comput. 27, 1119-1125 (1978).

2. G. M. Masson, Binomial switching networks for concentration and distribution. IEEE Trans. Communs 25, 873-883 (1977).

3. G. M. Masson and S. B. Morris, Expected capacity of (~') networks. IEEE Trans. Comput. 32, 649--657 (1983). 4. M. R. Garey, F. K. Hwang and G. W. Richards, Asymptotic results for partial concentrators. IEEE Trans: Communs

36, 214--217 (1988). 5. M. Pinsker, On the complexity of a concentrator. Proc. 7th Int. Teletraffic Conf., Stockholm, Sweden, pp. 318/1-318/4

(1973). 6. S. Nakamura and G. M. Masson, Lower bounds on crosspoints in sparse crossbar concentrators. IEEE Trans. Comput.

31, 1173-1179 (1982).

AUTHOR'S BIOGRAPHY

i~ili~i~ ¸̧~ i

Sanjay Klmnm---Sanjay Khanna is a Member of the Technical Staff at AT&T Bell Laboratories, Holmdel, New Jersey, U.S.A. He obtained his B.E. (Hons) from Birla Institute of Technology and Science, Pilani, India. He received his:M.S. and Ph.D. degrees in computer engineering from Syracuse University, Syracuse, New York, U.S.A. in 1987 and 1990, respectively. He was awarded the National Talent Search Scholarship in 1978 and the Northeast Parallel Architecture Fellowship in 1987. His current areas of research interest are computer networks, protocol verification and parallel computing.