Permutatin Capability of Optical Multistage

Journal of Parallel and Distributed Computing 60, 72�91 (2000)

Permutation Capability of Optical MultistageInterconnection Networks

Yuanyuan Yang1

Department of Electrical and Computer Engineering, State University of New York at Stony Brook,Stony Brook, New York 11794

Jianchao Wang

GTE Laboratories, 40 Sylvan Road, Waltham, Massachusetts 02254

and

Yi Pan2

Department of Computer Science, University of Dayton, Dayton, Ohio 45469

Received September 2, 1997; accepted August 24, 1999

In this paper, we study optical multistage interconnection networks(MINs). Advances in electro-optic technologies have made optical com-munication a promising networking choice to meet the increasing demands ofhigh-performance computing�communication applications for high channelbandwidth and low communication latency. Although optical MINs holdgreat promise and have demonstrated advantages over their electronic coun-terpart, they also hold their own challenges. Due to the unique properties ofoptics, crosstalk in optical switches should be avoided to make them workproperly. Most of the research work described in the literature is on elec-tronic MINs, and hence, crosstalk is not considered. In this paper, we intro-duce a new concept, semi-permutation, to analyze the permutation capabilityof optical MINs under the constraint of avoiding crosstalk, and we apply itto two examples of optical MINs, the banyan network and the Benesnetwork. For the blocking banyan network, we show that not all semi-per-mutations are realizable in one pass, and we give the number of realizablesemi-permutations. For the rearrangeable Benes network, we show that anysemi-permutation is realizable in one pass and any permutation is realizablein two passes under the constraint of avoiding crosstalk. A routing algorithm

Article ID jpdc.1999.1595, available online at http:��www.idealibrary.com on

720743-7315�00 �35.00Copyright � 2000 by Academic PressAll rights of reproduction in any form reserved.

1 Research supported by the US Army Research Office under Grant DAAH04-96-1-0234 and by theNational Science Foundation under Grants OSR-9350540 and MIP-9522532.

2 To whom should be adressed. E-mail: pan�udcps.cps.udayton.edu. Research supported by theNational Science Foundation under Grants CCR-9211621, OSR-9350540, and CCR-9503882.

for realizing a semi-permutation in a Benes network is also presented. Withthe speed and bandwidth provided by current optical technology, an opticalMIN clearly demonstrates superior overall performance over its electronicMIN counterpart. � 2000 Academic Press

Key Words: optical interconnects; guided wave technology; crosstalk; mul-tistage interconnection networks; permutations; parallel computing.

1. INTRODUCTION

Communications among processors in a parallel computing system are alwaysthe main design issue when a parallel system is built or a parallel algorithm isdesigned. With advances in silicon and Ga�As technologies, processor speed willsoon reach the gigahertz range. Traditional metal-based communication technologyused in parallel computing systems is becoming a potential bottleneck. Thisrequires either that significant progress be made in the traditional interconnects orthat new interconnect technologies, such as optics, be introduced in parallel com-puting systems.

Advances in electro-optic technologies have made optical communication apromising networking choice to meet the increasing demands of high-performancecomputing�communication applications for high channel bandwidth and low com-munication latency. Fiber optic communications offer a combination of highbandwidth, low error probability, and gigabit transmission capacity. They havebeen extensively used in wide-area networks and have received much attention inthe parallel processing community as well. In fact, many commercial massivelyparallel computers such as the Cray C90 use optical technology in their com-munication subsystems.

Using new optical technologies in parallel computers may require us to rethinkhow we design interconnection networks and write parallel algorithms. Fullyexploring the capabilities of optical technology requires careful analysis of theproperties of optics, proposal of new performance measures, and design of newinterconnection networks and routing algorithms and new parallel application algo-rithms. Some research results in this area are described in [10].

Multistage interconnection networks (MINs) have been an important intercon-necting scheme for parallel computing systems. A MIN can be blocking, such as abanyan network [6], rearrangeably nonblocking, such as a Benes network [1],nonblocking, such as a crossbar [16], or with variable connecting capabilities, fromrearrangeable for permutation to nonblocking for multicast, such as a Clos network[1, 4, 20], depending on the number of stages, the number of switches, the switchcapability, and the interconnection patterns used between stages. MINs have beenextensively studied in the literature; e.g., see [1, 4, 6, 19, 20]. However, most of theresearch reported is on electronic MINs.

As optical technology advances, there is considerable interest in using opticaltechnology to implement interconnection networks and switches. Although elec-tronic MINs and optical MINs have many similarities, there are some fundamentaldifferences between them. The major problem in optical MINs is the problem of

73OPTICAL NETWORK PERMUTATION CAPABILITY

crosstalk in the optical switches. Although the problem can be reduced by usingfree-space interconnects instead of fiber optics, the crosstalk problem can still besignificant due to the alignment to the problem, etc. To avoid the crosstalkproblem, traditional routing algorithms and results for electronic MINs are notapplicable here [11�13, 18]. New research is needed to address the issuesassociated with optical MINs.

In this paper, we will first review some previous related research. In particular,the optical switching model, the limitations and problems of optical MINs, and theapproaches to solving these problems will be discussed. Then permutation in opti-cal MINs will be studied. Parallel permutations of data are frequently needed inparallel processing. It is desirable to have a MIN that can realize necessary per-mutations in a minimum number of routing steps. Although extensive research onelectronic MINs has been done, the research results cannot be used directly in opti-cal MINs. In this paper, we will mainly consider the permutation capability ofblocking and rearrangeably nonblocking optical MINs.

2. PRELIMINARIES

In this section, we give some background and assumptions which are necessaryfor the analysis of optical MINs.

2.1. The Switching Model

An optical MIN can be implemented with either free-space optics or guided wavetechnology. Several optical crossbars and MINs have been proposed in theliterature [8, 17]. Some implementations have also been reported [7, 16]. Toexploit the huge optical bandwidth of fiber, the Wavelength Division Multiplexing(WDM) technique can also be used. With WDM, the optical spectrum is dividedinto many different logical channels, and each channel corresponds to a uniquewavelength [13]. In this paper, we consider optical implementation with guidedwave technology. The switching model can be described briefly as follows.

Optical switching, as the name implies, involves the switching of optical signals,rather than electronic signals as in conventional electronic systems. Two types ofguided wave optical switching systems can be identified. The first is a hybridapproach in which optical signals are switched, but the switches are controlled elec-tronically. With this approach, the use of electronic control signals implies that therouting will be carried out electronically. As such, the speed of the electronic switchcontrol signals can be much less than the bit rate of the optical signals beingswitched. The second approach is all-optical switching. This potentially overcomesthe problem associated with the hybrid approach. However, such systems will notbecome practical until far into the future [9], and hence only hybrid optical MINsare considered in this paper.

In hybrid optical MINs, the electronically controlled optical switches, such aslithium niobate directional couplers [5], can have switching speeds from hundredsof picoseconds to tens of nanoseconds. If packet switching is used, the address

74 YANG, WANG, AND PAN

information in each packet must be decoded in order to determine the switch state.In a hybrid MIN, such a requirement usually means conversions from opticalsignals to electronic ones, which could be very costly. For this reason, circuitswitching is usually preferred in optical MINs. In this paper, we assume that circuitswitching is used.

Wide-band optical signals can be switched under electronic control using direc-tional couplers between Ti:LiNbO3 waveguides on a planar LiNbO3 crystal [11].The basic switching element is a directional coupler with two active inputs and twoactive outputs. Depending on the amount of voltage at the junction of the twowaveguides which carry the two input signals, either of the two inputs can becoupled to either of the two outputs. Many architectures have been proposed toconstruct an n_n MIN using the 2_2 directional coupler as the basic component.These architectures are essentially analogs to similar architectures for electronicswitching and MINs.

2.2. Limitations of Optical MINs

Due to the difference in speeds of the electronic and optical switching elementsand the nature of optical signals, optical MINs also hold their own challenges. Oneproblem is path-dependent loss. Path dependent loss means that optical signalsbecome weak after passing through an optical path. In a large MIN, a substantialpart of this path-dependent loss is directly proportional to the number of couplersthat the optical path passes through. Hence, it depends on the architecture usedand its network size. Another problem is optical crosstalk. Optical crosstalk occurswhen two signal channels interact with each other. There are two ways in whichoptical paths can interact in a planar switching network. The channels carrying thesignals could cross each other in order to embed a particular topology. Alter-natively, two paths sharing a switch could experience some undesired coupling fromone path to another within a switch. For example, assume that the two inputs arey and z, respectively, the two outputs will have ly+lxz and lz+lxy, respectively,where l is path loss and x is signal crosstalk in a switch. Using the best devicereported in the literature [9], x=&35 dB and l=0.25 dB. For more practicallyavailable devices, it is more likely that x=&20 dB and l=1 dB [9]. Hence, whena signal passes many switches, the input signal will be distorted at the output dueto the loss and crosstalk introduced on the path. Experimental results [11] showthat it is possible to make the crosstalk from passive intersections of opticalwaveguides negligible by keeping the intersection angles above a certain minimumamount. Studies also indicate that the crosstalk problem is more severe than thepath-dependent loss problem with current optical technology [8, 17]. Thus, switchcrosstalk is the most significant factor that reduces the signal-to-noise ratio andlimits the size of a network. Luckily, first-order crosstalk can be eliminated byensuring that a switch is not used by two input signals simultaneously. Once themajor source of crosstalk disappears, crosstalk in an optical MIN will have a verysmall effect on the signal-to-noise ratio and thus a large optical MIN can be builtand effectively used in parallel computing systems. In the following discussion,switch crosstalk will be simply referred to as crosstalk.


3. APPROACHES TO SOLVING CROSSTALK PROBLEMS

One way to solve the crosstalk problem is a space domain approach, where aMIN is duplicated and combined to avoid crosstalk. Based on this idea, a dilatedBenes network has been proposed [11] to reduce the crosstalk level by allowingonly one signal to pass through a switch at any given time. The number of switchesrequired for the same connectivity in a dilated Benes network is slightly larger thantwice that for the regular Benes network. The dilated slipped banyan network(DSB) proposed by Thompson [18] is another example using this approach. ADSB(n) is constructed recursively from a DSB(n�2). Only one of the two ports ofswitching elements in the first and last columns is used to connect to the outside.This approach uses more than double the original network hardware to achieve thesame permutation capability.

Another way to implement this idea is a time domain approach [15]. A set ofconnections is partitioned into several subsets such that the connections in eachsubset can be established simultaneously in a network without crosstalk. These sub-sets can be used to determine a sequence of configurations that the network goesthrough in order to establish the set of connections. This approach makes sense inoptical MINs for two reasons. First, most multiprocessors use electronic processorsand optical MINs. There is a big mismatch between the slow processing speed inprocessors and the high communication speed in networks. Using time-divisionmultiplexing, even though we need several passes to satisfy the connection requestfrom the processors, the total time used to transfer all the messages is still smallcompared with the computation time of the processors. Second, there is a mismatchbetween the routing control and the fast signal transmission speed. Calculating therouting bits on-fly is impossible for optical MINs. A general method is to generatethese control bits before a computation through the analysis of the communicationpatterns in an application during its compiling. In the DSB network, both spacedomain and time domain approaches are used. Because a banyan network is ablocking network, even after its dilation, many permutations can still not berealized in one pass. In order to realize any permutation, a time domain approachis also used there. It emulates an n_n completely connected network by repeatedlyrealizing n different permutations. Each of these permutations is realized in one passfor the duration of a time slot, and every input is connected to a different outputeach time a different permutation is realized. Therefore, a completely connectednetwork is emulated without crosstalk using n passes. In other words, the DSBemulates a fully connected network in n passes. For undilated MINs, Qiao [13]has shown that an n node fully connected network can be emulated in 2n passes inan undilated Omega network (or its equivalent). A method for diagnosing crosstalkin an optical Benes network has also been proposed by Qiao [14].

In this paper, we will consider how to realize permutations in an undilated opti-cal MIN efficiently using the time domain approach. Because each switch can onlypass one signal at a given time, any permutation in an optical MIN requires at leasttwo passes. We will consider both blocking networks such as banyan networks andrearrangeable networks such as Benes networks under the avoiding crosstalk con-straint that for any switch in the network, only one signal is allowed to pass


through the switch at any time. Clearly, under this constraint, a 2_2 switch has themost efficient switch utilization. This is because that when a k_k switch is used(k�2) and each time we allow only one pair of inputs and outputs to be active,the rest of the k&1 inputs and outputs must be idle. Therefore, in the rest of thepaper we only discuss multistage networks consisting of 2_2 switches. For thistype of network, we will introduce a new concept, semi-permutation, which is a spe-cial type of partial permutation and has the maximum potential to be realized inan optical network without crosstalk.

4. SEMI-PERMUTATIONS

We consider an n_n multistage interconnection network with n inputs and noutputs where n=2m. A permutation for a network is a pairing of its inputs andoutputs such that each input appears in exactly one pair and each output appearsalso in exactly one pair. In other words, a permutation is a full one-to-one mappingbetween the network inputs and outputs. For an n_n network, suppose input xi ismapped to output yi , where xi=i and yi # [0, 1, ..., n&1] for i=0, 1, ..., n&1. Wedenote this permutation as

\x0

y0

x1

y1

} } }} } }

xn&1

yn&1 + .

In addition, we will refer to a one-to-one mapping between n$ network inputs andn$ network outputs (n$<n) as a partial permutation.

Apparently, a crosstalk-free optical network cannot realize a permutation in asingle pass, since at least the two input links on an input switch or the two outputlinks on an output switch cannot be active in the same pass. Then an interestingquestion we may ask is: What is the minimum number of passes required for realiz-ing a permutation in such a network? In other words, we are interested in whattypes of partial permutations could be possibly realized in an optical network underthe constraint of avoiding crosstalk. For an optical network consisting of 2_2switches, it is useful to introduce the following definition.

Definition 4.1. A partial permutation

\x0

y0

x1

y1

} } }} } }

xn�2&1

yn�2&1 + .

of an n-element set [0, 1, ..., n&1], where n is an even integer, x i , yi #[0, 1, ..., n&1], and x0<x1< } } } <xn�2&1 , is referred to as a semi-permutation ofthe n-element set, if

{\x0

2 �, \x1

2 �, ..., \xn�2&1

2 �==[0, 1, ..., n�2&1],


and

{\y0

2 �, \y1

2 �, ..., \yn�2&1

2 �==[0, 1, ..., n�2&1].

Example 4.2. For n=8, the partial permutation

\01

35

43

67+

is a semi-permutation, since we have

{\02�, \3

2�, \42�, \6

2�==[0, 1, 2, 3],

and

{\12�, \5

2�, \32�, \7

2�==[0, 2, 1, 3]=[0, 1, 2, 3].

Clearly, a semi-permutation is a partial permutation that ensures that there isonly one active link passing through each input switch and output switch, that is,it eliminates crosstalk in the first and last stages in the network, and thus it has thepotential to be realized in an optical network under the constraint of avoidingcrosstalk. Of course, to ensure that the entire network is crosstalk-free, we need toeliminate crosstalk in the switches in the intermediate stages as well.

4.1. Decomposition of a Permutation into Semi-permutations

In this section, we first show that there is a nice property for permutations thatany permutation can be decomposed into semi-permutations. We then give anefficient algorithm for such decompositions.

4.1.1. Decomposability

Theorem 4.3. Any permutation can be decomposed into two semi-permutations.

Proof. We will prove the theorem by using a combinatorial theorem of Hall [2].Let the permutation be

\x0

y0

x1

y1

x2

y2

x3

y3

} } }} } }

xn&2

yn&2

xn&1

yn&1+ , (1)

where xi=i for 0�i�n&1 and [ y0 , y1 , ..., yn&1]=[0, 1, ..., n&1]. Clearly, thispermutation can be decomposed into n�2 partial permutations

\x0

y0

x1

y1+ , \x2

y2

x3

y3+ , ..., \xn&2

yn&2

xn&1

yn&1+ .


Note that for the jth partial permutation

\x2j

y2j

x2j+1

y2j+1+ ,

where 0� j�n�2&1, we have

\x2j

2 �=\x2j+1

2 �= j. (2)

Denote [w y2 j �2x, w y2 j+1 �2x] as A j for j=0, 1, ..., n�2&1. It is easy to see thatthere are two 0's, two 1's, ..., two (n�2&1)'s distributed in the n�2 2-element setsA0 , A1 , ..., An�2&1 . Now for any k sets (1�k�n�2), Ai1 , Ai2 , ..., Aik , there are a totalof 2k elements in the k sets, and these 2k elements form a multiset with the multi-plicity of each element being no more than 2. Thus, the cardinality of the union ofthese k sets satisfies

|Ai1_ Ai2 _ } } } _ Aik |�k (3)

By Hall's distinct system representatives theorem [2], we know that (3) is thenecessary and sufficient condition for these n�2 subsets to have a set of distinctsystem representatives; that is, there exist a0 # A0 , a1 # A1 , ..., an�2&1 # An�2&1 suchthat ai {aj for any i{ j (0�i, j�n�2&1). This implies that

[a0 , a1 , ..., an�2&1]=[0, 1, ..., n�2&1]. (4)

Since aj is either w y2j �2x or w y2j+1 �2x, let aj be w ydj, 1�2x, and the other element

in Aj be w ydj, 2�2x. For the subscripts d j, 1 and d j, 2 , we have

[dj, 1 , dj, 2]=[2j, 2j+1]. (5)

Thus, the above permutation can be decomposed into two partial permutations,

\xd0, 1

yd0, 1

xd1, 1

yd1, 1

} } }} } }

xdn�2&1, 1

ydn�2&1, 1+ (6)

and

\xd0, 2

yd0, 2

xd1, 2

yd1, 2

} } }} } }

xdn�2&1, 2

ydn�2&1, 2+. (7)


Next, we will verify that both (6) and (7) are semi-permutations. Notice that

{\xd0, 1

2 �, \xd1, 1

2 �, ..., \xdn�2&1, 1

2 �==

by (2) {\d0, 1

2 �, \d1, 1

2 �, ..., \dn�2&1, 1

2 �==

by (5) [0, 1, ..., n�2&1]

{\yd0, 1

2 �, \yd1, 1

2 �, ..., \ydn�2&1, 1

2 �==[a0 , a1 , ..., an�2&1] =

by (4) [0, 1, ..., n�2&1]. (8)

Therefore, by Definition 4.1, (6) is a semi-permutation. Since

{\yd0, 1

2 �, ..., \ydn�2&1, 1

2 �, \yd0, 2

2 �, ..., \ydn�2&1, 2

2 �=is a multiset on [1, 2, ..., n�2&1] with the multiplicity of each element equal to 2,from (8) we have

{\yd0, 2

2 �, \yd1, 2

2 �, ..., \ydn�2&1, 2

2 �==[0, 1, ..., n�2&1].

Thus, (7) is also a semi-permutation. K

Example 4.4. The decomposition of a permutation into two semi-permutations.

\01

10

27

35

43

54

66

72+

O\01

35

43

66+ b \1

027

54

72+ .

4.1.2. The Algorithm

Theorem 1 guarantees the correctness of the decomposition. In the following, wedescribe a much simpler yet efficient algorithm to actually decompose a permuta-tion into two semi-permutations.

Given a permutation of the form (1), in which input xi is mapped to output yi ,where 0�i�n&1, define A[1]

j and A[2]j as the 2-element set [2j, 2j+1] for

0� j�n�2&1, where A[1]j and A[2]

j correspond to the inputs and the outputs,respectively.

Now we construct an undirected bipartite graph G=(V1 , V2 ; E). The vertex setsof G are defined as

V1=[A[1]0 , A[1]

1 , ..., A[1]n�2&1], V2=[A[2]

0 , A[2]1 , ..., A[2]

n�2&1],


and the edge set E of G is defined thus: for any one-pair mapping ( xiyi

) in the per-mutation (1), if xi # A[1]

j1 and yi # A[2]j2 , then there is an edge between vertex A[1]

j1

and vertex A[2]j2 in E. We also assign each edge in E a label representing the

corresponding one-pair mapping in the permutation. This bipartite graph has thefollowing properties:

1. |V1 |= |V2 |=n�2 and |E|=n,

2. the graph may have parallel edges,

3. each vertex of the graph has degree 2,

4. the graph may consist of more than one connected component.

From graph theory [3], we know that for a component of a graph in which eachvertex has even degree, there exists an Euler tour which traverses each edge exactlyonce. In particular, in our case, since each vertex has degree 2, the Euler tourbecomes a cycle.

We are now in a position to give a high-level description of the decompositionalgorithm.

Decomposition Algorithm.

Step 1: Construct a bipartite graph G for the given permutation.

Step 2: For each connected component of G, start from a vertex of this componentin V1 , traverse through an unvisited edge to the neighbor vertex in V2 , backand forth until returning to the starting vertex. (During the traversing, avisited edge is marked ``forward'' if the traverse direction on this edge is fromV1 to V2 and is marked ``backward'' if the direction is opposite.)

Step 3: Take all one-pair mappings corresponding to the edges marked with``forward,'' to form one semi-permutation; let the remaining one-pair map-pings, corresponding to the edges marked with ``backward,'' form anothersemi-permutation.

End

The above algorithm is correct because from the properties of the bipartite graphG listed above, the set of all edges marked with ``forward'' is a perfect matching ofthe bipartite graph G, and so is the set marked with ``backward.'' Also, it is easyto see that the time to construct the bipartite graph is proportional to the numberof pairs in the permutation, i.e., O(n), and the time to traverse all edges isO( |E| )=O(n). Hence, the complexity of the decomposition algorithm is O(n).

Let us apply the above algorithm to Example 4.4 in the last subsection. Thebipartite graph and edge traverses are shown in Fig. 1, where

e0=\01+, e1=\1

0+, e2=\27+, e3=\3

5+,

e4=\43+, e5=\5

4+, e6=\66+, e7=\7

2+.


FIG. 1. The bipartite graph and edge traverses of the decomposition.

Then the ``forward'' pairs e0 , e3 , e4 , and e6 form

\01

35

43

66+ ,

and the ``backward'' pairs e1 , e2 , e5 , and e7 form

\10

27

54

72+ ,

which completes the decomposition.

4.2. The Number of Semi-permutations

Another interesting question we may ask for semi-permutations is: What is thetotal number of semi-permutations for an even integer n?

Theorem 4.5. For any even integer n, the total number of semi-permutations foran n-element set is 2n } (n�2)!.

Proof. Let the n-element set be [0, 1, ..., n&1]. Define a 2-element set Aj

(0� j�n�2&1) as [2j, 2j+1]. Consider a permutation for an n�2-element set[A0 , A1 , ..., An�2&1]

\A0

Ai0

A1

Ai1

} } }} } }

An�2&1

Ain�2&1+ , (9)


where indexes (i0 i1 ... i(n�2)&1) are a permutation of the n�2-element set[0, 1, ..., n�2&1]. Take any xj # Aj and yj # Aij (0� j�n�2&1) in (9); then

\x0

y0

x1

y1

xn�2&1

yn�2&1+forms a semi-permutation for the n-element set [0, 1, ..., n&1]. We have a total of2n�2 } 2n�2=2n different such semi-permutations corresponding to one permutation ofthe form (9). On the other hand, any semi-permutation for the n-element set[0, 1, ..., n&1] corresponds to one permutation for the n�2-element set [A0 , A1 , ...,An�2&1] of some form (9). Note that there are a total of (n�2)! permutations of theform (9). Therefore, there are a total of 2n } (n�2)! semi-permutations for then-element set [0, 1, ..., n&1]. K

Now, the problem of realizing a permutation in a crosstalk-free network can betransformed into the problem of realizing semi-permutations in the crosstalkfreedom network.

However, it should be pointed out that introducing the semi-permutation con-cept in a network composed of 2_2 switches can only guarantee crosstalk freedomin the switches in the input stage and the output stage of the network. In fact,realizing a semi-permutation in a single pass implies that there is only one activeinput on each switch in the input stage and only one active output on each switchin the output stage. To ensure that the entire network is crosstalk free, we need toknow if there exists a proper routing that can eliminate crosstalk in the switches atthe intermediate stages along different active paths. In the subsequent sections, wewill look into this issue for two different types of networks.

5. REALIZING SEMI-PERMUTATIONS IN A BANYAN NETWORK

Banyan networks were first introduced by Goke and Lipovski [6] using graphmethods. In a simplifying approach, any multistage network for which there is aunique path from each network input to each network output is called a banyannetwork. The banyan network considered in this paper is an n_n network com-posed of 2_2 switches, where n=2m. The network has m stages of switches andthe switches in the consecutive stages are linked by recursively applying butterflyinterconnection patterns. Figure 2 shows a 16_16 banyan network composed of2_2 switches. Figure 3 illustrates the recursive construction of an n_n banyannetwork.

Due to the unique path nature of a banyan network, a semi-permutation isrouted through the network in a fixed switch setting. Consequently, some semi-per-mutations can be realized in a banyan network in a single pass, while others can-not. In Fig. 4, we give examples of such semi-permutations for an 8_8 banyannetwork. Fig. 4a shows that the semi-permutation ( 0

325

47

61) can be realized in a

single pass, and Fig. 4b shows that the semi-permutation ( 04

26

50

62) cannot be

realized in a single pass.


FIG. 2. A 16_16 banyan network composed of 2_2 switches.

Thus, it would be interesting to know how many semi-permutations can berealized in a banyan network in a single pass under the constraint of avoidingcrosstalk.

Theorem 5.1. The number of semi-permutations that can be realized in an n_nbanyan network in a single pass under the constraint of avoiding crosstalk is2(3�4) n } nn�4.

Proof. Let S(n) be the total number of semi-permutations that can be realizedin an n_n banyan network without crosstalk. Let F(n) be the total number of

FIG. 3. Recursive construction of an n_n banyan network.


FIG. 4. Realizing semi-permutations in an 8_8 banyan network. (a) A semi-permutation that canbe realized in a single pass. (b) A semi-permutation that cannot be realized in a single pass. Bold boxesindicate the switches with crosstalk.

semi-permutations for the given fixed active inputs in the n�2 switches in the inputstage. By the symmetry of semi-permutations, we have

S(n)=2n�2 } F(n). (10)

Consider the recursive definition of an n_n banyan network (Fig. 3). For the n�2switches in input stage u0 , u1 , ..., un�2&1 , we pair them off as follows:

[u0 , u1], [u2 , u3], ..., [un�2&2 , un�2&1].

There are a total of n�4 such pairs. Now, let us consider the interstage connectionsbetween the first stage of the network and the rest of the network. For each of sucha pair, the upper outputs of both switches are linked to a switch in the upper(n�2)_(n�2) banyan network, the lower outputs of both switches are linked to aswitch in the lower (n�2)_(n�2) banyan network, and only input switches in thepair are linked to those two switches in the next stage (one is in the upper sub-banyan network, and the other is in the lower subbanyan network). To guaranteecrosstalk freedom, partial permutations on the upper and lower subbanyannetworks must be semi-permutations. There are two possible ways for each inputswitch pair to be linked to (some fixed inputs) in the next stage, that is, letting theoutput of a switch go to the upper subnet and the output of the other switch go


to the lower subnet, or vice versa. Since there are a total of n�4 such pairs, weimmediately obtain a recurrence for F(n),

F(n)=2n�4[F ( n2)]2. (11)

Also note that F(2)=2. Therefore,

F(n)=2k } n�4 _F \ n2k+&

2k

=2(log n&1) } n�4 } 2n�2=2n�4 } nn�4.

Thus from (10), we have S(n)=23n�4 } nn�4. K

By comparing Theorem 5.1 and Theorem 4.5, we can see that there are a sub-stantial number of semi-permutations that cannot be realized in a banyan network,especially when n becomes larger.

6. REALIZING SEMI-PERMUTATIONS IN A BENES NETWORK

A Benes network can be constructed by concatenating a banyan network and areverse banyan network with the center stages overlapped. Figure 5 is an exampleof a 16_16 Benes network. It is symmetric about the central stage. A Benesnetwork can also be defined in a recursive fashion as shown in Fig. 6. In an n_nBenes network, the upper or lower output of each input switch is linked to an inputof the upper or lower (n�2)_(n�2) Benes network, respectively, and the upper orlower input of each output switch is linked from an output of the upper or lower(n�2)_(n�2) Benes network, respectively. Electronic Benes networks are well knownfor being capable of realizing all possible permutations [1]. In the following, wewill show that Benes networks also have good properties to support permutationsin optical networks.

Theorem 6.1. Any semi-permutation can be realized in a Benes network in asingle pass under the constraint of avoiding crosstalk.

FIG. 5. A 16_16 Benes network composed of 2_2 switches.


FIG. 6. Recursive construction of an n_n Benes network.

Proof. Consider the recursive definition of an n_n Benes network (Fig. 6). Letthe n�2 input switches be u0 , u1 , ..., un�2&1 , and let the n�2 output switchesbe v0 , v1 , ..., vn�2&1 . We pair all input switches into n�4 pairs as A[1]

j =[u2j , u2j+1]for 0� j�n�4&1. Similarly, we pair all output switches into n�4 pairs asA[2]

j =[v2 j , v2 j+1] for 0� j�n�4&1.Since a Benes network consists of a banyan network and a reverse banyan

network, the observations for banyan networks in the proof of Theorem 5.1 areapplicable here. That is, both input switches in A[1]

j have links to some twoswitches in the next stage (one is to the upper sub-Benes network and the other isto the lower sub-Benes network), and no other input switches have links to any ofthose two switches in the next stage. Symmetrically, both output switches in A[2]

j

have links from some two switches in the previous stage (one is from the upper sub-Benes network and the other is from the lower sub-Benes network), and no otheroutput switches have links from any of those two switches in the previous stage.

A pair of input�output mappings can be realized by starting from the corre-sponding input switch via either the upper or the lower subnetwork and endingat the corresponding output switch. To ensure crosstalk freedom in the entirenetwork, a proper arrangement is needed to provide the upper and lower sub-networks with semi-permutations, so that the routing can continue in a recursivefashion. To achieve this, for the pair of input switches in A[1]

j , we connect the onlyactive output of one switch to the upper subnetwork and the only active output ofthe other switch to the lower subnetwork. The arrangement for the pair of outputswitches in A[2]

j is similar.Not surprisingly, the idea of the decomposition algorithm described in Section

4.1.2 can also serve our purpose here. To achieve this, we build a bipartite graphG=(V1 , V2; E) as follows. The vertex sets are defined as

V1=[A[1]0 , A[1]

1 , ..., A[1]n�4&1], V2=[A[2]

0 , A[2]1 , ..., A[2]

n�4&1],


and the edge set E is defined as follows; for any one-pair mapping ( xiyi

) in the semi-permutation, if the input switch corresponding to xi belongs to A[1]

j1 and the outputswitch corresponding to yi belongs to A[2]

j2, then there is an edge between vertex

A[1]j1 and vertex A[2]

j 2in E. We assign each edge in E a label representing the corre-

sponding one-pair mapping in the semi-permutation. Note that the bipartite graphis of size |V1 |=|V2 |=n�4 and |E|=n�2. The idea here is that after running thedecomposition algorithm, for one-pair mapping ( xi

yi) marked with ``forward,'' we

make the input xi pass the corresponding input switch, link to the upper sub-network, then link to the corresponding output switch, and finally reach the outputyi ; on the other hand, for a one-pair mapping marked with ``backward,'' we makethe connection via the lower subnetwork.

Now the routing algorithm for a semi-permutation in an n_n Benes networkcan be obtained by slightly modifying the decomposition algorithm described inSection 4.1.2.

Routing Algorithm (integer n, SEMI�PERM semi� perm).

Step 1: If n is 2, make the connection in the 2_2 switch according to semi� perm;exit.

Step 2: Construct the bipartite graph G=(V1 , V2 ; E) corresponding to semi� permin the n_n Benes network.

Step 3: Same as Step 2 in the Decomposition Algorithm in Section 4.1.2.

Step 4: Take all one-pair mappings corresponding to the edges marked with``forward'' to form one semi-permutation semi� perm� upper, and for each ofthese one-pair mappings make a connection pass through the correspondinginput switch to the upper subnetwork, and from the upper subnetwork to thecorresponding output switch and to the output; take all one-pair mappingscorresponding to the edges marked with ``backward'' to form a semi-per-mutation semi� perm� lower, and for each of these one-pair mappings makea connection pass through the corresponding input switch to the lower subnetwork and from the lower subnetwork to the corresponding output switchand to the output.

Step 5: Recursively call Routing Algorithm (n�2, semi� perm� upper) in the uppersub-Benes network.

Step 6: Recursively call Routing Algorithm (n�2, semi� perm� lower) in the lowersub-Benes network.

End

The correctness of the algorithm is clear. Since Steps 1�4 take O(n) steps andSteps 5�6 recursively call the same algorithm for n�2, it is easy to see that the timecomplexity of the routing algorithm is O(n log n). K

In the proof of the above theorem, we have actually given an efficient routing-switch set up algorithm.


7. REALIZING PERMUTATIONS IN A BENES NETWORK

Finally, we consider how many passes are required to realize a permutation in aBenes network.

Theorem 7.1. Any permutation can be realized in a Benes network in two passesunder the constraint of avoiding crosstalk.

Proof. By Theorem 4.3, we know that any permutation can be decomposed intotwo semi-permutations. Then by Theorem 6.1, each semi-permutation can berealized in a Benes network in a single pass. Thus, in two passes we can realize anypermutation without crosstalk. K

It should be pointed out that a permutation requires at least two passes in anyn_n optical MIN due to the constraint of avoiding crosstalk in the input stage ofswitches. In other words, 2 is the lower bound on the number of passes for anyoptical permutation networks under the constraint of avoiding crosstalk. Theorem7.1 indicates that an undilated Benes network reaches this lower bound and realizespermutations optimally.

8. CONCLUSIONS

In this paper, we have analyzed the permutation capability of optical MINs. Weintroduced a new concept, semi-permutation, which ensures that there is only oneactive link passing through each input switch and output switch and thus has thepotential to be realized without crosstalk in an optical network. An efficient algo-rithm to decompose a permutation into two semi-permutations has also beendeveloped. For the blocking banyan network, we have shown that not all semi-per-mutations are realizable in one pass and have given the number of realizable semi-permutations. For the rearrangeable nonblocking Benes network, we have shownthat any semi-permutation is realizable in one pass and any permutation isrealizable in two passes under the constraint of avoiding crosstalk. A routing algo-rithm for realizing a semi-permutation in a Benes network has been given. Webelieve that the concepts and analytical methods developed in this paper can alsobe used to solve the crosstalk problem in other optical MINs.

REFERENCES

1. V. E. Benes, Heuristic remarks and mathematical problems regarding the theory of switchingsystems, Bell System Technical J. 41 (1962), 1201�1247.

2. K. P. Bogart, ``Introductory Combinatorics,'' 2nd ed., Harcourt Brace Jovanovich, San Diego, 1990.

3. J. A. Bondy and U. S. R. Murty, ``Graph Theory with Applications,'' Am. Elsevier, New York, 1976.

4. C. Clos, A study of non-blocking switching networks, Bell System Technical J. 32 (1953), 406�424.

5. J. Erickson and H. Hinton, Implementing a Ti:LiNbO3 4_4 nonblocking interconnection network,in ``SPIE Integrated Optical Circuit Engineering,'' Vol. 578, pp. 201�206, 1985.

6. L. R. Goke and G. J. Lipovski, Banyan networks for partitioning multiprocessor systems, in ``Proc.of the First Annual Symposium on Computer Architecture, 1973,'' pp. 21�28.


7. A. Himeno and M. Kobayashi, 4_4 Optical-gate matrix switch, J. Lightwave Tech. 3 (April 1985),230�235.

8. H. Hinton, A non-blocking optical interconnection network using directional couplers, in ``Proc.IEEE Global Telecom. Conference, 1984,'' pp. 885�889.

9. D. K. Hunter and I. Andonovic, Guided wave optical switch architectures, Internat. J. Optoelectron.9, 6 (1994), pp. 477�487.

10. K. Li, Y. Pan, and S. Q. Zheng, ``Parallel Computing Using Optical Interconnections,'' KluwerAcademic, Boston, 1998.

11. K. Padmanabhan and A. N. Netravali, Dilated networks for photonic switching, IEEE Trans.Commun. 35, 12 (December 1987), 1357�1365.

12. Y. Pan, C. Qiao, and Y. Yang, Optical multistage interconnection networks: New challenges andapproaches, IEEE Commun. Magazine 37, 2 (February 1999), 50�56. [Feature Topic on OpticalNetworks, Communication Systems and Devices]

13. C. Qiao, ``A High-Speed Interconnection Paradigm and Its Applications to Optically InterconnectedNetworks,'' Ph.D. dissertation, Department of Computer Science, University of Pittsburgh, 1993.

14. C. Qiao, A two-level process for diagnosing crosstalk in photonic dilated Benes networks, J. Paralleland Distrib. Comput. 41, 1 (1997), 53�66.

15. C. Qiao and R. Melhem, A time domain approach for avoiding crosstalk in optical blockingmultistage interconnection networks, J. Lightwave Tech. 12, 10 (October 1994), 1854�1862.

16. A. A. Sawchuck, B. K. Jenkins, C. S. Raghavendra, and A. Varma, Optical crossbar networks,Computer 20, 6 (June 1987), 50�60.

17. R. Spanke, Architectures for large non-blocking optical switches, IEEE J. Quantum Electron. 22, 6(June 1986), 964�967.

18. R. A. Thompson, The dilated slipped banyan switching network architecture for use in an all-opticallocal-area network, J. Lightwave Tech. 9, 12 (December 1991), 1780�1787.

19. Y. Yang and G. M. Masson, Broadcast ring sandwich networks, IEEE Trans. Comput. C-44, 10(1995), 1169�1180.

20. Y. Yang and G. M. Masson, Nonblocking broadcast switching networks, IEEE Trans. Comput.C-40, 9 (1991), 1005�1015.

YUANYUAN YANG received the B.Eng. and M.S. in computer engineering from Tsinghua Univer-sity, Beijing, china, in 1982 1nd 1984, respectively, and the M.S.E. and Ph.D. in computer science fromJohns Hopkins University, Baltimore, Maryland, in 1989 and 1992, respectively. She is currently anassociate professor of computer engineering at the State University of New York at Stony Brook. Beforejoining SUNY Stony Brook, Dr. Yang was a faculty member in the Department of Computer Science,University of Vermont in Burlington, from 1992 to 1999 (as an associate professor from 1998 to 1999).Dr. Yang's research interests include parallel and distributed computing and systems, high-speednetworks, optical networks, high-performance computer architecture, and fault-tolerant computing. Shehas published extensively in major journals and refereed conference proceedings related to these researchareas. Dr. Yang holds two U.S. patents in the area of multicast communication networks, with four morepatents pending. Her research has been supported by the U.S. Army Research Office and the NationalScience Foundation. She has served on the program�organizing committees of a number of internationalconferences. Dr. Yang is a senior member of the IEEE and a member of the ACM, IEEE ComputerSociety, and IEEE Communication Society.

JIANCHAO WANG is currently a principal member of technical staff at GTE Laboratories inWaltham, Massachusetts. He received the B.Eng. in computer engineering from Tsinghua University,Beijing, China, in 1982 and the M.S. and Ph.D. in computer science from the Institute of ComputingTechnology, Chinese Academy of Sciences, Beijing, China, in 1985 and 1988, respectively. Before hejoined GTE Labs, Dr. Wang worked at the institute of computing Technology, Chinese Academy ofSciences, John Hopkins University, and Legent Corporation in Marlboro, Massachusetts. Dr. Wang hasreceived a number of excellence�achievement awards from GTE and Legent and has four U.S. patents


granted or pending. Dr. Wang's research interests include IP telephony, databases, programminglanguages, computer communication networks, computer algorithms, and fault-tolerant computing.Dr. Wang is a member of the IEEE Computer Society and Association for Computing Machinery.

YI PAN was born in Jiangsu, China. He received his B.Eng. degree in computer engineering fromTsinghua University, China, in 1982, and his Ph.D. degree in computer science from the University ofPittsburgh in 1991. Dr. Pan joined the Department of Computer Science at the University of Dayton,Ohio, in 1991 and has been an associate professor since 1996. His research interests include parallel algo-rithms and architectures, optical communication and computing, distributed computing, image process-ing, and networking. He has published more than 80 research papers including over 30 papers in highlyrated journals such as IEEE Transactions on Parallel and Distributed Systems, IEEE Transactions onCircuits and Systems, IEEE Transactions on Systems, Man, and Cybernetics, IEEE Transactions onReliability, IEEE Communications Magazine, IEEE Computing in Science and Engineering, Journal ofParallel and Distributed Computing, Optical Engineering, and Journal of Supercomputing. He has receivedmany awards, including the Outstanding Scholarship Award of the College of Arts and Sciences atUniversity of Dayton (1999), a Japanese Society for the Promotion of the Science Fellowship (1998), anAFOSR Summer Faculty Fellowship (1997), a NSF Research Opportunity Award (1994, 1996), anAndrew Mellon Fellowship from the Mellon Foundation (1990), the best paper award from PDPTA '96(1996), and the Summer Research Fellowship from the Research Council of the University of Dayton(1993). His research has been supported by the NSF, the AFOSR, the U.S. Air Force, and state of Ohio.Dr. Pan is currently an editor of the Journal of Parallel and Distributed Computing Practices, anassociate editor of the International Journal of Parallel and Distributed Systems and Networks, and onthe editorial boards of the Journal of Supercomputing and the Journal of Information. He has also servedas a guest editor of special issues for several journals. He will be the general co-chair of the 11th Interna-tional Conference on Parallel and Distributed Computing and Systems in November 1999. He was theprogram chair of the 10th International Conference on Parallel and Distributed Computing and Systemsin 1998, the conference co-chair of the 4th International Conference on Computer Science and Infor-matics in 1998, and the program chair of the 3rd IPPS workshop on Optics and Computer Science in1999. He has also served as vice program chair, publicity chair, session chair, or a committee memberfor more than 20 international conferences. Dr. Pan is a senior member of the IEEE and a member ofthe IEEE Computer Society. He is listed in Men of Achievement and Marquis Who's Who in the Midwest.


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Permutatin Capability of Optical Multistage