realization of large-scale distributors based on batcher sorters

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 7, JULY 1999 1103

Realization of Large-Scale DistributorsBased on Batcher Sorters

Jeong Gyu Lee,Member, IEEE, and Byeong Gi Lee,Fellow, IEEE

Abstract—In this paper, we rigorously examine how to con-struct a large-scale distributor by taking a logical and systematicapproach. We first consider requirements for realization, whichinclude the independent operation of unit modules and the packetsequence integrity, and consider what kind of existing deviceshave large-scale growth potential. In this process, we take theBatcher sorting network as the candidate device and show thatthe distribution function can be incorporated to it by furnishingeach constituent sorting element with the state inversion capa-bility. Then we show that this is equivalent to replacing eachsorting element with a 2 � 2 distributor. As a consequence,the large-scale distributor design problem turns into a simplesubstitution problem for which we substitute a unit distributor foreach sorting element in the prototype Batcher sorting network. Inaddition, we show that this substitution approach can be extendedto larger scale unit distributor modules, thus enabling a large-scale distributor realization. The resulting large-scale distributordesign procedure can be summarized as follows: given a desiredsize L = MN for the design of a large-scale distributor, wefirst design anN �N Batcher sorter (BS) and then replace eachconstituent 2� 2 sorting element with a 2M � 2M distributormodule.

Index Terms—ATM switch, Batcher sorter, packet concentra-tor, packet distributor.

I. INTRODUCTION

A DISTRIBUTOR refers to a multi-input multi-output de-vice that is designed to distribute active packets entering

the input ports to the output ports in an equiprobabilistic androtational manner [1]–[3]. A distributor is normally realizedby incorporating the rotation capability to a concentrator thatconcentrates the active input packets to a particular part ofthe output ports. Conventionally, the rotation function hasbeen acquired by prefixing, in front of a reverse banyannetwork (RBN), a running adder and an additional registerthat generates dummy addresses of each active packet for arotational distribution [1].

In recent studies, simpler structured distributors such as theautonomous distributor [4], [5] and the balanced distributor [6]have been introduced. The autonomous distributor is composedof only a controlled switching element (CSE)-based RBN anda set of delays, and the rotation function is embedded inthe constituent CSE’s themselves. The balanced distributor is

Paper approved by P. E. Rynes, the Editor for Switching Systems of theIEEE Communications Society. Manuscript received March 24, 1997; revisedNovember 9, 1998 and January 29, 1999.

J. G. Lee is with the Switching and Transmission Technology Laboratory,Electronics and Telecommunications Research Institute, Taejon 305-350,Korea.

B. G. Lee is with the School of Electrical Engineering, Seoul NationalUniversity, Seoul 151-742, Korea (e-mail: [email protected]).

Publisher Item Identifier S 0090-6778(99)05234-4.

composed of a folded crossbar switch array and horizontal andvertical controllers, and the rotational function is achieved bytaking different sets of switching stages that are involved inconcentrating active packets.

The distributor structures that have been introduced todate are all adequate for small-scale distributor realizationsonly. This is because they require certain information transfermechanisms among the constituent unit elements, which is themajor obstacle to increasing the distributor size. For example,the operation of the switching elements in the conventionaldistributor is dictated by the running adder network in front.The operation of each CSE in the autonomous distributor isgoverned by those CSE’s in the upstream, and each switchingelement in the balanced distributor operates only after thestates of its left and upper switching elements are determined.Further, the size of distributors that can be built into a veryl.arge scale integration (VLSI) chip is limited in practiceto a certain number, for example 32 32 or 64 64.Therefore, a design procedure is needed to construct a large-scale distributor that cannot be built on a single VLSI chip.

In this paper, we present a new distributor structure thatenables realizing large-scale distributors. In devising this newstructure, we will take a logical and systematic approachinstead of the normally used divide-and-conquer approach.We will start with: 1) the reasoning of what requirementsare needed for a large-scale distributor design; 2) whetheror not it is possible to meet all the requirements; 3) whatkind of existing devices have large-scale growth potential; 4)how to embed the rotation capability to convert the device toa distributor; and 5) how to expand the distributor to largescale. In this process, we take the Batcher sorter (BS) as thecandidate device and exert rigorous reasonings to pave the wayfor its evolution to a large-scale distributor.

The paper is organized as follows. We consider the re-quirements for a large-scale distributor and examine what typeof existing devices will possibly meet the requirements afterbeing converted into a distributor in Section II. In Section III,we investigate how to incorporate rotation capability to theBS, the candidate device selected in Section II, in order toconvert it to a distributor. Based on the previous sections, wediscuss how to realize large-scale distributors in Section IV.

II. CONSIDERATIONS FORLARGE-SCALE

DISTRIBUTOR REALIZATIONS

A general approach to designing a large-scale switchingdevice is interconnecting multiple small unit modules in sucha way that the overall configuration can meet the desired

0090–6778/99$10.00 1999 IEEE

1104 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 7, JULY 1999

requirements. In this paper, we also take this approach toconstruct large-scale distributors.

We first consider the requirements for realizing large-scaledistributors. In designing a large-scale distributor it is desirableto take the structure that meets the following requirements: 1)information transfer is not necessary among the constituentunit distributor modules, that is, each unit distributor moduleoperates fully independently; 2) input packet sequence ispreserved at the output; and 3) the number of constituent unitmodules is minimal. Among the three, the first requirementis primarily important because independent operation of unitmodules is directly tied with simplified interconnection, lessstringent synchronization requirements, high-speed operation,and simplified maintenance of the whole system. In contrast,the second requirement is not compulsory for a distributor, asa distributor is intended for fair cell distribution rather thancell sequence integrity.

As well known [3], the distributor is a generalized versionof the concentrator in that the concentrated string of activeinputs may start at any desired output port in a rotationalmanner. In this regard, it is natural to take a large-scaleconcentrator as the seed to develop to a large-scale distributor.Therefore, in this paper, we first consider the realizationsof large-scale concentrators and then convert them to large-scale distributors. For realization of the desired large-scaleconcentrators, it is reasonable to take a nonadaptive sortingnetwork1 as the basis because the inputs to its constituentsorting element are compared and exchanged solely basedon their own relative values, independently of other inputs,thus satisfying the first requirement.2 Further, it is possible toexpand a sorting network of elements to a sorting networkof elements simply by replacing each 22 comparisonelement with an -way merger or a sorting network[8]. In this context, we consider the three nonadaptive sortingnetworks—the odd–even transposition sorter, the odd–evenmerging sorter, and the bitonic sorter [7]–[9]—and examinewhich of them are suitable candidates that can possibly meetthe requirements. In this examination, we include the rotatabil-ity check because the rotation of the concentrated string is thekey process for converting a concentrator into a distributor.

First, the odd–even transposition sorter, whichconsists of comparators arranged in an-stage brick-like pattern, renders an easy implementation andcan preserve the active input sequence at the output, butrequires a large number of unit elements. If a large-scaleconcentrator is built based on this sorter, it may possiblymeet the independent operation and packet sequence integrity

1Sorting networks may be classified into two categories—adaptive andnonadaptive sorting networks [7]. These two differ in that the latter isconstructed based on comparators (i.e., sorting elements) only, whereas theformer includes other components to check conditions for comparison androuting. For example, the adaptive binary sorting network proposed in [7]includes two-way and four-way swappers and other combinational circuits. Insuch an adaptive sorting network, the operation of a component is governedby the other components, so the independent operation of the unit elementcannot be guaranteed.

2Note that when a sorting network is used as a concentrator, its constituentsorting elements compare the activity bit values of input packets, not thedestination fields, with the value “1” assigned to each active packet and thevalue “0” to each inactive packet.

requirements. However, it will not be minimal and will nothave the rotational capability because it is aprimitive networkthat contains unit elements for adjacent input comparisonsonly [8]. Second, the odd–even merging sorter forms a singlesorted sequence out of two sorted sequences employing theodd–even mergingscheme. Based on this sorter, the large-scaleconcentrator may require a relatively small number of unitelements, but cannot preserve the cell sequence because theodd–even merging sorter is a nonsequence-preserved network3

[8]. To make it worse, the interconnection pattern of the sortingelements does not provide the rotational capability suitable fora distributor, as one can confirm using the 44 odd–evenmerging sorter, which cannot generate all the required outputpatterns. Third, the BS generates a single sorted sequenceout of a bitonic sequence4 employing thebitonic mergingscheme. The large-scale concentrator based on this BS requiresa moderately small number of unit elements (in between theprevious two networks) but does not preserve input packetsequence, as for the case of the previous odd–even mergingsorter [8]. Fortunately, however, the bitonic sorter possessesthe rotational capability suitable for the distribution function,which can be realized by changing the state polarity of someconstituent sorting elements, as will be detailed in Section III.

From these discussions, we find that the odd–even transpo-sition sorter, which is the only sequence-preserved sorter, isnot suitable for realizing large-scale distributors. This impliesthat it is impossible to realize an ideal large-scale distributorthat meets the three requirements as far as the existing sortersare concerned. Therefore, we are forced to either abandonthe second requirement or keep it at the sacrifice of thefirst in realizing a large-scale distributor. While it is rela-tively easy to realize a large-scale distributor based on thelatter approach,5 the required state control and synchronizationprocesses become exceedingly complicated in the resultinglarge-scale distributor. Therefore, in this paper we will takethe former approach, thus concentrating on realizations ofnonsequence-preserved large-scale distributors that guaranteethe unit module’s independence.

III. REALIZATION OF ROTATION CAPABILITY

IN BATCHER SORTERS

Based on the discussions in Section II, we now concentrateon realizations of nonsequence-preserved large-scale distrib-utors based on BS network structures. For this, we willfirst establish that it is possible to modify a BS so that itcan generate the sorted and rotated sequences needed for

3In the literature, the termstable is generally used to represent thesequence preservation property of a sorter, but in this paper we use the termsequence-preserved, instead, as it can be used commonly for both sorters anddistributors. Likewise, we use the termnonsequence-preservedto parallel theterm unstable.

4A sequencefa1; a2; � � � ; aNg is bitonic if it is ordereda1 � a2 �� aj � aj+1 � � � � � aN for an integerj in 1 � j � N or if it can becircularly shifted to this ordering.

5We can realize this type of large-scale distributor simply by taking a largenumber ofN for the N � N autonomous distributor structure [4] or byreplacing its constituent switching element with anM �M distributor (if theelement is in the front column) or anN � N distributor (otherwise). Notethat this switching element replacement process brings forth anMN �MN

distributor.

LEE AND LEE: REALIZATION OF LARGE-SCALE DISTRIBUTORS BASED ON BATCHER SORTERS 1105

Fig. 1. Anatomy of the2N � 2N (N = 8) BS which consists of twoN � N BS’s in the FU and FL positions and a2N � 2N BM.

distributor realization. In Section IV, we will be more specificon establishing how to modify a BS to provide the distributionfunction. For a rigorous discussion, we will put these twoestablishments in the form oftheorems, and in support of theirproofs, we will arrange the rotation properties of a BS in twoproperties.

In describing the rotation operation,we will use the term-rotated to describe the circular shift operation that moves thelargest input to output port

Fig. 1 depicts the anatomy of a BS thatwe will refer to throughout the discussions in this section. Itis composed of two BS in the front-end upper (FU)and front-end lower (FL) positions, namely BS/FUand BS/FL, and the bitonic merger (BM) atthe rearend. The BM is composed of a divider andtwo BM’s in the rear-end upper (RU) and rear-endlower (RL) positions, namely BM/RU andBM/RL. This dissection equally applies to the half-sized BS’sin the frontend, iteratively. The output port numbering is doneaccording to the convention that port number 0 is assigned tothe output port to which the largest numbered input packet istransferred (see the circled numbers in the figure). Notice thatthe output port numbering is reversed for the BS/FLin the front-end bottom because it is put upside down.

Property 1: Let andbe two sorted sequences

in decreasing order and, for a fixed integer inlet and

Then, a) subsequencesand as well astheir concatenation are all bitonic,b) subsequences and

are both bitonic with

the ordering

(1)

for integers and in andand c) .6

Proof: a) Let ,, , which is the -rotated and reversed se-

quence of We partition sequences and intoand respectively, such that the first elementsof and belong to and Then subsequencesand are both decreasingly ordered with andsubsequences and are both increasingly-ordered with

Since subsequences and are formed bytaking, respectively, the smaller and the larger of the elementpair for they consistof the elements

(2)

for integers and in and There-fore, and are both bitonic and their concatenationisalso bitonic because it is the-rotated sequence of the concate-nation of the subsequences and

b) Since subsequences and are formed by taking,respectively, the larger and the smaller of the element pair

and for they consist

6Notationh�iN denotes the modulo-N operation, notation(X; Y ) indicatesthe concatenated sequence of the two sequencesX andY; and the inequalityX � Y means thatX(i) � Y (j) for each elementX(i) in X and eachelementY (j) in Y:


Fig. 2. Demonstration of Property 1 forN = 8 andk = 5:

of the elements

(3)

for some of the integers and in andTherefore, and are both bitonic with the

ordering in (1).c) Subsequences and are originally defined

such that and Since sequences andare partitioned such that and we get

and Therefore, we getand hence

Property 1 describes the behavior of the divider in the frontend of the BM in Fig. 1, in which the upwardand downward polarities of the constituent sortingelements, respectively, carry out themax and min operationsin the property. From this divider’s point of view, the propertycan be interpreted as follows:given two sorted sequencesof length each, with the second sequence additionally-rotated, if the polarity of the sorting elements are arrangedsuch that first elements are upward aligned and theother elements are downward aligned, then the concatenatedsequence is directed to the BM/RU and theconcatenated sequence is directed to theRM/RL.

The sequences given in the left-hand and the right-handsides of the divider in Fig. 2 demonstrate the operation of thedivider, where {15, 11, 10, 8, 7, 5, 4, 2}, {14, 13,12, 9, 6, 3, 1, 0}, 5, {15, 13, 14}, {0, 1, 3,4, 2}, {12, 11, 10}, {8, 7, 5, 6, 9}.

Property 2: Let andbe two bitonic sequences that

satisfy the relations in (1) and the inequalityGiven a -rotated sequence of the concatenated sequence

for an arbitrary integer in thebitonic merger can convert it to the sorted and-rotated

sequence by inverting the polarity of the sorting element towhich one sequence element in enters.

Proof: Let be the difference between the largest el-ement in and the smallest element in Thenbecause Let

Then and the concatenatedsequence becomes bitonic due to the relationsin (1). If this bitonic sequence is given as the input to the

BM, then it outputs its sorted sequence with the largestelement in (i.e., the largest element in appearing atoutput port 0 regardless of the integer value ofNow, insteadof adding to each element of we invert the polarity ofeach sorting element in the BM which admitsone sequence element in Then it has the same sortingeffect with the largest element in still appearing at outputport 0, which, however, is no longer the largest element in

The largest element in is now the largest element inwhich appears at output port Therefore, the output

sequence thus obtained is the sorted and-rotated sequenceof the concatenated sequence

Property 2 describes the behavior of the BM/RUwhich admits the concatenated sequence Accordingto the property, the BM/RU arranges this sequence orits -rotated sequence to the-rotated sorted sequence if somepertinent sorting elements are inversely polarized.In contrast,the BM/RL does the normal bitonic merging operationwhich sorts the bitonic input sequence

Fig. 3 demonstrates the structure and operation of theBM/RU for and We can confirm from the twocases in the figure that the BM produces the five-rotatedsequences of the concatenated input sequences regardless ofthe values of

Based on Properties 1 and 2, we now establish how toconvert the BS to a distributor.

Theorem 1: For an arbitrary input sequence, the BSwith can generate the-rotated sorted sequence for

by inverting the state polarity of somepertinent sorting elements.

Proof: We prove the theorem by induction. To beginwith, it is clear that the 2 2 BS, which corresponds to the 2

2 sorting element, generates the one-rotated sorted sequenceif its state polarity is inverted.

We suppose that the theorem holds up to the (i.e.,) BS and prove it for the (i.e., )

BS. Since this assumption means that the two front-endBS’s (i.e., the BS/FU and the BS/FL in Fig.1) can convert their inputs to any sorted and rotated sequencepatterns, all we need to do is specify those output sequencepatterns that can be rearranged by the merger to thedesired sorted and-rotated output sequences.

We first consider the case when Accordingto Property 2, if an BM admits the concatenation


(a) (b)

Fig. 3. Demonstration of Property 2 forN = 8 and k = 5: (a) p = 0 and (b) p = 2:

Fig. 4. Demonstration of thek-rotation and sorting capability of the2N � 2N BS (k = 5; N = 8):

of two bitonic sequences and with the ordering in(1) and (i.e., the BM/RU in Fig. 1),then it can produce the-rotated sorted output sequence for

by inverting the polarity of somepertinent sorting elements. According to Property 1, it ispossible to arrange such a concatenated sequence of length

for the BM/RU by inverting the polarity of -sorting elements in the bottom of the divider provided thatthe two input sequences of length are both sorted with thesecond sequence additionally reversed and-rotated (i.e.,and in Property 1). In this case the other sequence of length

which is directed to the BM/RL is the bitonicsequence with which is sortedthere. But, by assumption, these two sequences,and areexactly what the two front-end BS’s can exactly provide. Thisproves the theorem for the case whenFor the other case when we canfollow a similar reasoning with the roles of BM/RUand BM/RL interchanged.

The sequences illustrated in Fig. 4 demonstrate how theBS can generate the-rotated sorted sequence out

of the two input sequences {10, 15, 7, 2, 5, 8, 4, 11} and{13, 9, 1, 14, 0, 6, 3, 12}. Note, however, that this is one of

possible arrangements: according to Property 2 the same-rotated sorted sequence is obtained even if the concatenated

sequence is -rotated before the bitonic mergingprocess, for Therefore, we get the sameresults out of the different arrangements in which theBM/FU output sequence is-rotated and the BM/FLoutput sequence is -rotated.

IV. REALIZATION OF LARGE-SCALE DISTRIBUTORS

In the previous section, Theorem 1 describes how to realizea distributor based on the BS.The BS becomes a distributorif an active packet counter and a state controller are attachedto it to aid the involved state polarity inversion process; itsconstituent sorting elements compare to the activity bit valuesof input packets, which are set to “1” (or other larger value)for active packets and “0” (or other smaller value) for inactivepackets.

The resulting BS-based distributor structure is very similarto the CSE-based distributor structure in [3]–[5]. This real-


ization, however, is not practical as it requires a complexstate controller and does not guarantee the independence ofoperations among constituent unit elements. Therefore, in thissection, we will introduce a simplified realization in which thestate polarity inversion process is carried outautonomouslywithout requiring any packet counter and state controller, andthen consider how to modify the realization to get a large-scaledistributor structure.

Theorem 2: An BS with can provide thedistribution function by inverting, at the end of each time slot,the state polarity of every sorting element that has served asingle active packet.

Proof: We prove the theorem by induction. In the case ofthe 2 2 BS, which generates the one-rotated sorted sequenceif its polarity is inverted, it is obvious that the active packetsget distributed if its state polarity is inverted after each singleactive packet service.

We suppose that the theorem holds up to the (i.e.,) BS and prove it for the (i.e., )

BS. If reflected on the BS, this assumption impliesthat the front-end BS/FU and BS/FL (see Fig. 1) canperform the distribution function by following the givenstateinversion scheme, or, equivalently, the two BS’s canconcentrate and rotate their active input packets to their desiredoutput ports without regard to their input port patterns. So, weprove that the BM at the rear end can distributethe combination of them to the desired final output ports byfollowing the given state inversion scheme.

Let the BS be initially arranged (i.e., the polarityof each constituent sorting element is aligned) accordingto Theorem 1 such that it can generate a-rotated sortedsequence, for For this, let the internal

BS/FU and BS/FL be arranged, respectively,to generate -rotated and -rotated sorted sequences for

and On this basis,we first prove the theorem for the single active packet arrivalby showing that the given-state inversion scheme arranges thesorting element states such that the subsequent active inputpackets can be properly distributed. For this, we consider thetwo cases that A) the and active packets all arrive attime slot and B) one active packet arrives, without loss ofgenerality, at the BS/FU at time slotand the otherand active packets arrive at time slot and prove thatthe state inversion process yields the same results for the twocases in terms of routes and sorting element states.7

For convenience of proof, we assign activity bit value 2to the first active packet, activity bit value 1 to the other

active packets, and activity bit value 0 toinactive packets. Then, in Case A, the active packetsare supposed to be distributed to output portsthrough

with the first active packet distributed toport because the sorting elements in the bitonic merger areinitially aligned that way. Likewise, in Case B, the first activepacket is supposed to be distributed to output portsin timeslot For the other active packets that arriveat time slot to be distributed to the same destination as

7Note that the two cases yield the same distributed output pattern if therelevant routes and the sorting element states are the same.

for Case A, the polarity of the related sorting elements in theBM should be arranged, at the end of time slotsuch thatthe routes of these active packets become identical to thosein Case A. In order to show that the given state inversionscheme yields this necessitated arrangement, it suffices to showthat each constituent sorting element makes a right polaritydecision in support of the desired routing. For this, we mayconsider only the particular type of sorting elements to whichan activity 2 packet and an activity 1 packet arrive, assumingthat the state was initially upward polarized, since the givenstate inversion scheme does not bring in any difference to bothCases A and B for other type of sorting elements. Then, inCase A, the two packets will pass through the sorting element,concurrently, with the activity 2 packet appearing at the upperoutput port and the activity 1 packet appearing at the loweroutput port. In Case B, the activity 2 packet will pass throughthe sorting element, in single, at time slotarriving at theupper output port, and if the state polarity were kept intact,the activity 1 packet entering the sorting element at time slot

would appear at the upper output port again. However,due to the state inversion scheme, the state polarity will beinverted downward at the end of time slotthus routing theactivity 1 packet to the lower output port, and the state willresume the upward state at the end of time slot Thisproves that the relevant routes are the same in both cases andthe final sorting element states after state inversion are alsothe same.

We now prove the theorem for the general case with multipleactive packet arrival. It is established above that the given stateinversion scheme arranges the route for the first active inputpacket and the states of the sorting elements on the route suchthat the subsequent active input packets can becorrectly routed to the desired output ports, forming the sameconcentrated and rotated output pattern that the concurrent

active input packets can generate “in group.” Ifapplied repeatedly, this establishes that a BS-based distributorcan distribute active packets in one group or indivision of (for multiple subgroups,consisting of single active packet services and a groupservice of active packets, without affecting thedistribution pattern. But the single active packet services,in turn, yield the same sorting element states as that one groupservice of active packets does. Therefore, this proves thatthe state inversion scheme arranges the sorting element statesafter the active packet service (in group) such that thesubsequent active input packets can get properly distributed.Therefore, the proof is complete.

From the theorem we learn that a BS works as a distributorif the constituent sorting elements are equipped with the stateinversion capability. This state inversion capability, or thecapability to invert the state polarity after each single activepacket service, is equally equipped if each sorting element ismodified to provide the alternate routing capability to eachactive packet. But, this is exactly what a 2 2 distributordoes.Therefore, if we replace each2 2 sorting element inthe BS with a2 2 distributor (e.g., a CSE proposedin [3] and a delay) then we obtain an distributor, whichis independent and autonomous in operation.Fig. 5 shows the


Fig. 5. A BS-based 8� 8 distributor structure.

resulting 8 8 BS-based distributor structure, which employsthe CSE’s as the constituent 2 2 distributors.

So far, we have considered how to convert BS’s intodistributors by incorporating the rotation capability and, conse-quently, have established that it becomes possible if we simplyreplace each constituent 2 2 sorting element with a 2 2distributor. This fact triggers the conjecture that we may beable to obtain a large-scale distributor if we replace each 22sorting element with a large-sized distributor (i.e.,for ), instead. In fact, this conjecture turns out to betrue, as will become clear in the following theorem.

Theorem 3: An sorter (with and) obtained by replacing each 2 2 sorting element of

the BS with a BS can provide the distributionfunction by inverting, at the end of each time slot, the statepolarity of every 2 2 sorting elements that has served asingle active packet.

We can prove the theorem by induction noting that if wereplace the sorting element in the 2 2 (i.e., ) BSwith a BS, then the resulting sorter becomes the

Batcher sorter itself. We omit the details of the proofas one can easily extend the proof of Theorem 2 for this.

As a consequence of the theorem, we can now generalizethe highlighted statement at the end of Theorem 2 as follows:if we replace each2 2 sorting element in the BSwith a distributor then we obtain andistributor, which is independent and autonomous in operation.This means that we can obtain an distributorjust by interconnecting

distributor modules according to the BS’sinternal interconnection pattern. Theorem 3 renders a simplebut powerful tool to generate large-scale distributors out ofsmall-scale ones in a multiplicative fashion. Fig. 6 illustrates

Fig. 6. A 64� 64 distributor that is composed of 32� 32 distributor chips.

an example of a 64 64 distributor which is obtained byreplacing each sorting element in the 44 BS (i.e., )with a 32 32 distributor (i.e., ).8

V. CONCLUSIONS

In this paper, we have considered how to construct a large-scale distributor that guarantees independent operation amongall constituent unit elements even at the sacrifice of packet se-quence integrity, since independence is fundamentally requiredfor realizing a large-scale distributor with high-speed operationand simplified maintenance. We exerted rigorous reasoning toestablish that the desired solution can be found in convertingBS’s to distributors by incorporating the rotational capabilityto them.

Throughout Theorems 1 and 2, we have investigated howto incorporate the rotational capability to BS’s and found

8Note that the arrows inside the 32� 32 distributors indicate the directionof packet distribution.


that it can be done by furnishing each constituent sortingelement with the state inversion capability or the capabilityto invert its state polarity after each single active packetservice. Then, we have recognized that this is equivalent toreplacing each sorting element with a 22 distributor. As aconsequence, the distributor design problem has turned into asimple substitution problem that we substitute a unit distributorfor each sorting element in the prototype BS. By Theorem 3,we have found that this substitution approach can be extendedto larger scale unit distributors, thus enabling a large-scaledistributor realization.

Therefore, the procedure to build a nonsequence-preservedlarge-scale distributor out of a BS can be put as follows: givena desired size for design of a large-scale distributor,we first design an BS and then replace each constituent2 2 sorting element with a small-scale distributormodule. In case is a very large number, we may furtherdecompose the composite numberand repeat the substitutionprocess accordingly.

REFERENCES

[1] H. S. Kim and A. Leon-Garcia, “A self-routing multistage network forbroadband ISDN,”IEEE J. Select. Areas Commun., vol. 8, pp. 459–466,Apr. 1990.

[2] , “Nonblocking property of reverse banyan network,”IEEE Trans.Commun., vol. 40, pp. 472–476, Mar. 1992.

[3] J. G. Lee and B. G. Lee, “A new distribution network based oncontrolled switching elements and its applications,”IEEE/ACM Trans.Networking, vol. 3, pp. 70–81, Feb. 1995.

[4] , “Autonomous distributor based on controlled switching ele-ments,” inProc. Asia–Pacific Conf. Commun., 1995, pp. 278–282.

[5] , “ECI value generations in CSE-based distributors,”IEEE Trans.Commun., vol. 46, pp. 738–742, June 1998.

[6] M. Kazemi-Nia and H. W. Alnuweiri, “Balanced multiport buffer designin silicon,” in Proc. ATM Workshop, 1996, pp. xx–xx.

[7] M. V. Chien and A. Y. Oruc, “Adaptive binary sorting schemes andassociated interconnection networks,”IEEE Trans. Parallel Distrib.Syst., vol. 5, pp. 561–572, June 1994.

[8] D. E. Knuth,The Art of Computer Programming: Sorting and Searching.New York: Addison Wesley, 1973.

[9] S. G. Akl, Parallel Sorting Algorithms. New York: Academic, 1985.

Jeong Gyu Lee (S’90–M’98) received the B.S.degree in electronics engineering from the KoreaAdvanced Institute of Science and Technology in1990. He received the M.S. and Ph.D. degrees inelectronics engineering from Seoul National Uni-versity in 1992 and 1997, respectively. Since 1997,he has been with ETRI as a Senior Researcher, whois currently involved in developing ATM switch-ing system in ETRI. His research interests includebroadband switching architectures and its perfor-mance evaluation, and B-ISDN.

Byeong Gi Lee(S’80–M’82–SM’89–F’97) for a photograph and biography,see page 765 of the May 1999 issue of this TRANSACTIONS.

realization of large-scale distributors based on batcher sorters

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