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Full-connection Multistage Networks In a two-stage FC network it makes no sense talking about rearrangeability, since each I/Oconnection between a network inlet and a network outlet can

Chapter 3 Rearrangeable Networks

The class of rearrangeable networks is here described, that is those networks in which it isalways possible to set up a new connection between an idle inlet and an idle outlet by adopt-ing, if necessary, a rearrangement of the connections already set up The class of rearrangeablenetworks will be presented starting from the basic properties discovered more than thirty yearsago (consider the Slepian–Duguid network) and going through all the most recent findings onnetwork rearrangeability mainly referred to banyan-based interconnection networks

Section 3.1 describes three-stage rearrangeable networks with full-connection (FC) stage pattern by providing also bounds on the number of connections to be rearranged.Networks with interstage partial-connection (PC) having the property of rearrangeability areinvestigated in Section 3.2 In particular two classes of rearrangeable networks are described inwhich the self-routing property is applied only in some stages or in all the network stages.Bounds on the network cost function are finally discussed in Section 3.3

inter-3.1 Full-connection Multistage Networks

In a two-stage FC network it makes no sense talking about rearrangeability, since each I/Oconnection between a network inlet and a network outlet can be set up in only one way (byengaging one of the links between the two matrices in the first and second stage terminatingthe involved network inlet and outlet) Therefore the rearrangeability condition in this kind ofnetwork is the same as for non-blocking networks

Let us consider now a three-stage network, whose structure is shown in Figure 3.1 A veryuseful synthetic representation of the paths set up through the network is enabled by thematrix notation devised by M.C Paull [Pau62] A Paull matrix has rows and columns, asmany as the number of matrices in the first and last stage, respectively (see Figure 3.2) Thematrix entries are the symbols in the set , each element of which represents one

92 Rearrangeable Networks

of the middle-stage matrices The symbol a in the matrix entry means that an inlet ofthe first-stage matrix i is connected to an outlet of the last-stage matrix j through the middle-stage matrix a The generic matrices i and j are also shown in Figure 3.1 Each matrix entrycan contain from 0 up to distinct symbols; in the representation of Figure 3.2 a connectionbetween i and j crosses matrix a and three connections between k and j are set up throughmatrices b, c and d

Based on its definition, a Paull matrix always satisfies these conditions:

• each row contains at most distinct symbols;

• each column contains at most distinct symbols

In fact, the number of connections through a first-stage (last-stage) matrix cannot exceedeither the number of the matrix inlets (outlets), or the number of paths (equal to the number

of middle-stage matrices) available to reach the network outlets (inlets) Furthermore, eachsymbol cannot appear more than once in a row or in a column, since only one link connectsmatrices of adjacent stages

Figure 3.1 Three-stage FC network

Figure 3.2 Paull matrix

# 1

# r1

# 1

# r2N

r2

1 1

i k

r1

bcd a

min n r( , 2)

min r( 2,m)

Full-connection Multistage Networks 93

The most important theoretical result about three-stage rearrangeable networks is due to

D Slepian [Sle52] and A.M Duguid [Dug59]

Slepian–Duguid theorem A three-stage network is rearrangeable if and only if

Proof The original proof is quite lengthy and can be found in [Ben65] Here we will follow asimpler approach based on the use of the Paull matrix [Ben65, Hui90] We assume without loss

of generality that the connection to be established is between an inlet of the first-stage matrix i

and an outlet of the last-stage matrix j At the call set-up time at most and nections are already supported by the matrices i and j, respectively Therefore, if

at least one of the symbols is missing in row i and column j Then

at least one of the following two conditions of the Paull matrix holds:

1. There is a symbol, say a, that is not found in any entry of row i or column j

2. There is a symbol in row i, say a, that is not found in column j and there is a symbol in umn j, say b, that is not found in row i

col-If Condition 1 holds, the new connection is set up through the middle-stage matrix a fore a is written in the entry of the Paull matrix and the established connections neednot be rearranged If only Condition 2 holds, the new connection can be set up onlyafter rearranging some of the existing connections This is accomplished by choosing arbi-trarily one of the two symbols a and b, say a, and building a chain of symbols in this way(Figure 3.3a): the symbol b is searched in the same column, say , in which the symbol a ofrow i appears If this symbol b is found in row, say, , then a symbol a is searched in this row

There-If such a symbol a is found in column, say , a new symbol b is searched in this column Thischain construction continues as long as a symbol a or b is not found in the last column or rowvisited At this point we can rearrange the connections identified by the chain

replacing symbol a with b in rows and symbol b with symbol a incolumns By this approach symbols a and b still appear at most once in any row orcolumn and symbol a no longer appears in row i So, the new connection can be routedthrough the middle-stage matrix a (see Figure 3.3b)

Figure 3.3 Connections rearrangement by the Paull matrix

r2≥max n m( , )

n–1 m–1

r2>max n( –1,m–1) r2

i j,( )

1 1

This rearrangement algorithm works only if we can prove that the chain does not end on

an entry of the Paull matrix belonging either to row i or to column j, which would make the

rearrangement impossible Let us represent the chain of symbols in the Paull matrix as a graph

in which nodes represent first- and third-stage matrices, whereas edges represent second-stage

matrices The graphs associated with the two chains starting with symbols a and b are sented in Figure 3.4, where c and k denote the last matrix crossed by the chain in the second

repre-and first/third stage, respectively Let “open (closed) chain” denote a chain in which the firstand last node belong to a different (the same) stage It is rather easy to verify that an open chaincrosses the second stage matrices an odd number of times, whereas a closed chain makes it aneven number of times Hence, an open (closed) chain includes an odd (even) number of edges

We can prove now that in both chains of Figure 3.4 In fact if , by tion of Condition 2, and since would result in a closed chain with an oddnumber of edges or in an open chain with an even number of edges, which is impossible.Analogously, if , by assumption of Condition 2 and , since would result

assump-in an open chaassump-in with an even number of edges or in a closed chain with an odd

It is worth noting that in a squared three-stage network the Slepian–Duguid rule for a rangeable network becomes The cost index C for a squared rearrangeable network

is

The network cost for a given N depends on the number n By taking the first derivative of

C with respects to n and setting it to 0, we find the condition providing the minimum cost

network, that is

(3.1)Interestingly enough, Equation 3.1 that minimizes the cost of a three-stage rearrangeablenetwork is numerically the same as Equation 4.2, representing the approximate condition forthe cost minimization of a three-stage strict-sense non-blocking network Applying

Equation 3.1 to partition the N network inlets into groups gives the minimum cost of athree-stage RNB network:

2

=

r1

3 2

=

Thus a Slepian–Duguid rearrangeable network has a cost index roughly half that of a Closnon-blocking network, but the former has the drawback of requiring in certain network statesthe rearrangement of some connections already set up

From the above proof of rearrangeability of a Slepian–Duguid network, there follows thistheorem:

Theorem. The number of rearrangements at each new connection set-up ranges up to

Proof. Let and denote the two entries of symbols a and b in rows i and j, respectively, and, without loss of generality, let the rearrangement start with a The chain will not contain any symbol in column , since a new column is visited if it contains a, absent in

by assumption of Condition 2 Furthermore, the chain does not contain any symbol in row

since a new row is visited if it contains b but a second symbol b cannot appear in row

Hence the chain visits at most rows and columns, with a maximum number ofrearrangements equal to Actually is only determined by the minimumbetween and , since rows and columns are visited alternatively, thus providing

Paull [Pau62] has shown that can be reduced in a squared network with byapplying a suitable rearrangement scheme and this result was later extended to networks witharbitrary values of

Paull theorem The maximum number of connections to be rearranged in a Slepian–Duguidnetwork is

Proof Following the approach in [Hui90], let us assume first that , that is columns areless than rows in the Paull matrix We build now two chains of symbols, one starting from sym-

bol a in row i and another starting from symbol b in column j

In the former case the chain is obtained, whereas in the other case the chain

is These two chains are built by having them grow alternatively, so that thelengths of the two chains differ for at most one unit When either of the two chains cannotgrow further, that chain is selected to operate rearrangement The number of growth steps is atmost , since at each step one column is visited by either of the two chains and the start-

ing columns including the initial symbols a and b are not visited Thus , as also theinitial symbol of the chain needs to be exchanged If we now assume that , the sameargument is used to show that Thus, in general no more than

rearrangements are required to set up any new connection request between an idle network

The example of Figure 3.5 shows the Paull matrix for a three-stage network with and The rearrangeability condition for the network requires ; letthese matrices be denoted by the symbols In the network state represented byFigure 3.5a a new connection between the matrices 1 and 1 of the first and last stage is

requested The middle-stage matrices c and d are selected to operate the rearrangement

accord-ing to Condition 2 of the Slepian–Duguid theorem (Condition 1 does not apply here) If the

rearrangement procedure is based on only one chain and the starting symbol is c in row 1, the

final state represented in Figure 3.5b is obtained (new connections are in italicized bold), with

a total of 5 rearrangements However, by applying the Paull theorem and thus generating two

alternatively growing chains of symbols, we realize that the chain starting from symbol d in

column 1 stops after the first step So the corresponding total number of rearrangements is 2

(see Figure 3.5c) Note that we could have chosen the symbol e rather than d since both of

them are missing in column 1 In this case only one connection would have been rearrangedrather than two as previously required Therefore minimizing the number of rearrangements inpractical operations would also require to optimal selection of the pair of symbols in the Paullmatrix, if more than one choice is possible, on which the connection rearrangement proce-dure will be performed

In the following for the sake of simplicity we will assume a squared network, that is

, unless specified otherwise

3.2 Partial-connection Multistage Networks

We have shown that banyan networks, in spite of their blocking due to the availability of onlyone path per I/O pair, have the attractive feature of packet self-routing Furthermore, it is pos-sible to build rearrangeable PC networks by using banyan networks as basic building block.Thus RNB networks can be further classified as

• partially self-routing, if packet self-routing takes place only in a portion of this network;

• fully self-routing, if packet self-routing is applied at all network stages.

These two network classes will be examined separately in the next sections

3.2.1 Partially self-routing PC networks

In a PC network with partial self-routing some stages apply the self-routing property, someothers do not This means that the processing required to set up the required network permu-tation is partially distributed (it takes place directly in the self-routing stages) and partiallycentralized (to determine the switching element state in all the other network stages)

Figure 3.5 Example of application of the Paull matrix

a b

b e

e f a

1 2 3 4 5

a b

b e

e f a

(c)

1 2 3 4

1 2 3 4 5

N = M n, 1 = m s = n

Two basic techniques have been proposed [Lea91] to build a rearrangeable PC networkwith partial self-routing, both providing multiple paths between any couple of network inletand outlet:

• horizontal extension (HE), when at least one stage is added to the basic banyan network.

• vertical replication (VR), when the whole banyan network is replicated several times;

Separate and joined application of these two techniques to build a rearrangeable network isnow discussed

3.2.1.1 Horizontal extension

A network built using the HE technique, referred to as extended banyan network (EBN), is obtained by means of the mirror imaging procedure [Lea91] An EBN network of size

with stages is obtained by attaching to the first network stage of a banyan

network m switching stages whose connection pattern is obtained as the mirror image of the permutations in the last m stage of the original banyan network Figure 3.6 shows a

EBN SW-banyan network with additional stages Note that adding m stages means

making available paths between any network inlet and outlet Packet self-routing takesplace in the last stages, whereas a more complex centralized routing control is

required in the first m stages It is possible to show that by adding stages to theoriginal banyan network the EBN becomes rearrangeable if this latter network can be builtrecursively as a three-stage network

A simple proof is reported here that applies to the -stage EBN networkbuilt starting from the recursive banyan topology SW-banyan Such a proof relies on a property

of permutations pointed out in [Ofm67]:

Ofman theorem It is always possible to split an arbitrary permutation of size N into two

subpermutations of size such that, if the permutation is to be set up by the work of Figure 3.7, then the two subpermutations are set up by the two non-blocking

central subnetworks and no conflicts occur at the first and last switching stage ofthe overall network

This property can be clearly iterated to split each permutation of size into two permutations of size each set up by the non-blocking subnetworks ofFigure 3.7 without conflicts at the SEs interfacing these subnetworks Based on this property itbecomes clear that the EBN becomes rearrangeable if we iterate the process until the “central”subnetworks have size (our basic non-blocking building block) This result is obtainedafter serial steps of decompositions of the original permutation that generate per-mutations of size Thus the total number of stages of switching elementsbecomes , where the last unity represents the “central” subnet-works (the resulting network is shown in Figure 3.6c) Note that the first and last stage of SEsare connected to the two central subnetworks of half size by the butterfly pattern

sub-If the reverse Baseline topology is adopted as the starting banyan network to build the

-stage EBN, the resulting network is referred to as a Benes network [Ben65] It is

interesting to note that a Benes network can be built recursively from a three-stage nection network: the initial structure of an Benes network is a Slepian–Duguidnetwork with So we have matrices of size in the first and third

Figure 3.6 Horizontally extended banyan network with N=16 and m=1-3

stage and two matrices in the second stage interconnected by an EGS patternthat provides full connectivity between matrices in adjacent stages Then each of the two

matrices is again built as a three-stage structure of matrices of size inthe first and third stage and two matrices in the second stage The procedure isiterated until the second stage matrices have size The recursive construction of a Benes network is shown in Figure 3.8, by shadowing the subnetworks recursively built

The above proof of rearrangeability can be applied to the Benes network too In fact, therecursive network used with the Ofman theorem would be now the same as in Figure 3.7 withthe interstage pattern replaced by at the first stage and at the last stage

This variation would imply that the permutation of size N performed in the network of

Figure 3.6c would give in the recursive construction of the Benes network the same setting ofthe first and last stage SEs but two different permutations of size The Benes network isthus rearrangeable since according to the Ofman theorem the recursive construction ofFigure 3.7 performs any arbitrary permutation

Thus an Benes network has stages of SEs, each stageincluding SEs Therefore the number of its SEs is

Figure 3.7 Recursive network construction for the Ofman theorem

N

inlets

N outlets N/2 x N/2

N/2 x N/2

N/4 x N/4 N/4 x N/4

=

with a cost index (each SE accounts for 4 crosspoints)

(3.3)

If the number of I/O connections required to be set up in an network is N, the connection set is said to be complete, whereas an incomplete connection set denotes the case of less than N required connections (apparently, since each SE always assumes either the straight

or the cross state, N I/O physical connections are always set up) The number of required nections is said to be the size of the connection set The set-up of an incomplete/complete

con-connection set through a Benes network requires the identification of the states of allthe switching elements crossed by the connections This task is accomplished in a Benes net-

work by the recursive application of a serial algorithm, known as a looping algorithm [Opf71], to

the three-stage recursive Benes network structure, until the states of all the SEs crossed by atleast one connection have been identified The algorithm starts with a three-stage net-work with first and last stage each including elements and two middle

networks, called upper (U) and lower (L) subnetworks By denoting with busy (idle) a network

termination, either inlet or outlet, for which a connection has (has not) been requested, thelooping algorithm consists of the following steps:

1. Loop start In the first stage, select the unconnected busy inlet of an already connectedelement, otherwise select a busy inlet of an unconnected element; if no such inlet is foundthe algorithm ends

Figure 3.8 Benes network

2. Forward connection Connect the selected network inlet to the requested network let through the only accessible subnetwork if the element is already connected to the othersubnetwork, or through a randomly selected subnetwork if the element is not yet con-nected; if the other outlet of the element just reached is busy, select it and go to step 3; oth-erwise go to step 1.

out-3. Backward connection Connect the selected outlet to the requested network inletthrough the subnetwork not used in the forward connection; if the other inlet of the ele-ment just reached is busy and not yet connected, select it and go to step 2; otherwise go tostep 1

Depending on the connection set that has been requested, several loops of forward andbackward connections can be started Notice that each loop always starts with an unconnectedelement in the case of a complete connection set Once the algorithm ends, the result is theidentification of the SE state in the first and last stage and two connection sets of maximumsize to be set up in the two subnetworks by means of the looping algo-rithm An example of application of the looping algorithm in an network is represented

in Figure 3.9 for a connection set of size 6 (an SE with both idle terminations is drawn asempty) The first application of the algorithm determines the setting of the SEs in the first andlast stage and two sets of three connections to be set up in each of the two central subnetworks The looping algorithm is then applied in each of these subnetworks and the resultingconnections are also shown in Figure 3.9 By putting together the two steps of the loopingalgorithm, the overall network state of Figure 3.10 is finally obtained Parallel implementations

of the looping algorithm are also possible (see, e.g., [Hui90]), by allocating a processing bility to each switching element and thus requiring their complete interconnection for themutual cooperation in the application of the algorithm We could say that the looping algo-rithm is a constructive proof of the Ofman theorem

capa-A further refining of the Benes structure is represented by the Waksman network [Wak68] which consists in predefining the state of a SE (preset SE) per step of the recursive network

construction (see Figure 3.10 for ) This network is still rearrangeable since, compared

to a Benes network, it just removes the freedom of choosing the upper or lower middle work for the first connection crossing the preset SE Thus the above looping algorithm is now

net-Figure 3.9 Example of application of the looping algorithm

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

U

0 - 1

2 - 3

3 - 2 L

0 1 2 3

Subnetwork L

0 1 2 3

0 1 2 3

modified in the loop start, so that the busy inlets of the preset element must have already beenconnected before starting another loop from non-preset elements Apparently, in the forwardconnection there is now no freedom of selecting the middle subnetwork when the loop startsfrom a preset element Figure 3.12 gives the Waksman network state establishing the same con-nection set of size 6 as used in the Benes network of Figure 3.10

It is rather easy to verify that the number of SEs in a Waksman network is

(3.4)with a cost index

(3.5)

Figure 3.10 Overall Benes network resulting from the looping algorithm

Figure 3.11 Waksman network

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

2 2( log2N–1) 2i

3.2.1.2 Vertical replication

By applying the VR technique the general scheme of a replicated banyan network (RBN)

is obtained (Figure 3.13) It includes N splitters , K banyan networks and N biners connected through EGS patterns In this network arrangement the packet self-routing takes place within each banyan network, whereas a more complex centralized control

com-of the routing in the splitters has to take place so as to guarantee the rearrangeability condition

A rather simple reasoning to identify the number K of banyans that guarantees the network rearrangeability with the VR technique relies on the definition of the utilization factor (UF)

Figure 3.12 Overall Waksman network resulting from the looping algorithm

Figure 3.13 Vertically replicated banyan network

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

N×N

K×1

1 0

N-1

1 0

[Agr83] of the links in the banyan networks The UF of a generic link in stage k is defined

as the maximum number of I/O paths that share the link Then, it follows that the UF is given

by the minimum between the number of network inlets that reach the link and the number ofnetwork outlets that can be reached from the link Given the banyan topology, it is trivial to

show that all the links in the same interstage pattern have the same UF factor u, e.g in a

banyan network for the links of stage , respectively(switching stages, as well as the interstage connection patterns following the stage, are alwaysnumbered 1 through from the inlet side to the outlet side of the network)

Figure 3.14 represents the utilization factor of the banyan networks with , inwhich each node represents the generic SE of a stage and each branch models the generic link

of an interstage connection pattern (the nodes terminating only one branch represent the SEs

in the first and last stage) Thus, the maximum UF value of a network with size N is

meaning that up to I/O connections can be requiring a link at the “center” of the work Therefore the following theorem holds

net-Theorem A replicated banyan network with size including K planes is rearrangeable

if and only if

(3.6)

Proof The necessity of Condition 3.6 is immediately explained considering that the connections that share the same “central” network link must be routed onto distinctbanyan networks, as each banyan network provides a single path per I/O network pair The sufficiency of Condition 3.6 can be proven as follows, by relying on the proof given in

[Lea90], which is based on graph coloring techniques The n-stage banyan network can be

Figure 3.14 Utilization factor of a banyan network

seen as including a “first shell” of switching stages 1 and n, a “second shell” of switching stages

2 and , and so on; the innermost shell is the -th Let us represent in a graph theSEs of the first shell, shown by nodes, and the I/O connections by edges Then, an arbitrarypermutation can be shown in this graph by drawing an edge per connection between the left-side nodes and right-side nodes In order to draw this graph, we define an algorithm that is aslight modification of the looping algorithm described in Section 3.2.1.1 Now a busy inletcan be connected to a busy outlet by drawing an edge between the two corresponding termi-nating nodes Since we may need more than one edge between nodes, we say that edges oftwo colors can be used, say red and blue Then the looping algorithm is modified saying that inthe loop forward the connection is done by a red edge if the node is still unconnected, by ablue edge if the node is already connected; a red (blue) edge is selected in the backward con-nection if the right-side node has been reached by a blue (red) edge

The application of this algorithm implies that only two colors are sufficient to draw thepermutation so that no one node has two or more edges of the same color In fact, the edgesterminating at each node are at most two, since each SE interfaces two inlets or two outlets.Furthermore, on the right side the departing edge (if the node supports two connections) hasalways a color different from the arriving edge by construction On the left side two cases must

be distinguished: a red departing edge and a blue departing edge In the former case the ing edge, if any, must be blue since colors are alternated in the loop and the arriving edge iseither the second, the fourth, the sixth, and so on in the chain In the latter case the departingedge is by definition different from the already drawn edge, which is red since the node wasinitially unconnected Since two colors are enough to build the graph, two parallel banyan

arriv-planes, including the stages 1 and n of the first shell, are enough to build a network where no two inlets (outlets) share the same link outgoing from (terminating on) stage 1 (n) Each of

these banyan networks is requested to set up at most connections whose specificationdepends on the topology of the selected banyan network

This procedure can be repeated for the second shell, which includes stages 2 and ,thus proving the same property for the links outgoing from and terminating on these twostages The procedure is iterated times until no two I/O connections share any inter-

stage link Note that if n is even, at the last iteration step outgoing links from stage are theingoing links of stage An example of this algorithm is shown in Figure 3.15 for areverse Baseline network with ; red edges are represented by thin line, blue edges bythick lines For each I/O connection at the first shell the selected plane is also shown (plane Ifor red edges, plane II for blue connections) The overall network built with the looping algo-rithm is given in Figure 3.16 From this network a configuration identical to that of RBNgiven in Figure 3.13 can be obtained by “moving” the most central splitters and combiners to the network edges, “merging” them with the other splitters/combiners and rep-licating correspondingly the network stages being crossed Therefore splitters and

combiners are obtained with K replicated planes (as many as the number of “central” stages at the end of the looping algorithm), with K given by Equation 3.6 Therefore the suffi-

The number of planes making a RBN rearrangeable is shown in Table 3.1

By means of Equation 3.6 (each SE accounts for 4 crosspoints), the cost index for arearrangeable network is obtained:

(3.7)where the last term in the sum accounts for the cost of splitters and combiners

Thus we can easily conclude that the HE technique is much more cost-effective than the

VR technique in building a rearrangeable network : in the former case the cost growswith (Equations 3.3 and 3.5), but in the latter case with (Equation 3.7)

Figure 3.15 Example of algorithm to set up connections in a reverse Baseline network

Figure 3.16 Overall RNB network resulting from t looping algorithm with N=16

8-13 9-12 10-8 11-15 12-1 13-5 14-10 15-6

I II I II I II II I

0-2 3-7 5-1 6-5 10-14 11-12 12-8 13-11

0-1 2-3 4-5 6-7 8-9 10-11 12-13 14-15

0-1 2-3 4-5 6-7 8-9 10-11 12-13 14-15

2 3

I

1-1 2-4 4-0 5-3 10-14 11-15 12-10 15-13

0-1 2-3 4-5 6-7 8-9 10-11 12-13 14-15

0-1 2-3 4-5 6-7 8-9 10-11 12-13 14-15

1

2 2

3.2.1.3 Vertical replication with horizontal extension

The VR and HE techniques can be jointly adapted to build a rearrangeable network [Lea91]: anon-rearrangeable EBN is first selected with additional stages and then some parts

of this network are vertically replicated so as to accomplish rearrangeability As in the case of anEBN network, the initial banyan topology is either the SW-banyan or the reverse Baseline net-work, so that the horizontally extended network has a recursive construction and the loopingalgorithm for network rearrangeability can be applied In particular, an EBN network

with m additional stages determines 2 “central” subnetworks of size of stages, 4 “central” subnetworks of size of stages, …, “central” subnet-works of size of stages

Theorem An extended banyan network with size and m additional stages is

rear-rangeable if and only if each of the “central” subnetworks of size isreplicated by a factor

and N splitters and N combiners are provided to providefull accessibility to the replicated networks An example of such rearrangeable VR/HE banyannetwork is given in Figure 3.17 for ,

Proof In a banyan network with m additional stages, applying recursively the Ofman theorem

m times means that the original permutation of size N can always be split into tations of size each to be set up by a network (in the example ofFigure 3.17 we can easily identify these four subnetworks) Then by applying the rear-rangeability condition of a pure VR banyan network, each of these subnetworks has to bereplicated

subpermu-times in order to obtain an overall rearrangeable network ❏

Table 3.1 Replication factor in a rearrangeable VR banyan network

Table 3.2 gives the replication factor of a rearrangeable VR/HE banyan network with and Note that the diagonal with gives the Benes net-work and the row gives a pure VR rearrangeable network.

Figure 3.17 Vertical replication of an extended banyan network with N=16 and m=2

Table 3.2 Replication factor in a rearrangeable VR/HE banyan network