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Chapter 3 Rearrangeable Networks The class of rearrangeable networks is here described, that is those networks in which it is always 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 rearrangeable networks will be presented starting from the basic properties discovered more than thirty years ago (consider the Slepian–Duguid network) and going through all the most recent findings on network rearrangeability mainly referred to banyan-based interconnection networks. Section 3.1 describes three-stage rearrangeable networks with full-connection (FC) inter- 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 are investigated in Section 3.2. In particular two classes of rearrangeable networks are described in which 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. 3.1. Full-connection Multistage Networks In a two-stage FC network it makes no sense talking about rearrangeability, since each I/O connection between a network inlet and a network outlet can be set up in only one way (by engaging one of the links between the two matrices in the first and second stage terminating the involved network inlet and outlet). Therefore the rearrangeability condition in this kind of network 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 very useful synthetic representation of the paths set up through the network is enabled by the matrix notation devised by M.C. Paull [Pau62]. A Paull matrix has rows and columns, as many as the number of matrices in the first and last stage, respectively (see Figure 3.2). The matrix entries are the symbols in the set , each element of which represents one r 1 r 3 12… r 2 ,, ,{} This document was created with FrameMaker 4.0.4 net_th_rear Page 91 Tuesday, November 18, 1997 4:37 pm Switching Theory: Architecture and Performance in Broadband ATM Networks Achille Pattavina Copyright © 1998 John Wiley & Sons Ltd ISBNs: 0-471-96338-0 (Hardback); 0-470-84191-5 (Electronic) 92 Rearrangeable Networks of the middle-stage matrices. The symbol a in the matrix entry means that an inlet of the 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 entry can contain from 0 up to distinct symbols; in the representation of Figure 3.2 a connection between i and j crosses matrix a and three connections between k and j are set up through matrices 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 exceed either 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, each symbol cannot appear more than once in a row or in a column, since only one link connects matrices of adjacent stages. Figure 3.1. Three-stage FC network Figure 3.2. Paull matrix #1 #r 1 #1 #r 2 N n x r 2 r 1 x r 3 #1 #r 3 M r 2 x m #j #i ij,() r 2 1 1 jr 3 i k r 1 bcd a min n r 2 ,() min r 2 m,() net_th_rear Page 92 Tuesday, November 18, 1997 4:37 pm 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 a simpler 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 con- 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 col- umn j , say b , that is not found in row i . If Condition 1 holds, the new connection is set up through the middle-stage matrix a . There- fore a is written in the entry of the Paull matrix and the established connections need not be rearranged. If only Condition 2 holds, the new connection can be set up only after 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 of row i appears. If this symbol b is found in row, say, , then a symbol a is searched in this row. If such a symbol a is found in column, say , a new symbol b is searched in this column. This chain construction continues as long as a symbol a or b is not found in the last column or row visited. 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 in columns . By this approach symbols a and b still appear at most once in any row or column and symbol a no longer appears in row i . So, the new connection can be routed through the middle-stage matrix a (see Figure 3.3b). Figure 3.3. Connections rearrangement by the Paull matrix r 2 max nm,()≥ n 1– m 1– r 2 max n 1– m 1–,()> r 2 ij,() ij– j 2 i 3 j 4 ij 2 i 3 j 4 i 5 …,,,,, ii 3 i 5 …,,, j 2 j 4 …,, ij– 1 1 jr 3 i r 1 a b j 2 j 4 j 6 i 5 i 3 b a ba 1 1 jr 3 i r 1 b b a b ab a (a) (b) net_th_rear Page 93 Tuesday, November 18, 1997 4:37 pm 94 Rearrangeable Networks 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 repre- sented in Figure 3.4, where c and k denote the last matrix crossed by the chain in the second and first/third stage, respectively. Let “open (closed) chain” denote a chain in which the first and last node belong to a different (the same) stage. It is rather easy to verify that an open chain crosses the second stage matrices an odd number of times, whereas a closed chain makes it an even 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 assump- tion of Condition 2, and since would result in a closed chain with an odd number 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 in an open chain with an even number of edges or in a closed chain with an odd number of edges, which is impossible. ❏ It is worth noting that in a squared three-stage network the Slepian–Duguid rule for a rear- 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 rearrangeable network is numerically the same as Equation 4.2, representing the approximate condition for the 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 a three-stage RNB network: (3.2) Figure 3.4. Chains of connections through matrices a and b kij,≠ ca= kj≠ ki≠ ki≠ C 1 C 2 cb= ki≠ kj≠ kj≠ C 1 C 2 bab ca baba c i j 2 i 3 j 4 i 5 k j 3 i 4 j 5 k j i 2 C 1 C 2 r 2 n= NM n, m r 1 , r 3 ===() C 2nr 2 r 1 r 1 2 r 2 + 2n 2 r 1 nr 1 2 + 2Nn N 2 n +=== n N 2 = r 1 C 22N 3 2 = net_th_rear Page 94 Tuesday, November 18, 1997 4:37 pm Full-connection Multistage Networks 95 Thus a Slepian–Duguid rearrangeable network has a cost index roughly half that of a Clos non-blocking network, but the former has the drawback of requiring in certain network states the rearrangement of some connections already set up. From the above proof of rearrangeability of a Slepian–Duguid network, there follows this theorem: 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 of rearrangements equal to . Actually is only determined by the minimum between and , since rows and columns are visited alternatively, thus providing . ❏ Paull [Pau62] has shown that can be reduced in a squared network with by applying a suitable rearrangement scheme and this result was later extended to networks with arbitrary values of . Paull theorem. The maximum number of connections to be rearranged in a Slepian–Duguid network is Proof. Following the approach in [Hui90], let us assume first that , that is columns are less 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 the lengths of the two chains differ for at most one unit. When either of the two chains cannot grow further, that chain is selected to operate rearrangement. The number of growth steps is at most , 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 the initial symbol of the chain needs to be exchanged. If we now assume that , the same argument 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 inlet and an idle network outlet. ❏ The example of Figure 3.5 shows the Paull matrix for a three-stage network with and . The rearrangeability condition for the network requires ; let these matrices be denoted by the symbols . In the network state represented by Figure 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 ϕ M 2min r 1 r 3 ,()2–= i a j a ,() i b j b ,() j b j b i b i b r 1 1– r 3 1– r 1 r 3 2–+ ϕ M r 1 r 3 ϕ M 2min r 1 r 3 ,()2–= ϕ M n 1 r 1 = r 1 ϕ M min r 1 r 3 ()1–= r 1 r 3 ≥ abab…,,,,() baba…,,,,() ij 2 i 3 j 4 i 5 …,,,,, j i 2 j 3 i 4 j 5 …,,,,, r 3 2– ϕ M r 3 1–= r 1 r 3 ≤ ϕ M r 1 1–= min r 1 r 3 ()1– 24 25× r 1 4= r 3 5= r 2 6= abcdef,,,,,{} net_th_rear Page 95 Tuesday, November 18, 1997 4:37 pm 96 Rearrangeable Networks 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 rearranged rather than two as previously required. Therefore minimizing the number of rearrangements in practical operations would also require to optimal selection of the pair of symbols in the Paull matrix, 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 only one 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, some others 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 partially centralized (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 b f c c c d f cd a d (a) 1 2 3 4 12345 a b b e e f a b f d c d d c f dc a c (b) 1 2 3 4 12345 a b b e e f a b f c d c c d f cd a d (c) 1 2 3 4 12345 NM n 1 , m s n=== net_th_rear Page 96 Tuesday, November 18, 1997 4:37 pm Partial-connection Multistage Networks 97 Two basic techniques have been proposed [Lea91] to build a rearrangeable PC network with partial self-routing, both providing multiple paths between any couple of network inlet and 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 is now 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 takes place 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 the original banyan network the EBN becomes rearrangeable if this latter network can be built recursively as a three-stage network. A simple proof is reported here that applies to the -stage EBN network built 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 net- 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 of the overall network. This property can be clearly iterated to split each permutation of size into two sub- permutations of size each set up by the non-blocking subnetworks of Figure 3.7 without conflicts at the SEs interfacing these subnetworks. Based on this property it becomes 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 obtained after serial steps of decompositions of the original permutation that generate per- mutations of size . Thus the total number of stages of switching elements becomes , 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 SEs are connected to the two central subnetworks of half size by the butterfly pattern. 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 full-con- nection network: the initial structure of an Benes network is a Slepian–Duguid network with . So we have matrices of size in the first and third NN× nm+ mn1–≤() 16 16× m 13–= 2 m nN 2 log= mn1–= 2 N 2 log 1–() N 2⁄ NN× N 2⁄ N 2⁄× N 2⁄ N 4⁄ N 4⁄ N 4⁄× 22× n 1– N 2⁄ N 2 n 1– ⁄ 2= 22× 2 n 1–()1+ 2n 1–= 22× β n 1– 2 N 2 log 1–() NN× n 1 m 3 2== N 2⁄ 22× net_th_rear Page 97 Tuesday, November 18, 1997 4:37 pm 98 Rearrangeable Networks Figure 3.6. Horizontally extended banyan network with N=16 and m=1-3 m=1 SW-banyan (a) m=2 SW-banyan (b) m=3 SW-banyan (c) net_th_rear Page 98 Tuesday, November 18, 1997 4:37 pm Partial-connection Multistage Networks 99 stage and two matrices in the second stage interconnected by an EGS pattern that 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 in the first and third stage and two matrices in the second stage. The procedure is iterated 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, the recursive network used with the Ofman theorem would be now the same as in Figure 3.7 with the 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 of the first and last stage SEs but two different permutations of size . The Benes network is thus rearrangeable since according to the Ofman theorem the recursive construction of Figure 3.7 performs any arbitrary permutation. Thus an Benes network has stages of SEs, each stage including 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 N/4 x N/4 N/4 x N/4 N 2⁄ N 2⁄× N 2⁄ N 2⁄× N 4⁄ 22× N 4⁄ N 4⁄× 22× 16 16× B Nn⁄ n 248,,=() β n 1– σ n 1– 1– σ n 1– N 2⁄ NN× s 2 N 2 log 1–= 22× N 2⁄ SN N 2 log N 2 –= net_th_rear Page 99 Tuesday, November 18, 1997 4:37 pm 100 Rearrangeable Networks 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 con- nections is said to be the size of the connection set. The set-up of an incomplete/complete connection set through a Benes network requires the identification of the states of all the 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 at least 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, the looping algorithm consists of the following steps: 1. Loop start. In the first stage, select the unconnected busy inlet of an already connected element, otherwise select a busy inlet of an unconnected element; if no such inlet is found the algorithm ends. Figure 3.8. Benes network B 8 B 4 B 16 B 2 22× C 4NN 2 log 2N–= NN× NN× NN× N 2⁄ N 2⁄ N 2⁄× net_th_rear Page 100 Tuesday, November 18, 1997 4:37 pm