Two-Pass Rearrangeability in Faulty Benes Networks

extra stages or multiple layers of subnetworks may offergood permutation capability in the presence of failures[1–4], but the additional hardware, in turn increases thecomplexity of routing. Another technique is to recirculatethe data through the network in order to reach the destina-tion in finite number of passes; it effectively increases thetransmission delay [5]. The third strategy is to reconfigurethe system in a degraded mode, such that full-access crite-rion is maintained among all the active modules (PE’s and/or memory units) [6, 7]. But this strategy results in anenormous wastage of resources and loss of parallelism;moreover, it might require redesign of algorithms to beexecuted by the system.

On the other hand, in the (2n 2 1)-stage rearrangeablenetworks, which can realize any N 3 N permutation in asingle pass, such as Benes, multiple paths originally existbetween any source–destination pair. Therefore, pure soft-ware approach of fault-tolerant routing is possible for thesenetworks, without any degradation in the performance ofthe system. Because of the structural redundancy existingin these networks, full-access property may be retainedeven in the presence of multiple faults, so that by applyingsuitable routing technique, a fault-free path can be set upbetween any source–destination pair.

Most of the existing fault-tolerant routing schemes forrearrangeable networks, investigated the Benes network,with very limited fault-tolerant capability. Agrawal usedan extra switch, attached on either side of the network, tocompensate for a single faulty switch [8]. A fault-tolerantrouting scheme, based on the finite state model of Benesnetwork [9] has been reported in [10]. Nassimi presenteda fault-tolerant self-routing technique for a general classof (2n 2 1)-stage networks [11] to tolerate a single faultin each stage and no fault in the centre stage. The seriouslimitation of all these fault-tolerant routing schemes is that,only the control-line fault in Benes network has been inves-tigated by them.

In [12], the looping algorithm [13] has been extendedto achieve fault-tolerance and graceful degradation in thesystem performance, in the presence of both switch faultsand control-line faults. Weighted permutation requestshave been considered there; source–destination pairs withhigher weight are serviced with higher priority. In [14], afault-tolerant routing scheme is proposed to tolerate multi-ple control-line faults in each stage; in the presence of

JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING 35, 191–198 (1996)ARTICLE NO. 0080

Two-Pass Rearrangeability in Faulty Benes Networks

NABANITA DAS1 AND JAYASREE DATTAGUPTA

Electronics Unit, Indian Statistical Institute, 203, B. T. Road, Calcutta 700 035, India

0743-7315/96 $18.00Copyright 1996 by Academic Press, Inc.

All rights of reproduction in any form reserved.

191

Existing fault-tolerant routing schemes for Benes networkseither consider only the control line stuck-at faults, or handlethe switch faults by some graceful degradation routing schemesthat reconfigure the network into a smaller system with minimalloss. Now, even in the presence of a single switch fault in anN 3 N Benes network B(n), (n 5 log2N), no N 3 N permutationcan be realized in a single pass. In this paper, we attempt tocharacterize the switch fault sets in B(n), in the presence ofwhich the network is always capable of realizing any arbitraryN 3 N permutation P in two passes, such that any source–destination path is set up in a single pass, no recirculation isneeded, but the whole set of N source–destination paths of Pis partitioned in two subsets and are realized in two successivepasses. We propose an algorithm that will detect if the switchfault set present in a B(n), belongs to this class; if it is yes,we present another algorithm that computes the fault-tolerantrouting to realize any arbitrary permutation P in two passes.This scheme enables us to make B(n) fault-tolerant in the pres-ence of a restricted class of multiple switch faults, without anyrecirculation through intermediate nodes, or any reconfigura-tion of the system. 1996 Academic Press, Inc.

1. INTRODUCTION

With the growing interest in large scale parallel pro-cessing systems, a variety of interconnection architectureshave been proposed. Multistage Interconnection Networks(MIN’s), consisting of multiple stages of switches and inter-connecting links, are used to support parallel communica-tion among the different modules of a medium/large scaleparallel processing systems. Now, the failure of a switchor a link in the MIN, can bring down the entire system ormay cause a severe degradation in the performance, unlesssufficient measures are provided to make the network tol-erant to such failures. By fault-tolerant routing through aMIN, we mean successful routing of messages betweensource-destination pairs, even in the presence of faults inthe network.

In the n-stage unique-path full-access N 3 N MIN’s(n 5 log2N), like baseline, omega, reverse-baseline, etc.,any input may access any output through a unique path.In these networks, even a single fault in a switching elementor a link, destroys the full-access property. Addition of

E-mail: [email protected].

free for practical purposes. With this assumption, we haveshown that the class of switch faults considered here coversall possible single faults, more than 90% of all possibledouble faults, and more than 85% of all possible triplefaults, for N 5 16 and 32. Moreover, the coverage increaseswith N. Therefore, this technique will be very much suitablefor use in systems with large values of N, that employa Benes network for interconnecting different modules.Moreover, it has been shown in [15], that for implementingour fault-tolerant routing technique, the testing procedurefor detecting and locating the fault set present in a B(n),is also very fast and simple. This obviously makes ourtechnique more attractive.

Section 2 presents some preliminaries. The two-passfault-tolerant routing technique is described in Section 3.Section 4 summarizes the results.

2. PRELIMINARIES

A MIN is basically an interconnection of switches andlinks. Here, we consider MIN’s consisting of 2 3 2 switchesonly. A 2 3 2 switch is set up by a control line c; it canbe configured either in a straight mode (for c 5 0), or incross mode (for c 5 1), as has been shown in Fig. 1.

A full-access unique-path MIN is one in which any inputcan be connected to any output by a unique path, e.g.,baseline, omega, or flip.

An N 3 N rearrangeable MIN is one which can realizeany N 3 N permutation in a single pass, e.g., a Benesnetwork.

In this section, we analyze the basic structure of anN 3 N Benes network B(n) to find all possible paths ex-isting in the network between a given source–destination pair.

2.1. Subnets of B(n)

The recursive structure of an N 3 N Benes networkB(n) is shown in Fig. 2; the switches are labelled as Si,j,where i denotes the stage and j represents the position ina stage, 0 # i , 2n 2 1 and 0 # j , N/2. In B(n), the twoB(n 2 1)’s are designated as B0(n 2 1) (the upper one)and B1(n 2 1) (the lower one) respectively, as shown inFig. 2.

FIG. 2. An N 3 N Benes network B(n), n 5 log2N .

switch faults, it develops a graceful degradation routingscheme to make the loss of resources minimal for weightedpermutation requests.

Here, we analyze the behavior of Benes networks in thepresence of multiple switch faults. Earlier studies on fault-tolerance in the presence of switch faults reported somerouting schemes that reconfigure the system in a degradedmode. But here, we propose a fault-tolerant routingscheme by which, in the presence of a class of multipleswitch faults, any N 3 N permutation can be realized intwo passes through the faulty B(n). By our technique, eachsource–destination path is realized in a single pass, i.e., norecirculation through intermediate modules is required,and also the system need not be reconfigured into a de-graded mode; the only degradation in performance resultsin the form of an additional pass. Here, we characterizethe switch fault sets in the presence of which the Benesnetwork is capable of realizing any permutation in twopasses; an algorithm is also presented that checks if a givenfault set belongs to this class. Given any arbitrary permuta-tion P, we assume that the routing of P through the fault-free B(n) is already known. Now, in the presence of switchfaults of this class, another algorithm is developed to findalternative paths for the unsuccessful communications suchthat all the remaining paths can be set up in the secondpass. As a result, in the presence of multiple switch faultsof this class, Benes network retains its rearrangeability intwo passes, instead of a single pass.

This routing technique is applicable to a restricted classof multiple switch faults. Since faults in the first and laststages of the network make some inputs (outputs) totallydisconnected, when reconfiguration of the system into asmaller one is the only way out, we assume that the switchesin the first and last stages are specially built and are fault-

FIG. 1. A 2 3 2 switch S and its two configurations.

192 DAS AND DATTAGUPTA

In B(n), at any stage i, 0 # i , n 2 1, the switches Si,j,0 # j , N/2, can be divided into 2i disjoint subsets SSi,k,0 # k , 2i, such that each subset SSi,k forms the first stageof a B(n 2 i), designated as Bk(n 2 i). Note that SSi,k 5{Si,j, such that k ? 2n2i21 # j , (k 1 1) ? 2n2i21}. The upperand lower outputs of all the switches of SSi,k , are the inputsto the B2k(n 2 i 2 1) (the upper one) and B2k11(n 2 i 21) (the lower one), respectively.

DEFINITION 1. In a B(n), a Bp(n 2 i), 0 # p , 2i,is said to be the subnet of a Bq(n 2 j), 0 # q , 2j andj , i, if and only if Bp(n 2 i) is completely included inBq(n 2 j).

For any i, a pair of subnets of B(n), namely B2k(n 2i 2 1) and B2k11(n 2 i 2 1), are said to be the conjugatesof each other if the inputs of both are the outputs of thesame switch subset SSi,k, and the outputs of both are theinputs of the switch subset SS2n222i,k .

Fig. 3 shows a Benes network B(4) and the subset ofswitches SS1,0, as well as the corresponding conjugate sub-nets B0(2) and B1(2).

2.2. Alternative Paths in B(n)

In a B(n), for any source–destination pair, there existN/2 different paths. Given an arbitrary N 3 N permutationP to be realized in a B(n), we apply the looping algorithm[13] to generate the conflict-free set-up for the switches.We assume that for a given P, each source–destinationpath is represented by a (2n 2 1)-bit tag, referred here asthe routing tag, and is formally defined below:

DEFINITION 2. In a B(n), any source–destination pathis represented by a (2n 2 1)-bit tag x2n22 x2n23 ??? x1 x0,such that at any stage-i, 0 # i , 2n 2 1, if the path traversesvia the upper link of the switch, then x2n222i 5 0; otherwisex2n222i 5 1. The bit string x2n22x2n23 ??? x1 x0 is referred toas the routing tag (R-tag) of the source–destination path.

FIG. 3. A Benes network B(4). The dashed blocks show two conjugate B(2)’s corresponding to a set of switches SS1,0 (shaded ones).

193REARRANGEABILITY IN A FAULTY BENES NETWORK

For example, note that in Fig. 6, the R-tag for the path14 R 1 is 1010001.

Remark. Whatever the routing strategy, the part xn ???x1x0 , of the R-tag of a path is always the destination tagof the path; the additional bits of it are actually determinedby the specific routing technique adopted, to make all theN source–destination paths conflict-free. j

LEMMA 1. For a source–destination path in B(n), if abit x2n222i of the R-tag, 0 # i , n 2 1, is complemented,the modified R-tag will set up the same source–destination connection.

Proof. In B(n), let the R-tag x2n22x2n23 ??? x1x0 representa given source–destination path. At stage-i, 0 # i , n 21, the path passes through a switch belonging to someBk(n 2 i). It is clear that if x2n222i 5 0, the path entersB2k(n 2 i 2 1), otherwise it goes through B2K11(n 2 i 21). If the bit x2n222i is complemented, the path changesfrom B2k(n 2 i 2 1) to B2k11(n 2 i 2 1), or vice versa.Since the part xn21xn22 ??? x1x0 remains unaltered, by thelast n-stages of B(n), which is a unique-path full-accessMIN, the path will ultimately lead to the same destination.This proves the lemma. j

COROLLARY 1. Given any set of conflict-free paths inB(n), if some bit x2n222i of each R-tag, 0 # i , n 2 1, iscomplemented, the modified paths will remain conflict-free.

Proof. Follows directly from Lemma 1. j

These ideas about the redundant paths existing in Benesnetwork, lead us to develop two-pass fault-tolerant routing,which requires no recirculation of data, as well as no recon-figuration of the existing system.

3. TWO-PASS FAULT-TOLERANT ROUTING IN B(n)

First let us present the fault model assumed by us. Threefault models are commonly used to represent all faulty

pass in B(n), under fault-free condition. In the presenceof faults, if P is routed according to these R-tags, somepaths will be unsuccessful. Now, by our proposed routingtechnique, we will modify the original R-tags of the faultypaths to route them all in one additional pass.

DEFINITION 6. For a single fault F 5 {Si, j}, for 0 # i ,n, a subnet Bk(n 2 i) is the cover of F, such that Si, j is aswitch at the first stage of Bk(n 2 i); otherwise, for i . n,Bk(i 2 n 1 2) is the cover of F, such that Si, j is a switchat the last stage of Bk(i 2 n 1 2).

DEFINITION 7. In B(n), given a noncritical fault set F,let MC be the minimal set of subnets, such that the coverof each member of F is either a member of MC or a subnetof a member of MC. Then MC is said to be the minimalcover of F.

By definition, for a given fault set F in B(n), MC isunique.

EXAMPLE 2. Figure 5 shows a B(4) with a fault setF 5 {S2,1, S3,0, S3,1, S3,3, S4,0}.

We find the covers; for S2,1 , B0(2); for S3,0 , B0(1) (asubnet of B0(2)); for S3,1 , B1(1) (a subnet of B0(2)); forS3,3, B3(1); for S4,0, B0(2).

Hence the minimal cover of F is given by MC 5{B0(2), B3(1)}.

DEFINITION 8. In a B(n), for each member Bk(p) ofthe minimal cover MC of a given noncritical fault set F, ifthe conjugate subnet Bk9(p) (or any of its subnets), belongsto MC, replace both Bk(p) and Bk9(p) (or its subnet) inMC, by the subnet Bk/2(p 1 1), to generate a new set ofsubnets SC, which is called the subnet cover for F.

EXAMPLE 3. For the fault set F 5 {S2,1, S3,0, S3,1, S3,3,S4,0} of Example 2, we find the minimal cover, MC 5{B0(2), B3(1)}.

Now, for B0(2) of MC, its conjugate is B1(2), and asubnet of it B3(1) [ MC.

Therefore, the subnet cover of F is SC 5 {B0(3)}.

Remark. Given a fault set F in B(n), for each memberBk(n 2 i) [ SC, the subnet cover of F, the conjugate subnetis fault-free, except the case where SC contains B0(n), thenetwork itself.

THEOREM 1. Two passes are sufficient to realize anypermutation P in B(n), in the presence of a fault set F, ifB0(n) Ó SC, the subnet cover of F.

Proof. Let us consider a permutation P. Let OR(P)denote the set of R-tags for routing P in fault-free B(n).In the first pass, route P through B(n), according to OR(P).Let a noncritical fault set F be present in B(n); SC be thesubnet cover of F, such that B0(n) Ó SC. Now, in thepresence of F, OR(P) will fail to set up some paths of P.Let FOR(P, F) denote the set of R-tags for the faulty paths.

Now, for every Bk(n 2 i) [ SC, let SORj(P, F) representthe subset of FOR(P, F), that includes the R-tags of all the

situations of a MIN. These are the stuck-at fault model,the link fault model, and the switch fault model. The switchfault model, is the strongest of the three. In fact, the effectof control-line stuck-at fault or the link fault can be sub-sumed by the effects of switch fault. Here, we will considerthe switch fault model, assuming that the switch is so de-signed that any physical defection of it causes both theoutputs of the switch to be faulty. In this fault model, afailure makes the switch totally unusable, i.e., both theoutputs of the switch are disconnected from its inputs.

The fault-tolerant routing technique developed here, isdesigned to tolerate the switch faults. In the presence ofswitch faults, it is evident that no N 3 N permutation canbe implemented in a single pass. We are interested infinding out alternative paths in the faulty network, suchthat all the faulty paths are realized in one additional passonly. Here follow some definitions.

DEFINITION 3. The set of faulty switching elements ina Benes network, is defined as the fault set F.

DEFINITION 4. A critical fault set (F) is one, under whicheach possible path for at least one source-destination pairin a B(n), passes through some switch in F, i.e., the full-access property of the network is lost in the presence of F.

DEFINITION 5. Any fault set F, in the presence of which,there exists at least one fault-free path for any source-destination pair in a B(n), is defined as a noncriticalfault set.

Here, we will consider the routing in the presence ofnoncritical fault set in Benes network.

EXAMPLE 1. In B(3) of Fig. 4, F1 5 {S1,0, S3,3} is a criticalfault set, since in the presence of it, no fault-free path existsbetween any input of the set {0, 1, 2, 3} and any output ofthe set {4, 5, 6, 7}.

Similarly, F2 5 {S1,1, S2,1} is a noncritical fault set, sinceit does not destroy the full-access property of B(3).

Remark. The first and last stages of a B(n) are assumedto be fault-free always, since any fault in any of these twostages disconnects some inputs or outputs from the rest ofthe network, such that even recirculation through interme-diate nodes can not help, the only way out is to reconfigurethe system into a smaller one.

We assume that given any N 3 N permutation P, theR-tag for each source–destination path is already com-puted, by looping algorithm [13], that realizes P in a single

FIG. 4. A critical fault set F1 and a noncritical fault set F2 in B(3).


paths passing through the faulty Bk(n 2 i). By definition,SORj(P, F) > SORm(P, F) 5 B, for j ? m.

For each R-tag of SORj(P, F), the bit x2n2i21 is comple-mented. These new R-tags will divert all the paths inSORj(P, F) from Bk(n 2 i) to Bk9(n 2 i). Since Bk9(n 2 i)is fault-free, by Lemma 1 and Corollary 1, all the paths inSORj(P, F) can be realized in the same pass.

The same technique is applied for each subset of FOR(P,F), corresponding to each member of SC, to find out alter-native paths for the faulty ones. By definition, no twomembers of SC are subnets of each other; i.e., all subnetsof SC are disjoint. This implies that all the alternativepaths, given by the modified R-tags of FOR(P, F) will beconflict-free and hence can be realized in the same pass.Therefore, it proves the theorem. j

The above theorem leads us to identify the fault sets, inthe presence of which we can always realize any N 3 Npermutation through a B(n), in two passes. The conditionstated in Theorem 1 is referred to as the condition fortwo-passability.

3.1. Algorithm for Finding the Minimal Cover

For a B(n), given a fault set F 5 {Si, j, 0 # i , 2n 2 1,0 # j , N/2}, the following algorithm finds the minimalcover MC for F. The variable C denotes the cardinalityof MC.

ALGORITHM. Minimal Cover

Input: the fault set FOutput: the minimal cover MC

1. MC :5 B; C :5 0;2. For each Si, j [ F, if i . n 2 1 then Si, j r S2n222i, j

3. Sort all Si, j’s of F in ascending order of i; switcheswith same i are sorted in ascending order of j, and multiple

FIG. 5. A fault set F and its minimal cover in B(4).


occurences of same element are replaced by a single occur-ence. Let F 5 { f0 , f1 , ..., fm} be the sorted list.

4. For q 5 0 to m repeat:4.1. Take fq 5 Si, j; k :5 j/2i;4.2. If Bk(n 2 i) [ MC then goto step 4.54.3. For r 5 1 to C repeat:

4.3.1. Take MC(r) 5 Bu(v);if v . (n 2 i) then if k/2v2(n2i) 5 u then

goto step 4.54.3.2. Next r

4.4. C :5 C 1 1; MC(C) :5 Bk(n 2 i);4.5. Next q

5. Terminate

It is evident from the above algorithm that if |F| 5 p,the worst case time complexity of the algorithm MinimalCover will be O(p2).

3.2. Algorithm to Check Two-Passability Condition

For a B(n), given a fault set F, the algorithm MinimalCover presented in the preceding subsection will constructthe set MC 5 {Bu(v), 1 # v # n and 0 # u , 2n2v}. Herewe present an algorithm which will construct the subnetcover SC from MC and test whether the SC satisfies thecondition for two passability, as has been stated in Theo-rem 1. The boolean variable success is true if the conditionis satisfied; otherwise it is false. C denotes the cardinalityof MC.

ALGORITHM. Two-Passability

Input: minimal cover MCOutput: subnet cover SC and the boolean variable success

1. success :5 0; SC r MC2. If B0(n) [ SC then terminate3. Sort all Bu(v)’s of SC in descending order of v.

ALGORITHM. Two-Pass Routing

Inputs: (i) array SC(S), each element SC(q) is a subnetBk(n 2 i), 1 # q # S, and(ii) array FOR(P, F)(m), each element is of theform Rp 5 x2n22x2n23 ??? x0 , 1 # p # m.

Output: array FOR(P, F).

1. For p 5 1 to m repeat:1.1. For q 5 1 to S repeat:

1.1.1. Take SC(q) 5 Bk(n 2 i);If Rp/22n222(i21) 5 k then complement thebit x2n2i21 of Rp and go to 1.2

1.1.2. next q1.2. Next p

2. Terminate

The time complexity of the above algorithm is O(S.m),where S and m are the cardinalities of SC and FOR(P,F) respectively.

EXAMPLE 4. Let F 5 {S2,1, S3,0, S3,1, S3,4, S3,7, S4,0} in aB(4). The minimal cover of F is given by MC 5 {B0(2),B4(1), B5(1),}.

The subnet cover is SC 5 {B0(2), B4(1), B7(1)}. SC satis-fies the condition for two-passability. The routing of a per-mutation P: (3 7 6 2 9 10 14 8 11 13 15 12 4 0 1 5) in twopasses through the faulty B(4), is shown in Figs. 6 and 7.

3.4. Fault Coverage

Now we are capable of routing any arbitrary permutationin two passes through a Benes network B(n) in the pres-ence of all switch fault sets satisfying the condition of two-passability. It is obviously true that the fault set satisfyingthe condition for two-passability is a restricted class ofmultiple switch faults. But we find here that it covers alarge number of all possible single faults, double faults,and triple faults as well, and also, the coverage increaseswith increase in n.

For our analysis purpose, we will assume that all theswitches in the first and last stages of a B(n) are robust

4. For x 5 1 to C repeat:4.1. Take SC(x) 5 Bk( j);4.2. For y 5 x 1 1 to C repeat:

4.2.1. Take SC(y) 5 Bu(v);4.2.2. If Bu(v) is a subnet of Bk( j) or its conjugate

thenreplace Bk( j) by Bk/2( j 1 1) and removeBu(v) in SC;C 5 C 2 1;if j 5 n terminate

4.2.3. Next y4.3. Next x

5. success :5 1 and terminate

For a fault set F with cardinality p, the maximum cardi-nality possible for its minimal cover MC is also p. It is easyto see that the worst case time complexity of the algorithmis of O(p2).

3.3. Algorithm for Two-Pass Fault-Tolerant Routing

Here, we assume that the given fault set F, present inB(n), is processed by the algorithm Minimal Cover, whichoutputs MC, the minimal cover of F. Next, Algorithm 3.2finds that SC satisfies the two-passability condition; i.e.,any arbitrary N 3 N permutation P is routable in twopasses through the faulty B(n).

Let us consider an N 3 N permutation P. Let OR(P),the set of R-tags to route P in fault-free B(n), be known.P is routed by OR(P) in the faulty B(n). Some paths willfail. Let FOR(P, F) be the set of R-tags of the faultypaths. Our algorithm will utilize SC to modify the R-tagsof FOR(P, F), so that the new R-tags can route all thefaulty paths in the second pass. In order to achieve that,we are to identify the set of faulty paths, that pass througheach subnet belonging to SC. A path with R-tag R 5x2n22x2n23 ??? x0, passes through a subnet Bk(n 2 i), 1 #i , n, if and only if R/22n222(i21) 5 k. The R-tags are thenmodified accordingly, as has been described in the proofof Theorem 1.

FIG. 6. Routing of permutation P in the first pass.


enough, so that they always remain fault-free. In fact, afailure in a switch of the first (last) stage, essentially discon-nects two inputs (outputs) from the rest of the network,and there is no way out except reconfiguration of theN 3 N Benes network into a (N 2 2) 3 (N 2 2) network.It is a critical fault, where recirculation also fails to connectthose detached inputs (outputs). As we are concerned withfault-tolerant routing without any reconfiguration and re-circulation as well, the assumption about the fault-free firstand last stages is essential here. With this assumption,Table I shows the fault-coverages for different values ofn, in cases with different number of switch faults. It isevident that our fault-tolerant routing technique will bevery much useful for large systems.

4. CONCLUSION

In this paper, fault-tolerant routing in Benes networksin the presence of multiple switch faults is considered. Inthe presence of a single noncritical fault, two passes arealways sufficient to route any permutation through thefaulty network. In the presence of multiple faults, a condi-tion for two-passability is presented here. When the faultset present satisfies the condition for two-passability, afault-tolerant routing is developed here that routes anyarbitrary permutation in two passes, without any recircula-tion or reconfiguration of the system. Assuming the firstand last stages of the network to be fault-free, it has been

TABLE 1

Fault-coverage (%) for

n 5 3 n 5 4 n 5 5

Single faults 100 100 100Double faults 79 93 97Triple faults 73 89 92

FIG. 7. Routing of faulty paths of P in the second pass.


found that this fault set covers all possible single faults,more than 90% of all possible double faults, and more than85% of all possible triple faults, for N 5 16 and 32. Theratio increases with N. Therefore, this routing techniquewill be very useful for large systems with high values ofN, using the Benes network for interconnection. Moreover,very fast and simple procedures also have been developedin [15], for detecting and locating multiple faults of thisclass, for implementing our fault-tolerant routing tech-nique in Benes network.

REFERENCES

1. Reddy, S. M., and Kumar, V. P. On fault-tolerant multistage intercon-nection networks. Proc. 1984 International Conf. on Parallel Pro-cessing (ICPP), 1984, pp. 155–164.

2. Varma, A., and Raghavendra, C. S. Realization of permutations ongeneralized Indra networks. Proc. 1985 International Conf. on ParallelProcessing (ICPP), 1985, pp. 328–333.

3. Raghavendra, C. S., and Varma, A. Fault-tolerant multiprocessorswith redundant path interconnection networks. IEEE Trans. Comput.C-35 (Apr. 1986), 307–316.

4. Kumar, V. P., and Reddy, S. M. Augmented shuffle-exchangemultistage interconnection networks. Computer (June 1987), 30–40.

5. Varma, A., and Raghavendra, C. S. Fault-tolerant routing inmultistage interconnection networks. IEEE Trans. Comput. 38 (Mar.1989), 385–393.

6. Nelson, V. P. Fault-tolerance in reconfigurable multiprocessor sys-tems. Proc. Compsac’80. 1980, pp. 372–380.

7. Pradhan, D. K. On a class of fault-tolerant multiprocessor networkarchitecture. Proc. 3rd International Conf. on Distributed ComputingSystems. 1982, pp. 302–311.

8. Agrawal, D. P. Testing and fault tolerance of multistage interconnec-tion networks. Computer (Apr. 1982), 41–53.

9. Huang, S. T., and Tripathy, S. K. Finite state model and computabilitytheory: New analysis tools for permutation networks. IEEE Trans.Comput. C-35 (July 1986), 591–601.

10. Huang, S. T., and Tung, C. H. On fault-tolerant routing of Benesnetwork, J. Inform. Sci. Eng. 4 (July 1988), 1–13.

11. Nassimi, D. A fault-tolerant routing algorithm for BPC permutationson multistage interconnection networks. Proc. 1989 InternationalConf. on Parallel Processing (ICPP). 1989, pp. I-278–I-287.

14. Yeh, Y., and Feng, T. Y. Fault-tolerant routing on a class of re-arrangeable networks. Proc. 1991 International Conf. on Parallel Pro-cessing (ICPP). 1991, pp. I-305–I-312.

15. Das, N., and Dattagupta, J. A fault location technique and alternaterouting in Benes network. Proc. 4th Asian Test Symposium (ATS),1995, pp. 71–77.

12. Feng, T. Y. Fault tolerance in rearrangeable networks. Proc. of IEEERegion 10 Conf. on Computer and Communication Systems. 1990,pp. 399–403.

13. Opferman, D. C., and Tsao-Wu, N. T. On a class of rearrangeableswitching networks. Bell System Technical J. 50 (May–June 1971),1579–1618.


Received April 14, 1993; accepted June 2, 1995

Documents

Two-Pass Rearrangeability in Faulty Benes Networks