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1 Permutation Admissibility in Shue-Exchange Networks with Arbitrary Number of Stages Nabanita Das Bhargab B. Bhattacharya Rekha Menon Indian Statistical Institute Calcutta, India Sergei L. Bezrukov Dept. of Math. & CS University of Paderborn 9 Paderborn Germany sb@uni-paderborn.de Abstract The set of input-output permutations that are routable through a multistage interconnection network without any conict (known as the admissible set), plays an important role in determining the capability of the network. Recent works on the permutation admissibity problem of shueexchange networks (SEN) of size N N, deal with (n + k) stages, where n = log N, and k denotes the number of extra stages. For k = or, O(Nn) algorithms exist to check if any permutation is admissible, but for k, a polynomial time solution is not yet known. The more general problem of nding the minimum number (m) of shueexchange stages required to realize an arbitrary permutation, m n?, is also an open problem. In this paper, we present an O(Nn) algorithm that checks whether a given permutation P is admissible in an m stage SEN, m n, and determines in O(Nnlogn) time the minimum number of stages m of shue-exchange, required to realize P. Thus, a single-stage shue-exchange network will be able to realize such a permutation with m passes, by recirculating all the paths m times through a single-stage, i.e., with minimum transmission delay, which, otherwise cannot be achieved with a xed-stage SEN. Furthermore, we present a necessary condition for permutation admissibility in an m stage SEN, where n < m n?. Introduction An n-stage multistage interconnection network (MIN) of size N N, where N = n, for n, is a minimal full-access unique-path structure, since it provides exactly one path from any source to any destination, e.g., omega, baseline, cube and reverse-baseline [-]. However, they are inherently blockinn nature, i.e., to connect more than one source-destination pairs simultaneously, a single link may be demanded by two or more paths, causing conicts. To overcome this problem, as well as to provide some fault-tolerance, k-extra-stage MIN's (i.e., with (n + k) stages), were proposed, which provide k paths between any inputoutput pair [-8]. An N N permutation from the set of N inputs (; ; :::; N? ) to the set of N outputs (; ; :::; N? ) is said to be admissible, if N conict-free paths (one for each input-output pair) can be set up simultaneously in the MIN. In [], it has been reported that a permutation is admissible in a k-extra-stage SEN if and only if the conict graph is k -colorable, and since the c-coloring (c > ) problem in graphs is NPcomplete, the complexity of determining permutation admissibility in a k-extra-stage MIN, for k, is not known. For k = or, O(Nn) algorithms were reported for checking permutation admissibility, but in general, the problem remains open [,]. In this paper, we address the following more general problem: given any N N permutation P, check whether P is admissible in an m stage SEN, for m n. Our proposed algorithm ascertains this in O(Nn) time, and also nds the minimum number of stages (m) of shue-exchange, for which P is admissible, in O(N nlogn) time. Our technique will be especially applicable to a network implemented with a single-stage shue-exchange, where permutations are realized by recirculation. In other words, if a permutation P is admissible in m stages, m n, recirculating the paths m times through a single-stage shue-exchange network, one can realize P, with minimum transmission delay. This is not possible otherwise, with a multistage shue-exchange network having a xed number of stages. We explore some new results on admissibility, for stages m, n < m n?. Since a (n? )-stage shue-exchange network is rearrangeable, one can realize any arbitrary permutation in (n? ) stages of shue-exchange []. We formulate a necessary condition that a permutation P must satisfy for being admissible on an m stage SEN, for n < m n?. This gives a lower bound on the number of recirculation stages of shue-exchange required to make P admissible. The paper is organized as follows. In Section, we analyze some properties of m-stage N N SEN, m n?, and characterize the set of admissible permutations for any m. In Section, we propose an algorithm to solve the permutation admissibility problem in an m-stage SEN, m n, and show how it can be applied to nd the lower bound of m required to make any arbitrary permutation admissible. Conclusions and further discussons appear in Section

2 . Input-Output Groups and Permutation Admissibility An N N multistage interconnection network, in general, consists of n stages, which is essentially a minimal structure that provides full-accessability with exactly one path for any input-output pair. Now, to provide some fault-tolerance, as well as to enhance the set of permutations admissible on the MIN, k-extra stages are added to it. Since, in a SEN, the connection patterns between adjacent stages are always the same, the sets of permutations admissible in an m-stage SEN, for m n? exhibit many elegant properties.. Motivating Examples Let us consider an (8 8) m-stage SEN, and dierent admissible permutations for various m, m. Example : The permutation P : SEN with m =. Fig. shows the paths for P on a single-stage SEN. It is to be noted that though P is admissible in a single-stage SEN, it is not admissible in SEN's with, or -stages. Figure : Paths for permutation P Figure : Paths for permutation P Remark The identity permutation is always admissible in an n-stage N N SEN. Example : The permutation P :, is admissible in a SEN with m =. Fig. shows the paths on a -stage SEN. Here stages are essential to make P admissible. Figure : Paths for permutation P Example : The permutation P : SEN with m =. Fig. shows the paths on a -stage SEN. Note that P is admissible neither in a single-stage SEN, nor in a -stage SEN. Example : The permutation P : SEN with m =. Fig. shows the paths on a -stage SEN. Note that P is the identity permutation, and minimum (= n) stages are necessary to make it admissible. Figure : Paths for permutation P Example : The permutation P : SEN with m =. Fig. shows the paths on a - stage SEN. Note that minimum stages are required to make P admissible. Therefore, it is clear from the above examples that

3 Figure : Paths for permutation P instead of using a xed-stage SEN, we can use a singlestage SEN, and recirculate the paths, for routing any arbitrary permutation P with minimum delay, provided m, the minimum number of stages of SEN, required to make P admissible, is known.. Group Structures Here, we will use the idea of group structures for analyzing the relations between input (output) groups and permutation admissibilty of a SEN. The concept of input (output) group structures for MIN's was introduced earlier in [9, ]. For completeness, we give a brief description of the input (output) group structures for SEN. In an m stage N N SEN, m n?, inputs (or outputs) are labeled as: ; ; :::; N? respectively, from top to bottom, and each is represented uniquely by an n bit binary string; the stages are labelled as: ; ; ::::; (m? ), from the input side towards the output side; the output links of each stage are labeled as: ; ; ::::; (N? ), from top to bottom. A SEN with N = 8 and m = is shown in Fig.. i.e., n?j < p, and r = or, for n < r n? j. Here gi (j; p) will be represented as ( j x n?j? :::x ), where j is a string of j 's, and (x n?j? :::x ) is the binary string representing p. Example : For a SEN an input group (; ) = ( ) = (; ; ; ) = (; ; ; ). Denition Similarly, for an N N SEN, an output group g o (j; p) denes a group of j outputs at level j, j n, given by x n? x n? :::x j j? j? :::, where (x n? :::x j ) is the binary representation of p i.e., n?j < p, and r = or, for < r j?, and will be represented by (x n? x n? :::x j j ). Example : For an SEN an output group g o (; ) = ( ) = (; ; ; ; = (; ; ; ). For an 88 SEN, the input and output group structures are shown in Fig.. a) b) (,,,,,,, ) (,) (,) (,,, ) (,,, ) (,) (,) (,) (,) (, ) (, ) (, ) (, ) (,,,,,,, ) Stage: Figure : A -stage 8 8 SEN with link labels Denition For an N N SEN, an input group (j; p) at level j, j n, denes a group of j inputs given by n? n? :::: n?j x n?j? x n?j? :::x x, where x n?j? ::::x x is the binary representation of p, (,) (,) (,,, ) (,,, ) (,) (,) (,) (,) (, ) (, ) (, ) (, ) Figure : (a) Input and (b) output groups of an 8 8 SEN Next we consider the permutation admissibility problem in an N N SEN, with m stages, for m n?. Denition For an N N SEN, with m stages, for m n?, given any N N permutation P, we construct an N (n+m) binary matrix M, where each

4 row x n? x n? :::x x :::: m represents one inputoutput path x! y; x n? x n? :::x x is the input x, and the bits :::: m, are determined by the output y, and the number of stages m; and a window W j ; j < m, is the set of n consecutive columns of M, x n?j? :::x x :::: j, and each row of W j is the link, the path follows at the output of stage j. The matrix M is dened as the path matrix of P, in an m-stage SEN. Fig. 8 shows the path matrix M for the permutation P, of Example, and the corresponding windows for a -stage 8 8 SEN. But this set is the output group g o (m; p), where p = (x n?m? :::x x ). Now it is evident that for any input in the group (m; p) = ( m x n?m? :::x x ), the reachable set will be the same group g o (m; p), for p < n?m : Hence the proof. Corollary In an m stage N N SEN, < m < n, an output y = y n? y n? :::y y is reachable from an input x = x n? x n? :::x x, if and only if y n?j = x n?m?j, for 8j; j n? m. Proof : Follows directly from Lemma. W W Denition In an m-stage N N SEN, < m n, if an output y is reachable from an input x, the path x! y is said to be a realizable path. M = x x x * = y * = y Figure 8: The path matrix for permutation P in a -stage 8 8 SEN Remark A permutation P is admissible in an m stage N N SEN, if and only if there exists a path matrix M, where all the rows are distinct in each window W j, j < m. We study the two cases separately: one for < m n and the other with n < m n?.. m-stage N N SEN, m n In this case, the network is not a full-access one, except when m = n. Starting from any input one can reach only a particular set of m ( N) outputs. However, given any input-output pair, if a path exists, it is always unique. Denition For an m-stage N N SEN, < m n, the set of outputs which can be reached from an input x is referred to as the reachable set of the input x. Lemma In an m-stage N N SEN, < m n, for any input x (m; p), the reachable set is the output group g o (m; p), where n?m < p. Proof : Let us consider an N N SEN with m stages, < m n. Now starting from any input, say x = x n? x n? :::x x, at each stage it may follow either a shue or a shue-exchange. Therefore, after the rst stage, x may reach any output represented by : (x n? :::x x ), where = x n? or x n?, i.e., = or. Now after m successive stages x may reach the set of outputs represented as (y = x n?m? :::x x m ). Remark In an m-stage N N SEN, m n, if an output y = y n? y n? :::y y is reachable from an input x = x n? x n? :::x x, in the path matrix M, the path x! y is represented by the string of (n+m) bits: x! y : x n? x n? :::x x y m? y m? :::y y, where the window (x n?j? x n?j? :::x x y m? y m? :::y m?j? ), is the binary representation of the link label, the path follows at any stage j, j < m. By Corollary, we can represent the same path as x! y : x n? x n? :::x n?m y n? :::y y. The above result conforms with similar results established in [,], for m = n only. Example 8 : In a -stage SEN, a path! is shown in Fig. 9a. The path is represented as ()! () :, the four bits from the left (right) represents the input, (output ), and the two overlap in the middle bit. The links followed by the path in stages ; ; are (), (), and () respectively, as given by the respective windows, are shown in Fig. 9b. Therefore, for an m-stage N N SEN, m n, given any permutation P, if all input-output paths are realizable, the set of N paths is represented by a N (n + m) path matrix M, where each row stands for one input-output path, as explained earlier. Now P will be admissible in the SEN, if and only if all the rows in every window W j, j < m, are distinct. Denition In an N N SEN, two inputs (outputs) x and x are said to be covered by an input (output) group (o) (j; p) if x ; x belong to the same input (output) group at level j, j n, but in dierent groups at level (j? ). Example 9 : For a SEN, two inputs 8() and () are covered by (; ), i.e., the group ( ). Remark In an N N SEN, two inputs (outputs) x and x covered by an input (output) group =o (j; p) must dier in bit x n?j (x j ).

5 a) conictinn stage (j? ), where they both need the same link (x n?j? ::x n?m? :::x x y m? ::y m?j ): They may also conict in a following stage (j + k? ), k < m? j, if ym?j?r = y m?j?r; 8r; r k: Only if: Let y and y belong to two dierent output groups at level (m? j), say g o (m? j; p), and g o (m? j; p ); p = p : Let p = (y n? :::y m?j ), and p = (y n? :::y m?j ); they will dier at least in one bit position. The two paths will never conict at any stage-k, for k < m. Hence the proof. Theorem In an m-stage N N SEN ( m n), a permutation P is admissible if and only if i) each input is mapped to a reachable output, and ii) the j inputs of an input group at level j, j m are mapped to outputs such that exactly one of them belongs to a unique output group at level (m? j). Proof: Follows directly from Lemma. b) W W W Figure 9: (a) The path! in a -stage SEN, and (b) the windows in each stage Lemma In an m-stage N N SEN, ( m n), two realizable paths x! y and x! y will be conictinf and only if for x; x covered by an input group at level j; j m, y; y belong to the same group at level (m? j): Proof : Let x = x n? x n? :::x x. Then by Remark, x = x n? ::x x n?j+ n?jx n?j? :::x x, since they are covered by an input group at level j; j < m. Now since y is reachable for x, by Corollary, y = x n?m? x n?m? :::x x y m? y m? :::y y. Similarly, y = x n?m? x n?m? :::x x y m? y m? :::y y : Now the two paths can be represented as: x! y : x n? ::x n?m? :::x x y m? y m? :::y y, and x! y :x n? ::x x n?j+ n?jx n?j? ::x n?m? ::x y m? ::y : It is evident that these two paths will never conict at any stage k, for k j?, since x n?j = x n?j. If part: If the corresponding outputs y and y are in the same output group at level (m? j), y = x n?m? x n?m? :::x x y m? ::y m?j y m?j? :::y y. The two paths under consideration become: x! y : x n? ::x n?m? :::x x y m? ::y m?j :::y y, and x! y : x n?::x x n?j+ n?jx n?j? ::x n?m? :::x y m? ::y m?j y m?j? :::y. Now it is obvious that these two paths will be always Denition For an N N SEN, the i-admissible set of permutations i is dened as the set of all permutations admissible in i stages, where, i n. We use a notation to represent input-output mapping rules as follows: The mapping f (j; p )j (j; p )j:::j (j; p r )g! fg o (m? j; q ); g o (m? j; q ); :::; g o (m? j; q j )g, implies that each element of any input group (j; p u ); u r, is mapped to an output of a unique group g o (m? j; q v ); v j. Example : In an 8 8 SEN, where n =, let us compute the i-admissible sets of permutations i, for i. From Theorem : For any permutation P : for (; p): f(; )g! f(); ()g, f(; )g! f(); ()g, f(; )g! f(); ()g, and f(; )g! f(); ()g, Hence, the permutation P :. For any permutation P : for (; p): f(; )j(; )g! f(; ); (; )g, f(; )j(; )g! f(; ); (; )g, for (; p):f(; ; ; )g! f(); (); (); ()g, f(; ; ; )g! f(); (); (); ()g e.g., the permutation P :. Note that P = : For any permutation P : for (; p): f(; )j(; )j(; )j(; )g! f(; ; ; ); (; ; ; )g for (; p): f(; ; ; )j(; ; ; )g!

6 f(; ); (; ); (; ); (:)g, for (; p): f(; ; ; ; ; ; ; )g! f(); (); (); (); (); (); (); ()g e.g., the permutation P :. Note that P ; P = : The path matrix for P, in a -stage SEN is shown in Fig. 8. Note that in each case, all the rows in every possible window are distinct, proving that all the paths are conict-free, i.e., the permutations are admissible. Corollary In an m-stage N N SEN, all the i- admissible sets of permutations are disjoint. Proof : Follows directly from Theorem. In the next section, an O(Nn) algorithm has been proposed to determine whether or not an arbitrary N N permutation P is admissible in an m-stage N N SEN ( m n). Formerly, an algorithm with the same complexity was reported only for an n-stage SEN [].. m-stage N N SEN, n m n? In this case, the network is full-access, i.e., each input can reach any output, but instead of a uniquepath there exist m?n paths between any pair of input-output. Now we may represent an inputoutput path x! y, as a sequence of (n + m) bits, (x n? x n? :::x x ::: n?m y n? ::y y ), where (x n? x n? :::x x ) is the binary representation of the input x, and (y n? ::y y ) represents the output y; the (m? n) bits between the two, represented as ( ::: m?n ) are arbitrary. In fact, each possible combination of these bits, represents a path from x to y. Therefore, a given permutation P, is admissible in an m stage N N SEN, n m n?, if and only if there exists an assignment for all the bits ( ::: m?n ), such that all rows in each window W j, j < m, of the path matrix M are distinct. Example : Fig. shows a possible path matrix for the permutation P :, on a -stage 8 8 SEN. Note that all the rows in each window W j, j, are distinct. Theorem If an N N permutation P is admissible in an m-stage N N SEN, n m n?, then in P, the j inputs of each input group at level j, m?n < j n are mapped to outputs such that exactly m?n of them belong to the same output group at level (m? j). Proof : Let us consider a permutation P, such that there is at least one input group at level j, m? n < j n, of which more than m?n inputs are mapped to outputs belonging to the same output M = W W W W x x x * * y y y Figure : Path matrix for permutation P group at level (m? j). Now let us consider the window W j?, which consists of the bit string x n?j? :::x :: m?n y n? :::y m?j for each path of P. For any input group at level j, all the inputs consists of an identical bit string x n?j? :::x. Now if more than m?n inputs of the group are mapped to the outputs in the same output group at level (m? j), they will have identical bit string y n? :::y m?j. This will result more than m?n rows of W j? with identical bit strings for the part x n?j? ::x, as well as for y n? ::y m?j. Now, by using the bits ; :::; m?n, we can make at most m?n rows dierent. Hence for P, in W j?, there will be more than one identical rows for any possible assignments of ::: m?n. Hence the proof. Algorithms First, we present an algorithm that will decide if a given permutation P is admissible in an m-stage N N SEN, m n. The permutation is given as an array out of length N, such that out(i) stores the output corresponding to the input i, i N?. Algorithm : Admissibility Check Input: n, m, out(n) Output: success Step : success := Step : for x = : : : N?, if xmod( n?m ) = out(x)div m then terminate (* to check reachability *); Step : for i = : : : m for p = : : : ( i?? ) for x = p: n?i+ : : : (p: n?i+ + n?i? ) if out(x) and out(x + n?i ) rst dier in bit x m?i, when compared from MSB, then if out(x) > out(x + n?i ), exchange them in array out;

7 Step : next x else terminate; next p next i success:=, terminate; The above algorithm is of time complexity O(N n). Now given any permutation P, admissible on an i- stage SEN, i n, to nd the minimum number of stages m necessary to make P admissible, we can perform a binary search over the interval to i by invoking the above algorithm at most logn times. Therefore, m can be determined in O(N nlogn) time. Conclusion We have shown that the permutation admissibility problem for an m-stage shue-exchange network can be solved for m n, in O(Nn) time, which was formerly solved only for m = n and (n + ) [,]. This algorithm also helps us to nd the minimum number of stages m; m n of a SEN, necessary to make such a permutation admissible, with a complexity O(N nlogn). Therefore, by using a single-stage SEN, we can recirculate the paths m times through the network, to realize such a permutation P. This technique enables us to route the permutation through the network with minimum transmission delay. For n < m n?, we establish a necessary condition that a permutation P must satisfy for being admissible in an m stage SEN. It gives a lower bound on the number of stages required for making P admissible. References [] X. Shen, M. Xu, and X. Wang, \An optimal algorithm for permutation admissibility to multistage interconnection networks," IEEE Trans. Computers, vol., no., pp. -8, Apr. 99. [] X. Shen, \An optimal O(NlgN) algorithm for permutation admissibility to extra-stage cube-type networks," IEEE Trans. Computers, vol., no. 9, pp. -9, Sep. 99. [] A. Abdendher and T. Feng, \On rearrangeability of Omega-Omega networks," Proc. 99 Int'l Conf. Parallel Processing, pp. I-9-, 99. [] C. L. Wu and T. Feng, \On a class of multistage interconnection networks," IEEE Trans. Computers, vol. C-9, no. 8, pp. 9-, Aug. 98. [] H. S. Stone, \ Parallel processing with the perfect shuf- e," IEEE Trans. Computers, vol. C-, no., pp.-, Feb. 9. [] C. L. Wu and T. Feng, \The universality of the shueexchange network,"ieee Trans. Computers, vol., no. 8, pp. -, May 98. [] C. S. Raghavendra and A. Varma, \Fault-tolerant multiprocessors with redundant path networks," IEEE Trans. Computers, vol. C-, no., pp. -, April 98. [8] N. Das, B. B. Bhattacharya and J. Dattagupta, \Isomorphism of conict graphs in multistage interconnection networks and its application to optimal routing," IEEE Trans. Computers, vol., no., pp. -, June 99. [9] N. Das, B. B. Bhattacharya and J. Dattagupta, \Hierarchical classication of permutation classes in multistage interconnection networks," IEEE Trans. Computers, vol., no., pp. 9-, Dec. 99.