Channel graphs of bit permutation networks
Li-Da Tong
1, Frank K. Hwang
2, Gerard J. Chang
∗;1Department of Applied Mathematics, National Chiao Tung University, 1001 TA Hsueh Road, Hsinchu 30050, Taiwan
Accepted April 2000
Abstract
Channel graphs have been widely used in the study of blocking networks. In this paper, we show that a bit permutation network has a unique channel graph if and only if it is connected, and two connected bit permutation networks are isomorphic if and only if their channel graphs are isomorphic. c 2001 Elsevier Science B.V. All rights reserved.
Keywords: Multistage interconnection network; Switching network; Channel graph
1. Introduction
Recently, Changet al. [2] de:ned a class of (m + 1)-stage d-nary bit permuta-tion networks which are multistage interconnecpermuta-tion networks using only d × d square switches. Such a network has N=d (N = dn+1 is the network size) switches in a stage
which are labeled by d-nary sequences of length n, and there is a link from switch x at stage i to switch y at stage i + 1 if the bits of y can be obtained from the bits of x, except one, by a permutation dependingonly on i. More precisely, the network has vertices (xn; xn−1; : : : ; x1)i in stage i, Si, where xj∈ {0; 1; : : : ; d−1} for 16j6n and
06i6m. And, there exist m permutations f1; f2; : : : ; fm on {0; 1; : : : ; n} with fi(0) = 0
for 16i6m such that (xn; xn−1; : : : ; x1)i−1 is adjacent to (xfi(n); xfi(n−1); : : : ; xfi(1))i,
where x0∈ {0; 1; : : : ; d−1} and 16i6m. We use Nd(n; f1; f2; : : : ; fm) to denote the
network de:ned above. For any vertex (xn; xn−1; : : : ; x1)i in the network, xj is called
the jth coordinate of the vertex. Note that the popular class of binary (n + 1)th-stage networks includingOmega, baseline, banyan, etc., their inverses and their k-extra-stage extensions are all bit permutation networks. The above-mentioned class of (n +
1)th-∗Correspondingauthor. Tel.: +886-3-573-1945; fax: +886-3-542-2682.
E-mail address: [email protected] (G.J. Chang).
1Supported in part by the National Science Council under grant NSC86-2115-M009-002. 2Supported in part by the National Science Council under grant NSC87-2119-M009-002.
0304-3975/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved.
Fig. 1. A bit permutation network and its channel graph.
stage networks are known [1, 7] to be topologically equivalent and will be referred to as the Omega-equivalent class. Fig. 1(a) shows a tertiary four-stage shuJe exchange network N3(2; f1; f2; f3) with f1= f2= f3= the cyclic permutation (0 2 1); i.e., switch
(x2; x1)i−1 is adjacent to switch (x1; x0)i for 16i63.
Switches in the 0th (respectively, mth) stage of an (m+1)th-stage network are called input (respectively, output) switches. For a switch xs in stage s and a switch xt in stage
t¿s, an xs− xt channel-path is a path (xs; xs+1; : : : ; xt), where each xj is in stage j for
s6j6t. The channel graph CG(I; O) between an input switch I and an output switch O is the union of all channel-paths in the network connectingthe pair. If CG(I; O) is independent of I and O, then it is called the channel graph of the network
Channel graphs have been widely used in the study of blocking networks (see [3] for a survey), and recently also used by Lea and Shyy [4–6] to determine the strict and rearrangeable nonblockingness of a network. In [5, 6], results were established for the k-extra-stage inverse banyan network, but implied to hold for all k-extra-stage Omega-equivalent networks. Hwanget al. [4] pointed out that the equivalence amongthe Omega-equivalent networks is not preserved under the extra-stage addition. However, the results of Lea and Shyy [5, 6] actually depend only on the channel graph of the network. Thus the question arises as to whether there exist nonisomorphic networks havingisomorphic channel graphs; of course, the prior condition is that the channel graph of a k-extra-stage network always exists. In this paper we give an aKrmative answer to the second question for the larger bit permutation class, and a negative answer to the :rst question.
2. The main results
A sequence (k1; k2; : : : ; km) is canonical if {k1; k2; : : : ; km} = {1; 2; : : : ; r} for some
pos-itive integer r6n, and for k ∈ {1; 2; : : : ; r−1}, the :rst appearance of k always precedes that of k + 1 in the sequence. A canonical sequence (k1; k2; : : : ; km) induces a bit
per-mutation network, denoted by Nd(n; k1; k2; : : : ; km), which is Nd(n; f1; f2; : : : ; fm) with fi
beingthe permutation (0 ki) for 16i6m. Note that in this case, a vertex in stage i − 1
is adjacent to a vertex in stage i if and only if all of their coordinates are identical ex-cept possibly the kith one. It was shown [2] that (m+1)th-stage d-nary bit permutation
networks can be characterized by canonical sequences.
Theorem 1 (Changet al. [2]). Any Nd(n; f1; f2; : : : ; fm) is isomorphic to Nd(n; k1; k2;
: : : ; km) for some canonical sequence over {1; 2; : : : ; n}. Moreover; two (m+1)th-stage
d-nary bit permutation networks are isomorphic if and only if their corresponding canonical sequences are the same.
Lemma 2. Suppose x = (xn; xn−1; : : : ; x1)s is a switch in stage s and y = (yn; yn−1; : : : ;
y1)t a switch in stage t¿s; and D is the set of subscripts i where xi and yi di6er.
Then; there is an x–y channel-path in Nd(n; k1; k2; : : : ; km) if and only if D ⊆ {ks+1;
ks+2; : : : ; kt}.
Proof. Immediate from the de:nition of Nd(n; k1; k2; : : : ; km).
Lemma 3. Nd(n; k1; k2; : : : ; km) is connected if and only if {k1; k2; : : : ; km} = {1; 2; : : : ; n}. Proof. Suppose the network is connected. Then there is a path from (0; 0; : : : ; 0)0 to
(1; 1; : : : ; 1)m. Since a move from a vertex to a neighbor can only change on the kith
coordinate, any j in {1; 2; : : : ; n} is some ki, i.e., {k1; k2; : : : ; km} = {1; 2; : : : ; n}.
Conversely, suppose {k1; k2; : : : ; km} = {1; 2; : : : ; n}. For any two switches x and y,
let x (respectively, y) be the input (respectively, output) switch whose coordinates
are the same as x (respectively, y). By Lemma 2, there exist x–x, x–y and y–y
channel-paths. Thus, the network is connected.
Note that for {k1; k2; : : : ; km} = {1; 2; : : : ; n}, {1; 2; : : : ; n} is the disjoint union of
sets {k1; k2; : : : ; ki} ∩ {ki+1; ki+2; : : : ; km}, {1; 2; : : : ; n} − {k1; k2; : : : ; ki}, {1; 2; : : : ; n} −
{ki+1; ki+2; : : : ; km}.
Lemma 4. Suppose x is an input switch and y an output switch in a connected
network Nd(n; k1; k2; : : : ; km). Then; the vertex set of the channel graph CG(x; y) is
06i6m{z ∈ Si: zj= xj for j ∈ {1; 2; : : : ; n} − {k1; k2; : : : ; ki} and zj= yj for j ∈ {1; 2;
: : : ; n} − {ki+1; ki+2; : : : ; km}}.
Proof. The lemma follows from Lemma 2 and the fact that z is a vertex of CG(x; y)
Theorem 5. The channel graph of a bit permutation network exists if and only if it
is connected.
Proof. The existence of the channel graph of a network necessarily implies the
con-nectivity of the network. Now, suppose CG(x; y) and CG(x; y) are two channel
graphs of a connected bit permutation network Nd(n; k1; k2; : : : ; km). De:ne a function
f : V (CG(x; y)) → V (CG(x; y)) accordingto Lemma 4 by f(z) = z, where z; z∈ S i and z j= zj if j ∈ {k1; k2; : : : ; ki} ∩ {ki+1; ki+2; : : : ; km}; x j if j ∈ {1; 2; : : : ; n} − {k1; k2; : : : ; ki}; y j if j ∈ {1; 2; : : : ; n} − {ki+1; ki+2; : : : ; km}:
For any two distinct vertices u and v in V (CG(x; y))∩Si, there exists some j ∈ {k1; k2;
: : : ; ki} ∩ {ki+1; ki+2; : : : ; km} such that uj = vj. Then, uj= uj = vj= vj and so u = v.
This proves that f is a one-to-one function. By Lemma 4, |V (CG(x; y))| = |V (CG (x; y))| and then f is a bijection.
For any edge uw ∈ E(CG(x; y)) with u ∈ Si−1 and w ∈ Si, uj= wj for all j = ki.
Then, u
j= uj= wj= wjor uj= xj= wjor uj= yj= wj, dependingon j ∈ {k1; k2; : : : ; ki}∩
{ki+1; ki+2; : : : ; km} or j ∈ {1; 2; : : : ; n}−{k1; k2; : : : ; ki} or j ∈ {1; 2; : : : ; n}−{ki+1; ki+2; : : : ;
km}. Hence, uw is an edge of E(CG(x; y)). Therefore, CG(x; y) and CG(x; y) are
isomorphic.
Theorem 6. Two connected bit permutation networks are isomorphic if and only if
their channel graphs are isomorphic.
Proof. The “only if ” part is trivial. We prove the “if” part. Let Nd(n; k1; k2; : : : ; km) and
Nd(n; k∗1; k∗2; : : : ; k∗m) be two nonisomorphic connected networks. Then, by Theorem 1,
there exists a smallest i such that ki = k∗i . By the de:nition of a canonical
se-quence, ki∈ {k1; k2; : : : ; ki−1} or k∗i ∈ {k∗1; k∗2; : : : ; k∗i−1}, say, the :rst case holds. Then,
there exist j¡i such that kj= ki and k∗i ∈ {k∗j ; k∗j+1; : : : ; k∗i−1}. Consider the switches
x = (0; 0; : : : ; 0)j−1 and y = (0; 0; : : : ; 0)i in stages j−1 and i, respectively. By Lemma 2,
in the network Nd(n; k1; k2; : : : ; km), there exist two internal vertex-disjoint x–y
channel-paths, whose vertices have coordinates 0 except the kjth coordinate of each internal
vertex of second path is 1. However, since k∗i ∈ {k∗j ; k∗j+1; : : : ; k∗i−1}, it is impossible to
:nd two vertices x∗ and y∗ in stage j and i, respectively, such that there exist two disjoint x∗–y∗ channel-paths isomorphic to the precedingones in Nd(n; k∗1; k∗2; : : : ; k∗m).
This proves the theorem.
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