THE (N,K)-STAR GRAPH - A GENERALIZED STAR GRAPH

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Information

Processing

Letters

ELSEVIER Information Processing Letters 56 ( 1995) 259-264

The (~2, Q-star graph: A generalized star graph

Wei-Kuo Chiang, Rong-Jaye Chen *

Department of Computer Science and Information Engineering, National Chiao Tung University, Hsinchu, 30050, Taiwan, ROC

Received 1 October 1994; revised I June 1995 Communicated by S.G. Akl

Keywords: Fault tolerance; Star graphs; Node symmetry; Routing; Diameter

1. Introduction

The n-star graph [ 1 ] is an attractive alternative to the n-cube. It has significant advantages over the n- cube, such as a lower degree and a smaller diameter. However, a major practical difficulty with the n-star graph is the restriction on the number of nodes: n! for an n-star graph. Since there is a large gap between n! and (n + 1) !, one may face the choice of either too few or too many available nodes.

The objective of this paper is to propose a new topology, called the (n, k)-star graph, such that it re- moves the restriction of the number of nodes n! in the n-star graph, and preserves many attractive properties of the n-star graph such as node symmetry, hierarchical structure, maximal fault tolerance, and simple shortest routing.

The (n, k) -star graph is a generalized version of the n-star graph. The two parameters n and k can be tuned to make a suitable choice for the number of nodes in the network and for the degree/diameter tradeoff. This allows more flexibility in designing network topology than the star graph. The (12, k) -star graph is regular of degree n - 1, the number of nodes n!/(n - k)!, and diameter 2k - 1 for k < [n/21 and [(n - 1) /2J + k for

* Corresponding author.

k 2 [n/2] + 1. In addition, the (n, n - 1 )-star graph is isomorphic to the n-star graph, and hence, all these properties can be derived for the n-star graph as it is a special case of the (n, k) -star graph. Moreover, many parallel algorithms [ 21 for the n-star graph may adapt to the (n, k)-star graph with a slight modification.

2. Network topology and basic properties For simplicity, denote (n) = { 1,2,. . . , n}.

Definition 1. An (n, k) -star graph, denoted by S,,k, is specified by two integers n and k, where 1 6 k < n. The node set of Sri,,, is denoted by {pIp2. . .pk 1

P; E (n> and pi f pj for i # i }. The adjacency is defined as follows: ~1~2.. .pi . . .pk is adjacent to ( 1) pip2 . . . PI . . pk through an edge of dimension i, where 2 6 i 6 k (SwappI withpi),and (2) xp;!. . .pk through dimension 1, where x E (n) - {pi 1 1 < i 6

k}. The edges of type ( 1) are referred to as i-edges, and the edges of type (2) are referred to as 1 -edges. A (4,2) -star graph is shown in Fig. 1.

Definition 2. A graph is node symmetric if and only if for its any pair of nodes u and U, there exists an automorphism of the graph that maps u to U.

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260 W.-K. Chiang, R.-J. Chenl Information Processing Letters 56 (1995) 259-264

Theorem 3. The (n, k) -star graph is node symmetric.

Proof. We need to show that given any two nodes in

Sri,,, there exists an automorphism that maps one node intotheother.Letp=pip2...pkandq=qiq2...qk

be the two nodes in &,k, P = {pl ,p2,. . . ,pk}, Q = {qr42v.. , qk}, and 1x1 denote the number of elements in the set X.

Define the one-to-one onto mapping pl in (n):

l ~1 (pi) = qi, for 1 < i < k, i.e., for p; E P; l ~1 (x) = y, one-to-one mapping for x E Q - P and

y E P - Q (since IPI = lQ[, IQ - P( = IQ1 - IP n

Ql=lpl-lPnQl=lf'-Ql);

l

p](z)

=z,forz E (n)-PUQ.

Let MI be the one-to-one onto mapping in Sn,k:

Fig. I. A (4,2)-star graph.

Ml(t) =Pl(tl)/‘l(t2)...Pl(tk)

for all t = t] t:! . . . tk in $,k.

M2CPIP2.. .Pn-IPn) =mp2.. .pn-I

for a node p = ~1~2.. .pn_lpn of S,. Clearly MI maps the node p into the node q. Fur-

ther, MI is an automorphism, for if two nodes s and t are adjacent in Sn,k. then MI (s) and MI (t) are adjacent. More precisely, let s = si s2 . . . Si . . . Sk, then t = SiS2. . . S] . . . Sk (SWapSI withsi) OrtlQ...Si...Sk (tl E (n) - {Si I 1 < i < k}). Consider

Moreover, h42 preserves adjacency. Let p and q are two nodes joined with an i-edge in S,. If 2 6 i 6 n- 1, then M2( p) and M2( q) are also joined with an i-edge in S,,,_ 1; if i = n, then A42 (p) and M2 (q) are joined with a 1 -edge in S,,,_ 1. 0

MI(S)=PI(SI)CLI(S~)...PI(S~)...PI(S~) 3. Hierarchical structure and fault tolerance

and Definition 5. Let &.i,k-i (i) denote a subgraph of

S,,k induced by all the nodes with the same last symbol i, for some 1 6 i 6 n.

PI(S . ..PI(SI) *.*pl(Sk)v so MI (s) and MI (t) are adjacent. 0

Note that Sri,,, could not be edge symmetric. For

Lemma 6. &,k can be decomposed into n sub-

graphs S,_ I.k_l (i) , 1 < i < n, and each subgraph $_l,k_l (i) is isomorphic to &_l,k_l.

instance, in Fig. 1, each 2-edge belongs to a cycle of length at least 6, but each l-edge may be within a cycle of length 3.

Lemma 4. The (n, n - 1 )-star graph S,,,,_ I is iso-

morphic to the n-star graph S,,.

Proof. If we remove the last symbol of all the

nodes in &_i.+i(n), we obtain an &_i,k_i. That is, Sn-l,k-l(n) is isomorphic to S”_i,k_i. So, we only need to show S,_i,k__l(i), 1 < i < n - 1, and S,_ 1 ,k_ I (n) are isomorphic. We define the one-to-one mapping pug in (n):

Proof. To prove that S,,,_ 1 and S, are isomorphic, we

remove the last symbol in all nodes of S,, and obtain an S,,,_ 1 by Definition 1. That is, we define a bijection M2 from the nodes of S,, to those of ,!&_i by:

p3(i) = n, rug(n) = i, p3(x) =x for x E (n) - {i,n}. We also define a bijection M3 by:

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W.-K. Chiang, R.-J. Chenllnformation Processing Letters 56 (1995) 259-264 261

M3hP2.. *Pk) = p3(PI )p3@2) . . ./-‘3h)

for a node p = p1p2 . . .pk in S,,k.

Obviously, M3 transforms the nodes of Sn_l,k_-l (i) into those of S,_l,+, (n) and preserves adjacency. 0

Since S,,k can be partitioned into n mutually dis- joint subgraphs S,_,,k_, (i), 1 < i < n, S,,k is hierarchical. Fig. 1 shows that $2 can be viewed as an interconnection of four SJJ ‘s through 2-edges. Lemma 7. There are (n - 2)!/(n - k)! k- edges between any two subgraphs Sn_l,k_l (i) and S,_l,k_l (j); each of these nodes of S,_l,k-, (i) is connected to exactly one node in S,_ 1 ,k_ 1 (j) . Proof. For any pair of two nodes jp2 . . .pk-I i E

S,+-],k_l (i) and ipz.. .Pk-lj E Ll,k-l(j), i + j, there is a k-edge connecting them. Each p2p3 . . . pk- 1

is a unique permutation of k - 2 distinct symbols chosen out of the n - 2 symbols of (n) - {i, j}. Thus, we get the result. 0

Definition 8. The fault tolerance of a graph G is defined as the maximum number f such that if any f nodes are deleted from G, the resulting subgraph is still connected.

Theorem 9. The fault tolerance of the (n. k)-star graph is n - 2.

Proof. Let f (&,k) denote the fault tolerance of &k. First, we claim f(&k) 2 f(Sn__l,k_l) + 1 for 2 < k < n. For brevity, let f = f ( Sn__l,k_l ). We prove it by showing that S,,k remains connected even if f + 1 of its nodes are faulty. We consider two cases. In the first case, suppose that all of the faults are within a subgraph S,_ 1 ,k_ 1 (i) . Since the fault tolerance of S,_ I ,k_ 1 is f, S,,_ I ,k_ 1 (i) could be disconnected because of the f + 1 faults. But each nonfaulty node in S,_, ,k_ , (i) has a k-edge to the other copy S,__ I ,k- ,.

All other S,_ 1 ,k- 1 ‘s remain connected since they have no faults. Thus, Sri,,, remains connected.

Proceeding to the second case, suppose that the f + 1 faults are distributed among more than one subgraph S,_ 1 ,k- I in Sn,k. Since there are at most f faults in any subgraph, each subgraph &_-l,k_, (i) remains

connected. We merely need to prove that any two of n subgraphs &_-l,k_l (i), 1 6 i 6 n, remain connected to each other. By Lemma 7, each &_.l,k_-l (i) has ( n - 2) ! / (n - k) ! adjacent nodes in each of other S+l,k_l(j)‘s, j f i. If we regard each S,_l,k_l(i) of S,,k as a supernode, then the resulting graph is a complete graph of n nodes. To disconnect it, we would have to remove at least n - 1 nodes. So, if S,,k could be disconnected because of the f + 1 faults, then f + 1 3

(n-l)!/(n-k)!.Butf+l <deg(S,_l.k_l)+l= (n-2)+1 =n-1 6 (n-l)!/(n-k)!,for2< k< n. This is a contradiction. Therefore, S,,k also remains connected.

Based on the preceding discussion, we have the following recurrence relation: f (&,k) 3 f (S,_ I ,k- I ) + 1, and the initial condition: f (S”_k+ ],I ) = n - k - 1 since &_k+, ,I iS isomorphic to a complete graph of n - k + 1 nodes. Thus, f (Sn,k) b n - 2. The degree of S,,k iS n - 1, which implieS f (Sn,k) 6 n - 2. There- fore, f (S,,k) = n - 2. 13

4. Routing path and diameter

Due to node symmetry of the (n, k) -star graph, any node can be mapped to the identity node Ik = 12. . . k by renaming the symbols. For the routing between any two arbitrary nodes s and t, the renaming function M maps the destination node to the identity node, i.e., M(t) = Ik. Then all the paths between nodes s and t in the original graph are isomorphic to those between M(s) and the identity Ik in the renamed graph. So, without loss of generality, considering the problem of the routing between two nodes in the (n, k)-star graph, the destination node is always assumed to be the identity node lk.

Before solving the problem of routing between an arbitrary node p and the identity node Ik, we define a cycle representation for the label of each node in (n, k) -star graphs similar to the well-known cycle structure of permutation for star graphs [ 11.

To clarify our presentation, we call a symbol E (n) - (k) an external symbol since it is not used in the label of the identity node (destination). On the con- trary, a symbol E (k) is called internal. Unless stated otherwise, we use Ci (C,!) to denote an internal (external) cycle and mi (ml) to denote the number of elements in Ci (C:).

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262 W.-K. Chiang, R.-J. Chenllnformation Processing Letters 56 (1995) 259-264 Let/J =p1p2-. . pk, where pi denotes the symbol in

the position i. The symbol in its correct position, i.e., pi = i, is called invariant. In the following discussion, we omit all invariants in the cycle representation since they will not be moved during the shortest routing.

We construct the cycle representation of node p as follows. First, for each external symbol x,,,: in p we construct an external cycle C! = (XI, x2, . . . , xm: ) such that the desired position of Xj in p is held by xi+, for 1 < j < mf - 1. Additionally, for the cycle C,! we define the desired symbol d,-; whose desired position is held by the first element, x,, of the cycle. When we have constructed the external cycles for all external symbols in p, the rest are all internal symbols. Then we construct the internal cycles for the rest as the same as those in the star graph. An internal cycle C; = (x1,x2,. . . , n,, ) of p means that the position of

xi+, inpisdesiredbyxjforl < j<mi-l.For ease of illustration, if there exists a cycle containing p, , we specially choose pI as the first element of the cycle since any cyclic shift of the sequence of symbols within each cycle is allowed.

In the cycle representation of a node p. cycles can appear in any order. So, the cycle representation of a node p with (Y internal cycles and p external symbols can be denoted as

C(p) = C,C2.. .C&Ci.. .CA and p 2 0. We now demonstrate the cycle representation of a node in the (n, k) -star graph through an example. Consider node p = 2968 134 in $7. The cycle representation of p is C(p) = C, C:Ci where the cycles C, = (3,6),CI = (2,9, I), and Ci = (4,8). The desired symbols of Ci and Cl are dl = 5 and d2 = 7, respectively.

The routing from an arbitrary node p to the identity node lk can be achieved by moving internal symbols to their correct positions and exchanging external symbols with desired symbols. Here, we correct the cycles of C(p) one by one. If there exists a cycle containing ~1, we certainly correct it first.

In general, we correct an internal cycle Cj =

(x1,x2,. . . , x,, ) as follows. If x, = PI, we directly move xl to its correct position (held by x2), while x2 is swapped to the first position. Then x2 is taken to its correct position (held by x3) and so on until x,,_, is taken to its correct position. Note that each element of a cycle except for the first element of the cycle will be

moved to the first position as a result of the correction of the previous element in the cycle representation. Since xm, = 1, its correction can be considered as a result of the correction of x,,,,_, . If x, f pl , it requires an additional step to take x, into the first position of the label since a symbol must be in the first position before taken to its correct position. Then, we move

x1,x2,. . . ,&n, to their correct positions in the same

order. So, the number of steps required to correct each internal cycle Ci of length m;, 1 6 i < a, is

i

T?Z- 1 ifp, ECi, mj+l ifp, $C;.

On the other hand, in order to correct an external cycle, we move the internal symbols into their correct positions by using the same argument as the preceding. But, when the only external symbol is taken to the first position, we exchange it with the desired symbol of any other uncorrected external cycle (if exists) or itself (otherwise). More precisely, let C: denote the first corrected but not completed external cycle. Only when all external cycles other than Ci have been corrected completely, the external symbol in the first position is exchanged with dl . Through this strategy, all the external cycles are corrected consecutively such that the additional steps to swap the first element of these cycles other than Ci to the first position are re- duced. Thus, the number of steps required to correct the p external cycles is

(

m’+P-1 if p, E some Cj, m/+/3+ 1 ifpl 4 any Cj, where m’ = Cf_, rn;.

Example. Consider the correction of node p = 2968134 in S~,J. Following the above strategy, we first correct the external cycles CiCi = (2,9,1) (4,8) along the path:

2968134-i;! 9268134--+,7268134-t7 4268137 d4 8264137 +, 5264137 +5 1264537. Then the internal cycle C, = (3,6) is corrected along the path:

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W.-K. Chiang, R.-J. Chenllnformation Processing Letters 56 (199.5) 259-264 263 Note that the subscripts indicate the dimensions of numbers of cycles, misplaced symbols, and external

passed edges. symbols in p(p’) with respect to ~,(I,_I).

Definition 10. The distance between two nodes in G is the length of a shortest path joining them. The diameter of G is the largest distance between nodes of G.

(a) p] = 1. If pn = n, then c’ = c, m’ = m, and e’ = 0; thus, d(p) = d(p’) = c + m. If pn f n, then c’=c,m’=m- l,ande’= l;thus,d(p) =d(p’) = c+m.

Let c denote the total number of cycles of C(p), where all the external cycles is regarded as a cycle in count. So, if p > 0, then c = LY + 1; if p = 0, then c = cy. Let e denote the number of external symbols in p, i.e., e = p. Let m denote the total number of misplaced symbols of p with respect to the identity Ik, i.e., m = CL, m; + ~~zr mi = CL1 I?zi f m’.

(b) p] # 1. If pn = n, then c’ = c, m’ = m, and e’ = 0; thus, d(p) = d(p’) = c + m - 2. If p,, # n, then c’ = c,m’ = m - 1, and e’ = 1; thus, d(p) =

d(p’) =c+m-2. Cl

From the above, the distributed routing algorithm may be described by using repeatedly one of the following three rules until Ik is reached:

(RI) Theorem 11. The distunce d(p) from a node p to

the identity node Ik in &k is given by

if symbol 1 is in the first position, then interchange it with any symbol not in its correct position, (R2) 1 CS m+e d(p) = ifpi = 1, c+m+e-2 ifp] f 1. (R3)

if symbol i ( 1 < i < k) is in the first position, then move it to its correct position, and if the external symbol of cycle Cl is in the first position, then exchange it with the desired symbol, dj, of any other uncorrected cycle Ci. Proof. (a) pl = 1. There is no cycle containing pl . If

p = 0, thend(p) = Cf, (mi+l) = cu+m = c+m+e. If /3 > 0, then d(p) = IfI (mi + 1) + {cf, (ml + l)+l}=a+m+~+l=c+m+e.

(b) p] # 1. There exists a cycle containing pl. If /? = 0, then p] E some Cj, and d(p) = (mj - 1)+{~~,(~i+1)-(~j+1)}=~+m-2=

c + m + e - 2. If /I > 0 and pi E some Cj, then d(p) =(~j-1)+{~~,(~i+1)-(~j+~)}+

{~~,(mj+1)+1} =m+a+p-1 =c+m+e-2. If /3 > 0 and p] E Cj, it takes {Et, (ml + 1) - 1) to correct all exteyl cycles of p. Hence, d(p) =

~~~~~!)~{~i_l(m~+l)-l}=n+m+~-l =

Note that in rule 2, when an internal symbol is in the first position, we can move it to its correct position or interchange it with any symbol within another internal cycle. In addition, in rule 3, all the external cycles other than the cycle containing p] can be corrected in any order. So, for any symbol i, 1 < i < n, in the first position, we has more than one choice for routing. This turns out to be an especially useful rule for routing in the presence of faults.

Theorem 13. The diameter D( Sn.k) of &,k is given by

2k- 1 D(Sn.k) =

k-i-[+] if[F]+l<k<n-1. Corollary 12. The distance d(p) from a node p to

the identity node Ik in S,, is given by

1 c+m

d(p) = ifp] = 1,

c + m - 2 ifp] # 1.

Proof. The diameter D(&k) is max{d(p) 1 p E

Sn,k}. We consider two cases as follows.

Proof. Based on Lemma 4, the distance from p = pip2...pn tol, = 12.. .n in S,, is equivalent to the distancefromp’=ptp2...pn_i toI,_, = 12...(n-

1) in S,,,_,. Let c(c’),m(m’), and e(e’) denote the

(a) For 1 < k < [n/21: The diameter is obtained from one of the following two: ( 1) pl = 1, c = 1, m = k-lande=k-1,and(2)pl # l,c=l,m=k and e = k. Therefore, D(S,,k) = 2k - 1.

(b) For [n/21 + 1 < k < n - 1: If II is odd, the diameter is obtained for p] = 1, c = (2k - n -

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264 Table 1

W.-K. Chiang, R.-J. Chenl Information Processing Letters 56 (1995) 259-264

Comparison of network topologies

Topology Size Sri n! AG & ₂ A n.k n! (n-k)! Degree n-l 2(n - 2) k(n - k) Cost factor M 3(“-1)2 2 Z 3(n - 2)2 z ik2(n - k) s n.k 2k- I n! n-l if 1 6 k < [;1 x (n - 1)(2k- 1) (n-k)! k+ly] if[t]+I<k<n-1 2 z (n- I)k+ f(n- l)2

k + [(rr - 1)/2J. If n is even, the diameter can be obtained from one of the following two: ( 1) PI = 1, c=(2k-n)/2,m=k-l,e=n-k,and(2)pr + 1,

c = (2k - n)/2 + 1, m = k, e = n - k. Therefore,

D(&k) is also k+ [(n - 1)/21. 0

Corollary 14. The diameter D( S,,) of the n-star

graph is [3(n - 1)/2J.

Proof. Since S,, is isomorphic to L&_-l, D( S,) =

D(&,,-I ). 0

5. Performance comparison

Finally, we compare the (n, k)-star graph with three well-known topologies: (1) the n-star graph (S,) [ 11, (2) the alternating group graph (AC,,) [5], and (3) the (n, k)-arrangement graph (An,k), another generalized version of the star graph proposed in [ 41. The numbers of nodes, degrees, and diameters of &, AC,, &,k, and &k are presented in Table 1.

The number of parameters that one may vary in order to specify a graph provides a crude feeling for whether there are large gaps in the number of nodes of successive graphs in the family. From this viewpoint, An,k and &k are more flexible than S, and AC,,. The cost factor (diameter x degree of a node) is a good criterion to measure the performance of a network [ 31. We list the approximate cost factors of these four topologies in the last column of Table 1. The results show that S,,k is better than A,,k.

References 111 121 [31 [41 IsI

S.B. Akers, D. Horel and B. Krishnamurthy, The star graph: An attractive alternative to the n-cube, in: Proc. Internat. Confi on Parallel Processing ( 1987) 393-400.

S.G. Akl, K. Qiu and 1. Stojmenovic, Fundamental algorithms for the star and pancake interconnection networks with applications to computational geometry, Networks 23 ( 1993) 215-225.

L.N. Bhuyan and D.P. Agrawal, Generalized hypercube and hyperbus structures for a computer network, IEEE Trans. Comput. 33 (4) ( 1988) 323-333.

K. Day and A. Tripathi, Arrangement graphs: A class of generalized star graphs, Inform. Process. Lett. 42 ( 1992) 235-241.

J.S. Jwo, S. Lakshmivarahan and S.K. Dhall, A new class of interconnection networks based on the alternating group, Nefworks 23 ( 1993) 3 I S-326.