Orienting Split-Stars and Alternating Group Graphs

Eddie Cheng, Marc J. LipmanDepartment of Mathematics and Statistics, Oakland University, Rochester, Michigan 48309

Akers et al. proposed an interconnection topology, thestar graph, as an alternative to the popular nnn-cube.Cheng et al. proposed the split-star as an alternativeto the star graph and a companion graph to the al-ternating group graph proposed by Jwo et al. Stargraphs, alternating group graphs, and split-stars areadvantageous over nnn-cubes in many aspects. Day andTripathi proposed an assignment of directions to theedges of the star graph and showed that the result-ing directed graph is strongly connected and has asimple routing algorithm. In this paper, we give sim-ple routing algorithms for a proposed orientation ofalternating group graphs and split-stars. The result-ing directed graphs are not only strongly connectedbut they have maximal arc-fault tolerance and a smalldiameter. © 2000 John Wiley & Sons, Inc.

Keywords: interconnection networks; connectivity; routing

1. INTRODUCTION

Distributed processor architectures offer the advan-tage of improved fault tolerance and reliability. An im-portant component of such a distributed system is thesystem topology, which defines the interprocessor com-munication architecture. Some of the more popular sys-tem topologies are Boolean n-cubes (see, e.g., [3, 12]),butterfly graphs (see, e.g., [2, 4]), and star graphs (see,e.g., [1, 7].) Jwo et al. [9] studied the alternating groupgraph An and Cheng et al. [5] studied a variant of thestar graph which can be viewed as a companion graphfor An; it is known as the split-star.

Day and Tripathi [7] proposed an assignment of direc-tions to the edges of the star graph and showed that theresulting directed graph is strongly connected and has asimple routing algorithm. In this paper, we studied thisorientation problem for the alternating group graphs andsplit-stars. We give a simple near-optimal routing algo-rithm for each resulting directed graph. We also showthat they are not only strongly connected, but, in fact,maximally-arc-connected.

Received March 24, 1999; accepted May 25, 1999Correspondence to: E. Cheng; e-mail: echeng@oakland.edu.Contract Grant Sponsor: Oakland University Foundation

c© 2000 John Wiley & Sons, Inc.

We assume that the reader is familiar with the basicnotions of graphs, specifically, r-regular, diameter, con-nectivity (a k a fault tolerance),1 edge-connectivity (a k aedge-fault tolerance), arc-connectivity (a k a arc-fault tol-erance), strong connectivity, and paths (see [13]). A pathbetween two vertices with minimum length is a shortestpath or an optimal routing between them. (If the graphis directed, then the path must be a directed path.) Anr-regular graph is maximally fault-tolerant (maximallyedge-fault-tolerant) if its fault tolerance (edge-fault tol-erance) is r. Let X be a nonempty proper subset of thevertex-set of a graph. Define δ(X) to be the number ofarcs leaving X and ρ(X) to be the number of arcs en-tering X. If X = {v}, we write δ(v) and ρ(v) instead ofδ({v}) and ρ({v}), respectively. Given a directed graph,if min{δ(v), ρ(v)} = r for every vertex v, then it is max-imally arc-fault-tolerant if its arc-fault tolerance is r. Adirected graph is r-regular if δ(v) = ρ(v) = r for everyvertex v.

A split-star has the set of n! permutations of an n-setas the vertex set. We use N = {1, 2, . . . , n} as the n-set, and write [a1, a2, . . . , an] to represent the permutationwhich is the bijection π(i) = ai for all i ∈ N. The per-mutation [a1, a2, . . . , an] can be physically represented byplacing n checkers (here, the checkers are from any set oflabeled objects) with labels a1, a2, . . . , an (n ≥ 3) on thevertices of a graph, known as the generator-graph, withn vertices so that ai is on vertex i. Suppose that the per-mutations are represented by checker placement on thegenerator-graph in Figure 1 (a star with the root split); aspecific example of a generator-graph on seven verticesis given in Figure 2. Two permutations are related if onecan be obtained from the other by either a 2-exchange ora 3-rotation. A 2-exchange interchanges the checkers onthe vertices 1 and 2. A 3-rotation rotates the checkers onthe vertices of a triangle, that is, the triangle with vertices1, 2, and k for some k ∈ {3, 4, . . . , n}. Let S2

n be the rela-tion graph of these instances. It is called a split-star. Forexample, the graph in Figure 2 generates S2

7 (with 7! ver-tices). A smaller split-star S2

4 is given in Figure 3. Let S2n,E

be the subgraph of S2n induced by the set of even permu-

tations. This is precisely the alternating group graph, An,introduced in [9]. Let S2

n,O be the subgraph of S2n induced

NETWORKS, Vol. 35(2), 139–144 2000

by the set of odd permutations. Then, S2n,O is isomor-

phic to S2n,E via φ([a1, a2, a3, . . . , an]) = [a2, a1, a3, . . . , an].

Moreover, the edges corresponding to the 2-exchangesinduce a perfect matching2 between the set of even per-mutations and the set of odd permutations. The followingtheorem gives some known properties of S2

n and An; itsproof can be found in [9] and [5]:

Theorem 1.1. Let n ≥ 3.

1. S2n is a (2n − 3)-regular undirected graph on n! ver-

tices. An is a (2n − 4)-regular undirected graph onn!/2 vertices.

2. S2n is vertex-transitive and has two equivalence classes

of edges. An is vertex-transitive and edge-transitive.3. An optimal routing between any two vertices in S2

n

(and, hence, An) can be found in O(n2) time.4. The diameter of S2

n is b(3n)/2c − 2, and the diameterof An is b(3n)/2c − 3.

5. Both An and S2n are maximally fault-tolerant and max-

imally edge-fault-tolerant.

In fact, the following greedy strategy produces anoptimal routing from πa = [a1, a2, . . . , an] to πb =[b1, b2, . . . , bn]: Let π, the current permutation, be πa; re-peat the following until π is equal to πb:

• If there is a symbol in the first two positions of π thatis neither b1 nor b2, then perform a 3-rotation to putthis symbol, say bi, to the ith position. Let π be theupdated permutation.

• If the symbols in the first two positions of π are b1 andb2, and there exists another symbol bj not in the jthposition, then perform a 3-rotation to put this symbolin one the first two positions. Let π be the updatedpermutation.

• If the symbols in the first two positions of π are b1

and b2 and every other symbol bi is in the ith position,then perform a 2-exchange move if b1 is in the 2ndposition (and, hence, b2 is in the first position). Letπ be the updated permutation, which, of course, willbe πb.

(See [9] or [5] for a proof.) Of course, the last step is notnecessary for a routing in An.

2. ORIENTATION, ROUTING ALGORITHMS,AND DIAMETER

In this section, we show that both An and S2n can be

oriented in a way that the resulting graph is maximallyfault-tolerant, has a small diameter, and has a simplenear-optimal routing algorithm. Throughout the paper,we use the following notations when it is convenient:Let π = [a1, a2, a3, . . . , an].

• Then, E(π) = [a2, a1, a3, . . . , an], that is, the 2-exchange.

• Let 3 ≤ i ≤ n. Then, Fai (π) = Fi(π) =[ai, a1, a3, . . . , ai−1, a2, ai+1, . . . , an] (forward rotation)and Rai (π) = Ri(π) = [a2, ai, a3, . . . , ai−1, a1, ai+1, . . . , an](reverse rotation).

FIG. 1. The generator-graph for split-stars.

Given An. We note that the ends of every edge are πand Fi(π) for some π and i = 3, 4, . . . , n. We replace itby the arc π → Fi(π). Let the resulting graph be ~An. Thefollowing proposition is obvious:

Proposition 2.1. Let n ≥ 3. Then, ~An is a (n−2)-regulardirected graph on n! vertices.

We will now consider a routing algorithm in ~An. Wewant a simple algorithm—our goal is to modify thesimple greedy algorithm. The next observation is thekey:

Lemma 2.2. Every arc in ~An belongs to a directed 3-cycle.

FIG. 2. The generator-graph on seven vertices.

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FIG. 3. S24.

Proof. Let this given arc be π → Fi(π) for somei = 3, 4, . . . , n. Then, ~An contains the following 3-cycle:π → Fi(π) → Fi(Fi(π)) → Fi(Fi(Fi(π))) = π.

Lemma 2.2 gives us the following algorithm for arouting from πa to πb in ~An:

1. Find a greedy routing from πa to πb in An.2. Replace each step in the routing of the form π →

Ri(π) [which is the wrong direction in ~An] by π →Fi(π) → Fi(Fi(π)).

This immediately shows that the length of the optimalrouting from πa to πb in ~An is at most twice the lengthof the optimal routing from πa to πb in An.

We now turn our attention to orienting S2n. The ends of

every edge generated by a 3-rotation are π and Fi(π) forsome π and i = 3, 4, . . . , n; we orient it from π to Fi(π), asbefore. The ends of an edge generated by a 2-exchangeare of the form [a1, a2, a3, . . . , an] and [a2, a1, a3, . . . , an].We orient it from [a1, a2, a3, . . . , an] to [a2, a1, a3, . . . , an]if a1 > a2. We denote the resulting graph by ~S2

n. Figure4 gives ~S2

4 .

Proposition 2.3. Let n ≥ 3. Then, ~S2n is a directed graph

on n! vertices where δ(v) ∈ {n − 2, n − 1} and ρ(v) ∈{n − 2, n − 1} for every vertex v.

Lemma 2.4. Every arc generated by a 3-rotation in ~S2n

belongs to a directed 3-cycle. Every arc generated by a2-exchange in ~S2

n belongs to a l-cycle with l ≤ 6.

Proof. We have already proved the first statement.Consider an arc, ~e: [a1, a2, a3, . . . , an] → [a2, a1, a3, . . . , an],generated by a 2-exchange. So, a1 > a2. We consider twocases:

1. a1 = n and a2 = n − 1: Then, it is on a di-rected 6-cycle, namely, [n, n − 1, a3, a4, . . . , an] to[n − 1, n, a3, a4, . . . , an] to [a3, n − 1, n, a4, . . . , an] to[n, a3, n − 1, a4, . . . , an] to [a3, n, n − 1, a4, . . . , an] to[n − 1, a3, n, a4, . . . , an] to [n, n − 1, a3, a4, . . . , an].

2. a1 ≠ n or a2 ≠ n − 1. Then, there is an ai with3 ≤ i ≤ n such that ai > a2. Without lost of general-ity, assume that i = 3. Then, ~e is on a directed 4-cycle,namely, [a1, a2, a3, a4, . . . , an] to [a2, a1, a3, a4, . . . , an]to [a3, a2, a1, a4, . . . , an] to [a2, a3, a1, a4, . . . , an] to[a1, a2, a3, a4, . . . , an].

Lemma 2.4 gives us the following algorithm for arouting from πa to πb in ~S2

n:

1. Find a greedy routing from πa to πb in Sn.2. Replace each step in the routing of the form π →

Ri(π) [which is the wrong direction in ~S2n] by π →

Fi(π) → Fi(Fi(π)).3. If there is a step in the routing of the form π → E(π)

that is in the wrong direction, replace it by a directedpath of length 3 or 5 according to Lemma 2.4.

This immediately shows that the length of the optimalrouting from πa to πb in ~S2

n is at most twice the length ofthe routing from πa to πb in Sn plus 4. This is because

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FIG. 4. Orientation of S24.

a 2-exchange occurs at most once in a greedy routingin S2

n; if it occurs in the wrong direction, we replace itby a directed path of length at most 5—hence, the netdifference is 4.

Theorem 2.5. Let n ≥ 3. Then, ~An is strongly con-nected with diameter at most 2b(3n)/2c − 6, and ~S2

n isstrongly connected with diameter at most 2b(3n)/2c.

3. CONNECTIVITY

Theorem 2.5 already showed that both ~An and ~S2n are

strongly connected. In this section, we show that theyare, in fact, maximally arc-fault-tolerant.

Proposition 3.1. Let n ≥ 3. Then, ~An is vertex-transitive.

Proof. Since ~An is strongly connected, it is enoughto show that if πa → πb is an arc then there is an auto-morphism mapping πa to πb. Let t be any even permuta-tion on N. Define φt: V(~An) → V(~An) by φt(π) = tπ.This is a well-defined bijection since t is even. Con-sider an arc πc → πd. Since it is an arc, we haveπd = Fi(πc) = πc(1, i, 2)3 for some 3 ≤ i ≤ n. Then,φt(πd) = tπd = tπc(1, i, 2) = Fi(tπc) = Fi(φt(πc)), soφt(πc) → φt(πd)) is an arc. Hence, φt is an automor-phism. To get φt(πa) = πb, that is, tπa = πb, we lett = πbπ−1

a , which is even.

We note that the proof of Proposition 3.1 actuallyshowed that every even permutation is an element of theautomorphism group of ~An. To show that ~An is maximallyarc-fault-tolerant, that is, ~An has fault tolerance (n − 2),we need the following result where the proof for theundirected version can be found in [10]. We need thedirected version. The two proofs are essentially identical.For completeness, we include its proof here.

Theorem 3.2. Let D = (V, E) be a strongly connectedvertex-transitive r-regular directed graph. Then, D is r-arc-connected.

Proof. Let k be the arc-fault tolerance of D. Let X ⊆V be the smallest set that yields δ(X) = k. Since D isregular, δ(X) = ρ(X) = δ(X̄); hence, |X| ≤ |V|/2 asX is smallest. Let v1 and v2 be vertices in X. Let φ bean automorphism that maps v1 to v2. Let X′ = φ(X).Then, δ(X′) = k. Since v2 ∈ X ∩ X′, X ∩ X′ ≠ �. Since|X′| = |X| ≤ |V|/2 and X ∩ X′ ≠ �, X ∪ X′ ≠ V. Since

δ(X) + δ(X′) = δ(X ∩ X′) + δ(X ∪ X′) + d(X, X′),

where d(X, X′) is the number of arcs between X and X′(regardless of the directions), we have

k + k = δ(X) + δ(X′) ≥ δ(X ∩ X′) + δ(X ∪ X′) ≥ k + k

as X ∩ X′ ≠ �, X ∪ X′ ≠ V and k is thearc-fault tolerance. Hence, δ(X ∩ X′) = δ(X ∪ X′) = k.

142 NETWORKS–2000

Since X is the smallest, X∩X′ = X, that is, X = X′. Sincev1 and v2 are arbitrary, |Nδ(v) ∩ X| is a constant for allv ∈ X, where Nδ(v) = {w ∈ V: v → w is an arc.}; let thisconstant be n. Since D is strongly connected, r ≥ n + 1.Clearly, n ≤ |X| − 1. So, k = δ(X) = (r − n)|X| ≥(r−n)(n+1). So, k ≥ (r−n)(n+1) = n(r−n−1)+r ≥ rsince r ≥ n + 1. Since k ≤ r, we have k = r.

Theorem 3.3. Let n ≥ 3. Then, ~An is maximally arc-fault-tolerant.

We remark that another way to prove Theorem 3.3without using Proposition 3.1 and Theorem 3.2 is to usethe fact that An is 2(n − 2)-regular with edge-fault toler-ance 2(n − 2) and the following result: “Let H = (V, E)be a (2k)-regular graph with edge-fault tolerance 2k. LetG be an orientation of H. If G is k-regular, then G hasarc-fault tolerance k.”

To complete our objective, we need the arc-fault-tolerant version of the Fan Lemma [8] whose proof isincluded here for completeness.

Theorem 3.4. Let G = (V, D) be a directed graph witharc-fault tolerance r. Let v ∈ V and U ⊆ (V\{v}) with|U| = r. Then, there exist r arc-disjoint directed pathsfrom v to U and r arc-disjoint directed paths from U to v.

Proof. Let U = {u1, u2, . . . , ur}. Construct a new di-rected graph G′ by adding a new vertex z and adding 2rnew arcs, namely, z → ui and ui → z for i = 1, 2, . . . , r. Itis easy to see that G′ has arc-fault tolerance r as δ(X) ≥ rand ρ(X) ≥ r for any X, a nonempty proper subset ofV(G′). Then, there exist r arc-disjoint paths from v toz. Observe that these paths are arc-disjoint; moreover,ρ(z) = r and these r arcs are (ui, z) for i = 1, 2, . . . , r.Therefore, these r arc-disjoint paths induce r arc-disjointpaths from v to U. Similarly, there exist r arc-disjointpaths from U to r.

Proposition 3.5. Let n ≥ 3. Then, δ(V(S2n,E)) ≥ n − 2

and ρ(V(S2n,E)) ≥ n − 2.

Proof. It is easy to check the result for n = 3, 4, 5.So, assume that n ≥ 6. Consider π = [2, 1, 3, 4, . . . , n],which is odd. Consider the following 2(n − 4) odd per-mutations: ai = π(3, 4, i) and bi = π(3, i, 4) for i =5, 6, . . . , n. It follows from the arcs ai → E(ai) and bi →E(bi), i = 5, 6, . . . , n that ρ(V(S2

n,E)) ≥ 2(n − 4) ≥ n − 2.Consider the following n − 2 even permutations: ci =π(3, i) for i = 4, 5, . . . , n and d = π(3, i)(n − 2, n − 1, n).It follows from the arcs ci → E(ci) for i = 4, 5, . . . , n andd → E(d) that δ(V(S2

n,E)) ≥ n − 2.

Theorem 3.6. Let n ≥ 3. Then, ~S2n is maximally arc-

fault-tolerant.

Proof. Since δ(V(S2n,E)) ≥ n − 2 and ρ(V(S2

n,E)) ≥n − 2, we can apply Theorem 3.3, Theorem 3.4, and

Proposition 3.5, if necessary, to conclude that there arer arc-disjoint paths from πa to πb for any permutationπa and πb.

4. CONCLUDING REMARKS

We have presented orientations for An and S2n to obtain

the directed graphs ~An and ~S2n, respectively. We showed

that the diameter of the oriented graph is roughly twicethe diameter4 of the corresponding undirected graph andthe arc-connectivity is roughly half the edge-connectivityof the corresponding undirected graph. This is essentiallythe best one can hope for. Moreover, the near-optimalrouting algorithm is just a slight modification of the opti-mal greedy routing algorithm for An and S2

n, so it is veryefficient. We note that by a theorem of Nash-Williams[11] there exists an orientation transforming a 2n-edge-connected graph into an n-arc-connected directed graph,but such an orientation may not give a good bound on thediameter. In fact, [6] showed that the following questionis NP-complete: Given a graph G = (V, E) and a posi-tive integer K ≤ |V|, can the edges of G be oriented insuch a way that the resulting directed graph is stronglyconnected and has diameter no more than K? As a finalremark, we, in fact, have a lot of freedom in the orien-tation of the 2-exchange edges. The criteria are that (1)every 2-exchange edge should be in a directed cycle ofsmall length, and (2) a sufficient number, namely (n −2),of these 2-exchange edges are oriented into S2

n,E and outof S2

n,E, respectively. In this paper, we presented one suchorientation. In fact, this orientation gives something evenstronger, namely, ~An and ~S2

n are super arc-fault-tolerant.5

The proofs are omitted.

Acknowledgments

We are grateful to the anonymous referee for a num-ber of comments.

Notes

1. Some authors distinguish the terms connectivity and fault toleranceby defining fault tolerance to be one less than the connectivity.

2. Given a graph G = (V, E), M ⊆ E is a perfect matching if everyvertex of the graph H = (V, M) has degree one.

3. Here, (1, i, 2) is the typical cyclic notation for permutations.

4. Since there is a formula for the length of a shortest path in An andS2

n, we remark that one can use this to improve this estimate toabout 2n rather than 3n.

5. A directed graph is super-arc-fault-tolerant if the only minimumarc-disconnecting sets are one of the form Nδ(v) or Nρ(v) forsome vertex v; here, Nρ(v) is defined analogous to Nδ(v).

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