1180IEICE TRANS. FUNDAMENTALS, VOL.E97–A, NO.6 JUNE 2014

PAPER Special Section on Discrete Mathematics and Its Applications

Queue Layouts of Toroidal Grids

Kung-Jui PAI†, Jou-Ming CHANG††, Yue-Li WANG†††, Nonmembers, and Ro-Yu WU††††a), Member

SUMMARY A queue layout of a graph G consists of a linear order ofits vertices, and a partition of its edges into queues, such that no two edgesin the same queue are nested. The queuenumber qn(G) is the minimumnumber of queues required in a queue layout of G. The Cartesian productof two graphs G1 = (V1, E1) and G2 = (V2, E2), denoted by G1 × G2, isthe graph with {〈v1, v2〉 : v1 ∈ V1 and v2 ∈ V2} as its vertex set and an edge(〈u1, u2〉, 〈v1, v2〉) belongs to G1 × G2 if and only if either (u1, v1) ∈ E1

and u2 = v2 or (u2, v2) ∈ E2 and u1 = v1. Let Tk1 ,k2 ,...,kn denote the n-dimensional toroidal grid defined by the Cartesian product of n cycles withvaried lengths, i.e., Tk1 ,k2 ,...,kn = Ck1 ×Ck2 × · · · ×Ckn , where Cki is a cycleof length ki � 3. If k1 = k2 = · · · = kn = k, the graph is also called the k-aryn-cube and is denoted by Qk

n. In this paper, we deal with queue layoutsof toroidal grids and show the following bound: qn(Tk1 ,k2 ,...,kn ) � 2n − 2if n � 2 and ki � 3 for all i = 1, 2, . . . , n. In particular, for n = 2 andk1, k2 � 3, we acquire qn(Tk1 ,k2 ) = 2. Recently, Pai et al. (Inform. Process.Lett. 110 (2009) pp.50–56) showed that qn(Qk

n) � 2n−1 if n � 1 and k � 9.Thus, our result improves the bound of qn(Qk

n) when n � 2 and k � 9.key words: Queue layout, Toroidal grids, Cartesian product, Archedleveled-planar graphs

1. Introduction

Let G be a graph with vertex set V(G) and edge set E(G).A vertex ordering σ of G is a bijection from V(G) to{1, 2, . . . , |V(G)|}. For u, v ∈ V(G), we write u <σ v if σ(u) <σ(v). A k-queue layout of a graph G consists of a vertexordering σ and a partition of its edges into k queues suchthat no two edges in the same queue are nested (i.e., twoedges (u, v), (x, y) ∈ E(G) are nested if u <σ x <σ y <σ vor x <σ u <σ v <σ y). The queuenumber of a graph G, de-noted by qn(G), is the minimum k such that G has a k-queuelayout. A graph G is a k-queue graph if qn(G) � k.

Queue layouts were first introduced by Heath etal. [13], [16]. There are many applications of queue layoutsin computer science, including sorting permutations [7],[18], [22], [28], [32], parallel process scheduling [1], matrix

Manuscript received September 3, 2013.Manuscript revised November 12, 2013.†The author is with the Department of Industrial Engineering

and Management, Ming Chi University of Technology, New TaipeiCity, Taiwan, ROC.††The author is with the Institute of Information and Decision

Sciences, National Taipei College of Business, Taipei, Taiwan,ROC.†††The author is with the Department of Information Manage-

ment, National Taiwan University of Science and Technology,Taipei, Taiwan, ROC.††††The author is with the Department of Industrial Management,

Lunghwa University of Science and Technology, Taoyuan, Taiwan,ROC.

a) E-mail: eric@mail.lhu.edu.twDOI: 10.1587/transfun.E97.A.1180

computations [27] and graph drawing [5], [9], [33]. In par-ticular, queue layouts of interconnection networks have ap-plications to the Diogenes approach to testable fault-tolerantarrays of processors [30]. Heath and Rosenberg [16] showedthat recognizing a k-queue graph is NP-complete even ifk = 1. Thus, further investigations tended to study boundson queuenumber for certain families of graphs [5], [6], [8],[11]–[16], [24]–[26], [29], [33], [34].

This paper deals with queue layouts of a family ofgraphs called toroidal grids. Let ki � 3 be integers fori = 1, 2, . . . , n. The n-dimensional toroidal grid, denotedby Tk1,k2,...,kn , is a graph consisting of N = k1 × k2 ×· · · × kn vertices, each of which is associated with a la-bel x = 〈x1, x2, . . . , xn〉 where xi ∈ {0, 1, . . . , ki − 1} fori = 1, 2, . . . , n, and two vertices x = 〈x1, x2, . . . , xn〉 andy = 〈y1, y2, . . . , yn〉 are adjacent if and only if there exists aninteger j ∈ {1, 2, . . . , n} such that x j ≡ y j ± 1 (mod k j) andxi = yi for all i ∈ {1, 2, . . . , n} \ { j}. For example, Fig. 1(a),1(b) and 1(c) respectively depict T6, T4,3 and T3,3,3, where avertex 〈x1, x2, . . . , xn〉 is written as x1x2 · · · xn for notationalconvenience.

The class of Tk1,k2,...,kn includes k-ary n-cubes as a sub-

Fig. 1 Toroidal grids: (a) T6; (b) T4,3; (c)T3,3,3.

Copyright c© 2014 The Institute of Electronics, Information and Communication Engineers

PAI et al.: QUEUE LAYOUTS OF TOROIDAL GRIDS1181

GraphConditions

Bound ofRef.

Classes Queuenumber

Qn

n � 2 � n − 1 [16]

n � 4 � n − 2 [12], [24]

n � 8 � n − 3 [25]

n � 3 n − �lg(n − �lg(n − 1)�)� [10]

Q3n n � 3 � 2n − 3 [26]

Qkn

n � 2 and 4 � k � 8 � 2n − 2[26]

n � 1 and k � 9 � 2n − 1

n = 2 and k1, k2 � 3 = 2

Tk1 ,k2 ,...,kn n � 3 and ki � 3� 2n − 2

This paper

for 1 � i � n

class. The k-ary n-cube Qkn is defined as Tk1,k2,...,kn with the

restriction k1 = k2 = · · · = kn = k � 3. In particu-lar, the class of Q3

n is called the ternary n-cubes. The n-dimensional hypercube Qn is a graph with 2n vertices suchthat each vertex is associated with a label x = 〈x1, x2, . . . , xn〉where xi ∈ {0, 1} and two vertices are linked by an edge ifand only if they differ in exactly one coordinate. We notethat most current multicomputers (e.g., Cray T3D [23] andCray T3E [31], iWARP [3], Intel Teraflops system [4], Fu-jitsu AP1000 [17], and Chaos system [2], [20]) are basedon the 2- and 3-dimensional toroidal grids for low-latencyand high-bandwidth inter-processor communication. Also,Qn has resulted in several experimental and commercial ma-chines including Ncube 6400 [21] and Intel iPSC [19].

In this paper, we establish an upper bound onqn(Tk1,k2,··· ,kn ) by using induction. For the base step, we pro-vide a construction scheme to produce a 2-queue layout ofTk1,k2 for k1, k2 � 3. In the following table, we summarizesome recent results of queue layouts for the aforementionedclasses of graphs.

2. Arched Leveled-Planar Graphs

Because our construction scheme relies on a vertex order-ing yielded from a particular planar embedding introducedby Heath and Rosenberg [16], we now present the conceptof such planar embeddings. A graph G = (V, E) is leveled-planar if V can be partitioned into subsets V1,V2, . . . ,Vm sothat G has a planar embedding in which all vertices of Vi areon a vertical line �i defined by �i = {(i, y)|y ∈ R}, and eachedge in E is embedded as a straight-line segment wholly be-tween �i and �i+1 for some i. Such a planar embedding iscalled a leveled-planar embedding [16]. A leveled-planargraph under this embedding has a natural ordering on ver-tices, called the induced order, by scanning line �i from bot-tom to top for each i = 1, 2, . . . ,m and labeling the vertices1, 2, . . . , |V | as they are encountered. Heath and Rosenberg[16] further showed that every leveled-planar graph is a 1-queue graph and the induced order of its vertices yields a1-queue layout.

To completely characterize the class of 1-queue graphs,Heath and Rosenberg [16] introduced a wider class ofgraphs called arched leveled-planar graphs, which contains

leveled-planar graphs as a subclass. Consider a leveled-planar graph G. For 1 � i � m, let bi be the first vertex(i.e., the bottom vertex) and ti be the last vertex (i.e., the topvertex) in line �i. Let si be the first vertex in �i, which is adja-cent to some vertex in �i+1, or, let si = ti if there are no edgesbetween lines �i and �i+1. An arched leveled-planar graphis an augmentation of G that adds some new edges (zeroor more) to G by connecting vertex ti with vertex j, wherebi � j � min{ti−1, si}, and each added edge is called an archof G. An arched leveled-planar graph can be embedded inthe plane by drawing the arches around �1 such that archesdo not cross each other. Under this embedding, the inducedorder of an arched leveled-planar graph is the same as that ofthe corresponding leveled-planar graph without arches. Theedges that are not arches are called leveled edges. An archedleveled-planar graph is maximal if it cannot be augmentedwith further arches or leveled edges. A maximal archedleveled-planar graph on n vertices contains exactly 2n − 3edges [16]. This is useful for establishing lower boundson queuenumber. For example, Fig. 2(a) shows a leveled-planar graph, Fig. 2(b) shows an arched leveled-planar graphwith arches (4, 5) and (6, 8), and Fig. 2(c) shows a maximalarched leveled-planar graph.

Our construction scheme of queue layout is based onthe following known results.

Theorem 1 (Heath and Rosenberg [16]): A graph G is a 1-queue graph if and only if G is an arched leveled-planargraph. In particular, the induced order of vertices in anarched leveled-planar graph yields a 1-queue layout.

Corollary 2: A graph G is a k-queue graph if and only if Gcan be partitioned into k edge-disjoint spanning subgraphssuch that all subgraphs have arched leveled-planar embed-dings with the same induced order.

Also, from Theorem 1 and the concept of maximalarched leveled-planar graphs, we immediately obtain thefollowing lower bound on queuenumber for general graphs.

Corollary 3: For any graph G with |V(G)| > 2, the maxi-mum number of edges that can be assigned to a single queueis 2|V(G)| − 3. That is, qn(G) � � |E(G)|

2|V(G)|−3 �.

3. Queue Layout of Tk1,k2

The Cartesian product of two graphs G1 = (V1, E1) andG2 = (V2, E2), denoted by G1 × G2, is the graph with{〈v1, v2〉 : v1 ∈ V1 and v2 ∈ V2} as its vertex set and an edge(〈u1, u2〉, 〈v1, v2〉) belongs to G1 × G2 if and only if either(u1, v1) ∈ E1 and u2 = v2 or (u2, v2) ∈ E2 and u1 = v1. Them-th power of a graph G = (V, E) is a graph with the samevertex as G, denoted by Gm = (V, Em), such that (u, v) ∈ Em

if and only if there is a path with length no more than m be-tween u and v in G. A generalized n-dimensional toroidalgrid is defined by Cm

k1×Cm

k2× · · ·×Cm

kn, where Cm

kiis the m-th

power of a cycle of length ki � 2m + 1. In [34], Wood stud-ied queue layouts of graph products and graph powers, and

1182IEICE TRANS. FUNDAMENTALS, VOL.E97–A, NO.6 JUNE 2014

Fig. 2 (a) A leveled-planar graph; (b) an arched leveled-planar graph; (c) a maximal arched leveled-planar graph.

showed the following bounds on queuenumber of a general-ized toroidal grid.

Theorem 4 (Wood [34]): The queuenumber of a graphG = Cm

k1× Cm

k2× · · · × Cm

knsatisfies

nm2< qn(G) � (2n − 1)m.

A graph is unicyclic if every connected component hasat most one cycle. Heath and Rosenberg [16] proved thatany unicyclic graph has a 1-queue layout. In particular, ev-ery cycle is a 1-queue graph. Since Tk is isomorphic to acycle of length k, qn(Tk) = 1 for all k � 3. Also, by The-orem 4, qn(Tk1,k2 ) is either 2 or 3 for any k1, k2 � 3. In thefollowing three lemmas, we provide construction schemesto determine the queuenumber of Tk1,k2 . Without loss of gen-erality we consider k2 � k1 � 3.

Lemma 5: qn(T3,k) = 2 for k � 3.

Proof: Obviously, T3,k contains 3k vertices and 6k edges.To show that there exists a 2-queue layout of T3,k, by Corol-lary 2, we need to partition T3,k into two edge-disjoint span-ning subgraphs, say G1 and G2, such that both subgraphshave arched leveled-planar embeddings with the same in-duced order. For G1, its arched leveled-planar embeddingconsists of k + 1 levels such that all vertices are arrangedaccording to the following rules:

1. The vertex 〈2, j〉 is located in the top of line � j+2 forj = 0, 1, . . . , k − 1.

2. The vertex 〈0, j〉 is located in the middle of line � j+1 forj = 0, 1, . . . , k − 1.

3. The vertex 〈1, j〉 is located in the bottom of line � j+2 forj = 0, 1, . . . , k − 1.

Clearly, we have leveled edges (〈0, j〉, 〈0, j + 1〉),(〈1, j〉, 〈1, j+1〉) and (〈2, j〉, 〈2, j+1〉) for j = 0, 1, . . . , k−2,and (〈0, j〉, 〈1, j〉) and (〈0, j〉, 〈2, j〉) for j = 0, 1, · · · , k−1. Itis easy to check that no leveled edges cross each other. Also,we have arches (〈1, j〉, 〈2, j〉) for j = 0, 1, · · · , k − 1. As a

Fig. 3 Arched leveled-planar embeddings of T3,5: (a) G1; (b) G2.

result, G1 has 6k − 3 edges and is an arched leveled-planargraph (see Fig. 3(a) for such an embedding of G1 for T3,5).

Let σ be the induced order of G1 under the aboveleveled-planar embedding. Also, we know that G2 containsonly three edges (〈0, 0〉, 〈0, k − 1〉), (〈1, 0〉, 〈1, k − 1〉) and(〈2, 0〉, 〈2, k − 1〉). To configure an arched leveled-planarembedding of G2, we partition all vertices into two levels,where vertices from 〈0, 0〉 to 〈1, k−2〉 in σ are consecutivelyarranged in line �1 and all remaining vertices are consecu-tively arranged in line �2. Since 〈0, 0〉 <σ 〈1, 0〉 <σ 〈2, 0〉and 〈0, k − 1〉 <σ 〈1, k − 1〉 <σ 〈2, k − 1〉, the three edges donot cross each other (see Fig. 3(b) for such an embedding ofG2 for T3,5). �

For ease of description, all edges contained in Tk1,k2 aredivided into two types: wrapped edges and mesh edges. Theformer contains edges (〈i, 0〉, 〈i, k2−1〉) and (〈0, j〉, 〈k1−1, j〉)for i = 0, 1, . . . , k1 − 1 and j = 0, 1, . . . , k2 − 1, while the lat-ter contains all non-wrapped edges. In particular, since ourconstruction scheme partitions Tk1,k2 into two edge-disjointspanning subgraphs, say G1 and G2, the mesh edges dealtout to G2 are called splitting edges.

PAI et al.: QUEUE LAYOUTS OF TOROIDAL GRIDS1183

Lemma 6: qn(T2h+4,k) = 2 for h � 0 and k � 2h + 4.

Proof: Obviously, T2h+4,k contains (2h + 4)k vertices and4(h + 2)k edges. By Corollary 2, we partition T2h+4,k intotwo edge-disjoint spanning subgraphs G1 and G2, such thattheir leveled-planar embeddings have a common induced or-der. For G1, its leveled-planar embedding consists of 2k + hlevels. In each level, the indicated line prepares h + 3 posi-tions for arranging vertices and all positions are ranked in atop-down fashion (i.e., the first position is on the top in theline and the last position is on the bottom in the line.) Wenow arrange vertices in two parallelograms according to thefollowing rules:

1. For the first parallelogram, the vertex 〈i, j〉 is located inthe (i+2)-th position of line �i+ j+1 for i = 0, 1, . . . , h+1and j = 0, 1, . . . , k − 1.

2. For the second parallelogram, the vertex 〈i, j〉 is locatedin the (2h + 4 − i)-th position of line �k+2h−i+ j+3 for i =h + 2, h + 3, . . . , 2h + 3 and j = 0, 1, . . . , k − 1.

Thus, we have the following leveled edges: (〈i, j〉, 〈i, j + 1〉)for i = 0, 1, . . . , 2h+3 and j = 0, 1, . . . , k−2, and (〈i, j〉, 〈i+1, j〉) for i = 0, 1, . . . , h, h + 2, h + 3, . . . , 2h + 2 and j =0, 1, . . . , k − 1. It is easy to verify that G1 contains (2h +4)(k − 1)+ (2h+ 2)k = 4(h+ 2)k − (2k + 2h+ 4) mesh edgesand all these edges do not cross each other (see Fig. 4(a) forsuch an embedding of G1 for T6,7).

From the above construction, we know that G2 contains2k + 2h + 4 edges, where (〈i, 0〉, 〈i, k − 1〉) and (〈0, j〉, 〈2h +3, j〉) for i = 0, 1, . . . , 2h + 3 and j = 0, 1, . . . , k − 1 arewrapped edges, and (〈h+1, j〉, 〈h+2, j〉) for j = 0, 1, . . . , k−1are splitting edges. Let σ be the induced order of G1 underthe above leveled-planar embedding. Then, we partition allvertices of G2 into three levels, where vertices from 〈0, 0〉to 〈1, k − 2〉 in σ are consecutively arranged in line �1, ver-tices from 〈0, k − 1〉 to 〈2h+ 2, k − 2〉 in σ are consecutivelyarranged in line �2, and all remaining vertices are consecu-tively arranged in line �3. Note that 〈1, k − 2〉 (respectively,〈2h + 2, k − 2〉) is the second rightmost vertex in the sec-ond topmost row of the first parallelogram (respectively, thesecond parallelogram). Hence, there are 2k leveled edgesbetween lines �1 and �2 of three types, as follows:

(i) wrapped edges (〈0, j〉, 〈2h+3, j〉) for j = 0, 1, . . . , k−2,(ii) wrapped edges (〈i, 0〉, 〈i, k − 1〉) for i = 0, 1, . . . , h + 1,

and(iii) splitting edges (〈h+1, j〉, 〈h+2, j〉) for j = 0, 1, . . . , k−

h − 2.

We first observe that for two wrapped edges(〈0, j〉, 〈2h + 3, j〉) and (〈0, j′〉, 〈2h + 3, j′〉) where 0 � j <j′ � k− 2, we have 〈0, j〉 <σ 〈0, j′〉 and 〈2h+ 3, j〉 <σ 〈2h+3, j′〉. This shows that all type-(i) edges do not cross eachother. Moreover, for two type-(ii) edges (〈i, 0〉, 〈i, k − 1〉)and (〈i + 1, 0〉, 〈i + 1, k − 1〉) where i = 0, 1, . . . , h, thereis a type-(i) edge (〈0, i〉, 〈2h + 3, i〉) such that 〈i, 0〉 <σ〈0, i〉 <σ 〈i + 1, 0〉 (particularly, 〈i, 0〉 = 〈0, i〉 if i = 0)and 〈i, k − 1〉 <σ 〈2h + 3, i〉 <σ 〈i + 1, k − 1〉. Sim-ilarly, for two splitting edges (〈h + 1, j〉, 〈h + 2, j〉) and

(〈h + 1, j + 1〉, 〈h + 2, j + 1〉) where j = 0, 1, . . . , k − h − 3,there is a type-(i) edge (〈0, h + j + 1〉, 〈2h + 3, h + j + 1〉)such that 〈h + 1, j〉 <σ 〈0, h + j + 1〉 <σ 〈h + 1, j + 1〉 and〈h + 2, j〉 <σ 〈2h + 3, h + j + 1〉 <σ 〈h + 2, j + 1〉. That is, ifwe consider edges between �1 and �2 in a bottom-up fashion,then type-(i) edges and type-(ii) edges (respectively, type-(iii) edges) appear alternately, and all type-(ii) edges turn upbefore type-(iii) edges. Therefore, all edges between �1 and�2 do not cross each other.

We now consider the remaining 2h + 4 edges betweenlines �2 and �3 of three types, as follows:

(i’) wrapped edge (〈0, k − 1〉, 〈2h + 3, k − 1〉);(ii’) wrapped edges (〈i, 0〉, 〈i, k − 1〉) for i = h + 2, h +

3, . . . , 2h + 3;(iii’) splitting edges (〈h + 1, j〉, 〈h + 2, j〉) for j = k − h −

1, k − h, . . . , k − 1.

It is clear that for two wrapped edges (〈i, 0〉, 〈i, k − 1〉) and(〈i′, 0〉, 〈i′, k − 1〉) where h + 2 � i < i′ � 2h + 3, we have〈i′, 0〉 <σ 〈i, 0〉 and 〈i′, k − 1〉 <σ 〈i, k − 1〉. Thus, all type-(ii’) edges do not cross each other. Moreover, if we con-sider edges between �2 and �3 in a bottom-up fashion, it iseasy to see that (〈0, k − 1〉, 〈2h + 3, k − 1〉) is the first edge,and type-(ii’) edges and type-(iii’) edges appear alternately.Consequently, all edges between �2 and �3 do not cross eachother (see Fig. 4(b) for such an embedding of G2 for T6,7). �

Lemma 7: qn(T2h+5,k) = 2 for h � 0 and k � 2h + 5.

Proof: Obviously, T2h+5,k contains (2h + 5)k vertices and2(2h + 5)k edges. By Corollary 2, we partition T2h+5,k intotwo edge-disjoint spanning subgraphs G1 and G2, such thattheir leveled-planar embeddings have a common induced or-der. For G1, its leveled-planar embedding consists of 2k + hlevels. In each level, the indicated line prepares 2h+ 5 posi-tions for arranging vertices and all positions are ranked in atop-down fashion. We now arrange vertices in two parallel-ograms according to the following rules:

1. For the first parallelogram, the vertex 〈i, j〉 is locatedin the (h + i + 3)-th position of line �i+ j+1 for i =0, 1, . . . , h + 2 and j = 0, 1, . . . , k − 1.

2. For the second parallelogram, the vertex 〈i, j〉 is locatedin the (i − h − 2)-th position of line �k+2h−i+ j+4 for i =h + 3, h + 4, . . . , 2h + 4 and j = 0, 1, . . . , k − 1.

Thus, we have the following leveled edges: (〈i, j〉, 〈i, j + 1〉)for i = 0, 1, . . . , 2h+4 and j = 0, 1, . . . , k−2, and (〈i, j〉, 〈i+1, j〉) for i = 0, 1, . . . , h, h + 1, h + 3, . . . , 2h + 3 and j =0, 1, . . . , k − 1. It is easy to verify that G1 contains (2h +5)(k−1)+ (2h+3)k = 2(2h+5)k− (2k+2h+5) mesh edgesand all these edges do not cross each other (see Fig. 5(a) forsuch an embedding of G1 for T5,5).

From the above construction, we know that G2 contains2k + 2h + 5 edges, where (〈i, 0〉, 〈i, k − 1〉) and (〈0, j〉, 〈2h +4, j〉) for i = 0, 1, . . . , 2h + 4 and j = 0, 1, . . . , k − 1 arewrapped edges, and (〈h+2, j〉, 〈h+3, j〉) for j = 0, 1, . . . , k−1are splitting edges. Let σ be the induced order of G1 under

1184IEICE TRANS. FUNDAMENTALS, VOL.E97–A, NO.6 JUNE 2014

Fig. 4 Arched leveled-planar embeddings of T6,7: (a) G1; (b) G2.

Fig. 5 Arched leveled-planar embeddings of T5,5: (a) G1; (b) G2.

the above leveled-planar embedding. Then, we partition allvertices of G2 into three levels, where vertices from 〈0, 0〉 to〈1, k − 2〉 in σ are consecutively arranged in line �1, verticesfrom 〈0, k − 1〉 to 〈h + 3, k − h − 3〉 in σ are consecutively

arranged in line �2, and all remaining vertices are consecu-tively arranged in line �3. Note that 〈1, k − 2〉 is the secondrightmost vertex in the second topmost row of the first par-allelogram, and 〈h + 3, k − h − 3〉 is the (h + 3)-th rightmostvertex in the top row of the second parallelogram. Hence,there are 2k leveled edges between lines �1 and �2 of threetypes, as follows:

(i) wrapped edges (〈0, j〉, 〈2h+4, j〉) for j = 0, 1, . . . , k−2,(ii) wrapped edges (〈i, 0〉, 〈i, k − 1〉) for i = 0, 1, . . . , h + 2,

and(iii) splitting edges (〈h+2, j〉, 〈h+3, j〉) for j = 0, 1, . . . , k−

h − 3.

We first observe that for two wrapped edges(〈0, j〉, 〈2h + 4, j〉) and (〈0, j′〉, 〈2h + 4, j′〉) where 0 � j <j′ � k− 2, we have 〈0, j〉 <σ 〈0, j′〉 and 〈2h+ 4, j〉 <σ 〈2h+4, j′〉. This shows that all type-(i) edges do not cross eachother. Moreover, for two type-(ii) edges (〈i, 0〉, 〈i, k − 1〉)and (〈i + 1, 0〉, 〈i + 1, k − 1〉) where i = 0, 1, . . . , h + 1,there is a type-(i) edge (〈0, i〉, 〈2h + 4, i〉) such that 〈i, 0〉 <σ〈0, i〉 <σ 〈i + 1, 0〉 (particularly, 〈i, 0〉 = 〈0, i〉 if i = 0)and 〈i, k − 1〉 <σ 〈2h + 4, i〉 <σ 〈i + 1, k − 1〉. Sim-ilarly, for two splitting edges (〈h + 2, j〉, 〈h + 3, j〉) and(〈h + 2, j + 1〉, 〈h + 3, j + 1〉) where j = 0, 1, . . . , k − h − 4,there is a type-(i) edge (〈0, h + j + 2〉, 〈2h + 4, h + j + 2〉)such that 〈h + 2, j〉 <σ 〈0, h + j + 2〉 <σ 〈h + 2, j + 1〉 and〈h + 3, j〉 <σ 〈2h + 4, h + j + 2〉 <σ 〈h + 3, j + 1〉. That is, ifwe consider edges between �1 and �2 in a bottom-up fashion,then type-(i) edges and type-(ii) edges (respectively, type-(iii) edges) appear alternately, and all type-(ii) edges turn upbefore type-(iii) edges except (〈h + 2, 0〉, 〈h + 3, 0〉) before(〈h + 2, 0〉, 〈h + 2, k − 1〉). However, the exception does notproduce crossing edges. Therefore, all edges between �1 and�2 do not cross each other.

We now consider the remaining 2h + 5 edges betweenlines �2 and �3 of three types, as follows:

(i’) wrapped edge (〈0, k − 1〉, 〈2h + 4, k − 1〉);(ii’) wrapped edges (〈i, 0〉, 〈i, k − 1〉) for i = h + 3, h +

3, . . . , 2h + 4;(iii’) splitting edges (〈h + 1, j〉, 〈h + 2, j〉) for j = k − h −

2, k − h − 1, . . . , k − 1.

It is clear that for two wrapped edges (〈i, 0〉, 〈i, k − 1〉) and(〈i′, 0〉, 〈i′, k − 1〉) where h + 3 � i < i′ � 2h + 4, we have〈i′, 0〉 <σ 〈i, 0〉 and 〈i′, k − 1〉 <σ 〈i, k − 1〉. Thus, all type-(ii’) edges do not cross each other. Moreover, if we con-sider edges between �2 and �3 in a bottom-up fashion, it iseasy to see that (〈0, k − 1〉, 〈2h + 4, k − 1〉) is the first edge,and type-(ii’) edges and type-(iii’) edges appear alternately.Consequently, all edges between �2 and �3 do not cross eachother (see Fig. 5(b) for such an embedding of G2 for T5,5). �

We remark that a web site containing more instances ofarched leveled-planar embeddings of 2-dimensional toroidalgrids is available at URL http://poterp.iem.mcut.edu.tw/torus. From Lemmas 5, 6, and 7 and the symmetry of di-mensions in a toroidal grid, we have the following result.

PAI et al.: QUEUE LAYOUTS OF TOROIDAL GRIDS1185

Theorem 8: qn(Tk1,k2 ) = 2 for k1, k2 � 3.

4. Upper Bounds on qn(Tk1 ,k2,...,kn)

In this section, we provide upper bounds on queuenumberfor high dimensional toroidal grids. Given a vertex order-ing σ of a graph G, the length of an edge (u, v) ∈ E(G)is defined to be �σ(u, v) = |σ(u) − σ(v)|. Note that if|�σ(u, v)− �σ(x, y)| � 1, then (u, v) and (x, y) do not nest. Letσ1 and σ2 be two vertex orderings with no common vertex.The concatenation of σ1 and σ2, written (σ1) ◦ (σ2), is theordering σ1 followed by the ordering σ2.

For the n-dimensional toroidal grid Tk1,k2,...,kn , we de-note (Tk1,k2,...,kn )i, i = 0, 1, . . . , kn − 1, as the subgraphof Tk1,k2,...,kn induced by the set of vertices with i as thelast digit in their labels. For notational convenience, ifa vertex v is contained in Tk1,k2,...,kn , we simply write v ∈Tk1,k2,...,kn instead of v ∈ V(Tk1 ,k2,...,kn ). Clearly, each subgraph(Tk1,k2,...,kn)

i is isomorphic to Tk1,k2,...,kn−1 under the isomor-phism ϕ(〈x1, x2, . . . xn−1, i〉) = 〈x1, x2, . . . xn−1〉. The follow-ing lemma provides an induction to derive upper bounds.

Lemma 9: qn(Tk1,k2,...,kn ) � qn(Tk1,k2,...,kn−1 ) + 2 if n � 2 andki � 3 for i = 1, 2, . . . , n.

Proof: Let σ be a vertex ordering of Tk1,k2,...,kn−1 under whichTk1,k2,...,kn−1 can be laid out using exactly qn(Tk1,k2,...,kn−1 )queues. From the isomorphism of (Tk1,k2,...,kn )i andTk1,k2,...,kn−1 , let σi be the vertex ordering of (Tk1,k2,...,kn )i cor-responding to σ in Tk1,k2,...,kn−1 . Also, define Π = (σ0) ◦(σ1) ◦ . . . ◦ (σkn−1) to be the vertex ordering of Tk1,k2,...,kn .We now show that if vertices of Tk1,k2,...,kn are arranged asΠ, then all edges can be partitioned into qn(Tk1,k2,...,kn−1 ) + 2queues without nested edges. Obviously, edges containedwithin a subgraph (Tk1,k2,...,kn )i for all i = 0, 1, . . . , kn −1 can be laid out using qn(Tk1,k2,...,kn−1 ) queues. More-over, the remaining edges have two different lengths, oneis the set of edges {(u, v) : u ∈ (Tk1,k2,...,kn )i and v ∈(Tk1,k2,...,kn)

i+1 for i = 0, 1 . . . , kn − 2} and the other is the set{(u, v) : u ∈ (Tk1,k2,...,kn )0 and v ∈ (Tk1,k2,...,kn )kn−1}. Therefore,they can be placed in two additional queues. �

Applying Theorem 8 and Lemma 9, we immediatelyobtain the following theorem.

Theorem 10: qn(Tk1,k2,...,kn) � 2n − 2 if n � 2 and ki � 3for i = 1, 2, . . . , n.

Since the family of toroidal grids contains k-ary n-cubes as a subfamily, we have the following corollary.

Corollary 11: qn(Qkn) � 2n − 2 for n � 2 and k � 3.

Note that the result of Corollary 11 improves Wood’supper bound of generalized toroidal grids when m = 1 inTheorem 4. Also, if n � 2 and k � 9, the result is alsoan improvement over a recent result of [26] shown thatqn(Qk

n) � 2n − 1 for n � 1 and k � 9. In addition, sinceit has been shown in [26] that qn(Q3

n) � 2n − 3 if n � 3 by

constructing a 3-queue layout of Q3n, a natural way to im-

proving the upper bound on queuenumber of toroidal gridsis to settle the following conjecture.

Conjecture 1: qn(Tk1,k2,k3 ) � 3 if k1, k2, k3 � 3.

Acknowledgment

The research was partially supported by National Sci-ence Council under the Grants NSC101-2221-E-131-039, NSC102-2221-E-141-001-MY3, NSC101-2221-E-011-038-MY3 and NSC102-2221-E-262-013.

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Kung-Jui Pai received his B.S. and M.S. de-grees in Information Management Departmentof National Taiwan University of Science andTechnology in 1996 and 1998, respectively. In2009, he received his Ph.D. degree in Infor-mation Management from the National TaiwanUniversity of Science and Technology. Now, heis an assistant professor in the Department ofIndustrial Engineering and Management at theMing Chi University of Technology. Dr. Pai’smajor research interest includes graph theory

and algorithm analysis.

Jou-Ming Chang received his B.S. de-gree in Applied Mathematics from Chinese Cul-ture University in 1987, an M.S. degree in In-formation Management from National ChiaoTung University in 1992, and a Ph.D. degreein Computer Science and Information Engineer-ing from National Central University in 2001.Currently, he is a professor in the Institute ofInformation and Decision Sciences of NationalTaipei College of Business. Dr. Chang’s ma-jor research interest includes algorithm analysis,

graph theory, and parallel and distributed computing.

Yue-Li Wang received his B.S. and M.S. de-grees both in Information Engineering Depart-ment of TamKang University in 1975 and 1979,respectively. In 1988, he received his Ph.D. de-gree in Information Engineering from NationalTsing Hua University. Currently, he is a distin-guished professor in the Department of Informa-tion Management of National Taiwan Universityof Science and Technology. Dr. Wang’s ma-jor research interests include graph theory, al-gorithm analysis and parallel computing.

Ro-Yu Wu received his B.S. degree in In-dustrial Engineering from Tunghai University in1985, an M.B.A. degree in Industrial Manage-ment from National Cheng Kung University in1987, and a Ph.D. degree in Information Man-agement from National Taiwan University ofScience and Technology in 2007. Currently, heis an associate professor in the Department ofIndustrial Management of Lunghwa Universityof Science and Technology. Dr. Wu’s major re-search interest includes reliability, graph theory,

and combinatorial algorithms.