Topological Properties of SomeInterconnection Network GraphsRobert CimikowskiComputer Science DepartmentMontana State UniversityE-mail: cimo@cs.montana.eduAbstractInterconnection networks play a vital role in parallel computingarchitectures. We investigate topological properties of some networksproposed for parallel computation, based on their underlying graphmodels. The vertices of the graph correspond to processors and theedges represent communication links between processors. Parame-ters such as crossing number and thickness strongly a�ect the arearequired to lay out the corresponding circuit on a VLSI chip. Inparticular, we give upper bounds for the skewness, crossing number,and thickness of several networks including the mesh of trees, reducedmesh of trees, 2-dimensional torus, butter y, wrapped butter y, andBene�s graph.1 IntroductionMany di�erent interconnection networks have been proposed for parallelcomputer architectures. Examples include the linear array, mesh (grid),torus, mesh of trees, hypercube, star, butter y, and pancake. The textby Leighton [6] describes a variety of parallel network topologies and al-gorithms. A network is modelled as an undirected graph where the ver-tices denote the processing elements and the edges denote the bidirectionalcommunication channels (wires). Some desirable features for a good in-terconnection network include low degree, regularity, small diameter, largebisection width, and high fault tolerance (connectivity). These parametersa�ect the computational performance of the network. In addition, it isimportant to minimize the amount of area consumed by the circuit layout,which contributes largely to the overall cost of fabricating the chip. In [7]it is shown that the crossing number of a graph strongly in uences the lay-out area required. Also, the thickness of a graph indicates the minimum1

number of planar layers required to lay out the graph. Finally, when a sin-gle planar layer is required, it is desirable to remove the minimum number(skewness) of \nonplanar" edges from the original graph.In this paper, we investigate the above-mentioned topological proper-ties of several networks proposed as models of parallel architectures. Foreach network, we give upper bounds for the given parameters, based on acombinatorial analysis of the adjacency structure of the underlying graph-theoretic model of the network.2 Topological InvariantsLet G = (V;E) be a graph. We will assume that G is simple and undi-rected. The crossing number of G, �(G), is the minimum number of edgecrossings in any planar drawing of G. The thickness of G, �(G), is theminimum number of planar subgraphs whose union is G. The skewnessof G, �(G), is the minimum number of edges whose removal from G resultsin a planar graph.For arbitrary G, determining �(G), �(G), and �(G) are NP -hard (see[3]). Hence, from a computational standpoint, it is infeasible to obtain exactvalues for these parameters for graphs, in general. It is natural, then, toexplore bounds for the parameters. In the following sections of the paper,we derive upper bounds for several di�erent networks.3 Two-Dimensional Toroidal NetworkThe 2-dimensional torus, Tm;n, is formed by an m � n grid (mesh) with\wraparound" edges joining vertices in the �rst and last row and �rst andlast column (see Figure 1). Tm;n has mn vertices and 2mn edges and is 4-regular and hamiltonian. Observe that Tm;n = Cm�Cn, where `�' denotesthe cross product of two graphs.Based on the drawing of T4;4 in Figure 1, we obtain the following upperbound for the skewness of Tm;n:Theorem 1. �(Tm;n) � min(m;n).Proof: Wlog let m = max(m;n). Now draw the m `wraparound' edgesas concentric arcs outside the m� n grid, producing 0 crossings. Then then interior `wraparound' edges, when removed, eliminate all crossings. 2Figure 2 illustrates the proof for T4;4. The skewness is no more than 4as indicated by removing the four bold `wraparound' edges in the �gure.2

Figure 1: Torus T4;4.We conjecture, in fact, that �(T4;4) = 4.Corollary 1.1. �(Tn;n) � n.From Figure 2 and a similar argument, an upper bound for the numberof edge crossings in Tm;n can be derived:Theorem 2. �(Tm;n) � min(m;n) � (max(m;n)� 2).Proof: When Tm;n is drawn as shown in Figure 2, each of the min(m;n)interior `wraparound' edges intersects with exactly max(m;n) � 2 verticaledges, including max(m;n) � 2 curved arcs in the bottom row, for a totalof min(m;n) � (max(m;n)� 2) crossings. 2Corollary 2.1. �(Tn;n) � n(n� 2).The thickness of Tm;n can now be inferred from Figure 2 and the pre-vious arguments.Theorem 3. �(Tm;n) = 1 for m;n < 3, and �(Tm;n) = 2, otherwise.Proof: Draw all of the grid edges and the n vertical `wraparound' edgesin layer 1 and the m vertical wraparound edges in layer 2. 23

Figure 2: Torus T4;4 with \nonplanar" edges indicated.4 Mesh of TreesThe n� n mesh of trees, Mn, has a complete binary tree in each row andcolumn. Hence, Mn is only de�ned for n a power of 2. Mn has 3n2 � 2nvertices and 4n2� 4n edges and is nonhamiltonian. M4 is shown in Figure3. Although M2 is planar, Mn is nonplanar for n � 4, despite the relativesparsity of edges. Trees in interior rows and columns obstruct planarity,and by removing a small subset of these edges we can remove all planarobstructions.Theorem 4. �(Mn) � (n�2)22 .Proof: The trees in column 1 and row n can be ipped to the leftand down, resp., to eliminate some crossings. The remaining crossings arecaused by interior trees, which can be eliminated by removing one edgefrom the root of each row tree. For larger trees, single edges from subtreeroots must also be removed. In general, if (n�2)=2 edges are removed fromeach of the n� 2 interior row trees, all crossings are eliminated. Hence,�(Mn) � (n� 2) log2n�2Xi=0 2i = (n� 2)2log2n�1 � 14

Figure 3: 2-dimensional mesh of trees M4.= (n=2� 1)(n� 2) = (n� 2)22 : 2Theorem 5. �(Mn) � (n� 2)2.Proof: Consider the drawing of M4 in Figure 3. For each interior rowtree, the two edges incident with each root vertex cross with edges of thecorresponding column trees. The same holds true for any subtree root ver-tices. Hence, the number of crossings in Mn, when drawn in this fashion,is the product of the number of interior row trees (n� 2) and the numberof crossings caused by each tree (n� 2). 2Theorem 6. �(Mn) = 2 for n � 4.Proof: Note that all row trees of Mn can be drawn in one layer andall column trees can be drawn in another layer without crossings. Hence,�(Mn) � 2. Also, note that Mn is nonplanar for n � 4 since M4 can bereduced to K4;4 by a series of edge contractions. Initially, each bivalentvertex is replaced by a single edge. Subsequent contractions result in thefour row tree roots of M4 in one partition and the four column tree rootsin another. Hence, �(Mn) � 2. 2 5

Figure 4: 2-dimensional reduced mesh of trees RM4.5 Reduced Mesh of TreesThe n � n reduced mesh of trees, RMn, is a subgraph of Mn with onlynlog2n row and column trees. In particular, RMn consists of an n� n arraywith complete binary trees added to the (i log2n+1)st row and column for0 � i < nlog2n . RMn has n2 + 2n(n�1)log2n vertices and 4n(n�1)log2n edges. Figure4 shows a drawing of RM4, which is planar. The reduction in the numberof trees in the rows and columns leads to upperbounds for skewness andcrossing number for RMn.Theorem 7. �(RMn) � (n�2)22log2n .Proof: Since there are 1=log2n as many row and column trees, crossingsinvolving some of the edges are eliminated where they existed inMn; hence,the upper bound is immediate. 2Theorem 8. �(RMn) � (n�2)2log2n .Proof: By the same argument as before, there are 1=log2n as manycrossings as in Mn; hence, the bound is again immediate. 2RM8 is edge contractible to K3;3. Hence, RMn is nonplanar for n � 8,and the following result is immediate.Theorem 9. �(RMn) = 1 for n � 4, and �(RMn) � 2 for n � 8.6

6 Hypercubic NetworksThe hypercube has received considerable attention in the literature. A sur-vey of its graph-theoretic properties can be found in [4]. Recent crossingnumber results are given in [9, 10]. The power and utility of the hyper-cube are well known. A drawback, however, is that its vertex degree growslogarithmically with the size of the network, which can present intercon-nection problems for machines with a very large number of processors. Tocircumvent this problem, several bounded-degree derivatives of the hyper-cube have been proposed. They are commonly referred to as hypercubicnetworks.6.1 Butter y NetworkThe butter y network, also known as the \FFT network," was originally de-vised to implement FFT algorithms. The r-dimensional butter y, BFr, has(r+1)2r vertices, r2r+1 edges, maximum degree 4, and is nonhamiltonian.The vertices correspond to pairs hw; ii where i is the level or dimension ofthe vertex (0 � i � r) and w is an r-bit binary number denoting the rowof the vertex. Two vertices hw; ii and hw0; i0i are adjacent i� i0 = i+1 andeither w and w0 are identical, or w and w0 di�er in precisely the ith bit.Figure 5 shows drawings of BF2 and BF3.Note the highly recursive structure of BFr , e.g., BF3 contains two BF2subgraphs joined together by an extra column of vertices on the left witha set of `connecting' diagonal edges to the two subgraphs.Although BF2 is planar, BFr is nonplanar for r > 2. We can eliminateall crossings by removing a subset of the r2r diagonal edges of BFr. Notethat the number of `connecting' diagonal edges in BFr is 2r. Althoughtwo of the `connecting' diagonals of BFr, one at the top and one at thebottom (edges fa; bg and fc; dg of Figure 5), can be redrawn as exteriorarcs to eliminate some of the crossings, in BFr+1 one of these edges mustbe retained as a `straight' diagonal edge in each copy of BFr , which has acancelling e�ect in the resulting recursive formula for skewness. Hence, weobtain the recurrence �(BFr) � 2 � �(BFr�1) + 2r.With initial conditions �(BF2) � 2 and �(BF3) � 12, we solve therecurrence to obtain the following result giving an upper bound for theskewness of BFr:Theorem 10. �(BFr) � r2r � 3 � 2r�1.7

BF

2

3

b

c

d

a

BF

Figure 5: (a) Butter y BF2 (b) Butter y BF3.An upper bound for the number of crossings in BFr can be obtainedby counting the crossings in each level, based on the `straight-edge' draw-ings of Figure 5, and using the fact that some crossings can be eliminatedby redrawing some of the diagonal edges as exterior arcs. The numberof crossings in the two copies of BFr�1 is � 2(2r�1 � 1). The numberof `connector' diagonal crossings with horizontal edges is � 2r(2r�1 � 1).The number of crossings between the `connector' edges alone is � (2r�1)2.Finally, we can reduce the last set of `connector' crossings by the sum2 � 2r�1 � 1 by drawing two of the edges as exterior arcs. Thus we have�(BFr) � 2 ��(BFr�1)+3 �22r�2�2r�1 = 2 ��(BFr�1)+3 �22r�2�2r�1.The initial conditions are �(BF2) � 5, �(BF3) � 49, and �(BF4) � 273.Solving the recurrence, we obtain the following result:Theorem 11. �(BFr) � 3�4r2 � 3 � 2r � r2r + 1.8

It is easy to �nd minimum planar decompositions for BF2 and BF3.Therefore, �(BF2) = 1 and �(BF3) = 2. It is not known, however, if BF4is biplanar. The following result, however, is easy to show:Theorem 12. �(BFr) � 3.Proof: Place all horizontal edges in layer 1, all diagonal edges going onedirection in layer 2, and all diagonal edges going the opposite direction inlayer 3. Hence, we have a triplanar decomposition for BFr. 26.2 Wrapped Butter y NetworkFor computational purposes, the �rst and last levels of the butter y aresometimes merged into a single level with the respective vertices in eachrow merged into one vertex. The result is an r-level `wrapped' butter y,WBFr, with r2r vertices each of degree 4 and r2r+1 edges. Note thatWBFr is equivalent to BFr with the �rst and last columns of nodes merged.The `wrapped' edges join vertices in levels 1 and r. The `wrapped' butter ynetwork has been studied with regard to hamiltonian paths and cycles [1, 12]and VLSI layout [5]. Figure 6 shows a drawing of WBF3.WBFr also has a recursive structure, i.e.,WBFr contains twoWBFr�1subgraphs with an extra column (level 1) of 2r vertices on the left and aset of outer `wrapped' arcs joining the level 1 and level r vertices of the twosubgraphs. WBF2 is planar, but WBFr is nonplanar for n � 3.We can eliminate all crossings in WBFr by removing a subset of theinner diagonal and outer `wrapped' edges. Note that the number of outer`wrapped' edges in WBFr is 2r. By removing half of these edges (2r�1),the 2r crossing diagonal edges in level 2, and one diagonal edge from eachof the 2r�1 subgraphs in level r, we obtain a planar graph. The 2r�1outer `wrapped' edges removed from WBFr are not present in WBFr+1and hence have a cancelling e�ect when combined with the 2r�1 level rdiagonal edges removed. This leads to the recurrence �(WBFr) � 2 ��(WBFr�1) + 2r.With the initial conditions �(WBF3) � 16 and �(WBF4) � 48, wesolve the recurrence to obtain the following result:Theorem 13. �(WBFr) � 2r(r � 1).From the drawings in Figure 6, an upper bound for the number ofcrossings in WBFr can be readily obtained. We consider each of the twosubgraphs consisting of the top (bottom) 2r=2 rows of vertices. In each sub-graph, the number of crossings involving `wrapped' edges between verticesin the same row is 2r�2(2r�2)+2r�2(2r�2�1), and the number of crossings9

involving interior `straight' diagonal edges is 2r�1(2r�2�1)+(2r�2)2+2r�2.The number of crossings involving the outer `wrapped' edges between thetwo subgraphs is (2r�1)2. Finally, the previous outer `wrapped' edge cross-ings of WBFr�1 in each of the subgraphs must be subtracted, giving therecurrence �(WBFr) � 2�(WBFr � 1) + 2[2r�1(2r�2 � 1) + (2r�2)2 +2r�2 + (2r�2)2 + 2r�2(2r�2 � 1)] + (2r�1)2 � 2(2r�2)2. This can be simpli-�ed to �(WBFr) � 2�(WBFr�1) + 3 � 22r�2 � 2r. The initial conditionsare �(WBF3) � 48, �(WBF4) � 272, and �(WBF5) � 1280. Solving therecurrence, we obtain the following result:Theorem 14. �(WBFr) � 3�4r2 � 3 � 2r � r2r.

3WBFFigure 6: Wrapped butter y WBF3.10

3

B2

BFigure 7: (a) Bene�s graph B2 (b) Bene�s graph B3.By placing subsets of di�erent edge types in di�erent layers, we canobtain a multi-layer planar embedding of WBFr. As noted before, WBFris nonplanar for r � 3.Theorem 15. �(WBFr) � 3.Proof: Place all horizontal edges, all `wrapped' edges between verticesin the same row, and one set of outer `wrapped' edges in layer 1 (i.e. theset joining level 1 vertices of the top half of WBFr with the level r verticesof the bottom half). Place the second set of outer `wrapped' edges and alldiagonal edges drawn in one direction in layer 2. Place the remaining setof diagonal edges in layer 3. 26.3 Bene�s NetworkThe Bene�s network, Br, was introduced in [2]. Br consists of back-to-backbutter ies as shown in Figure 7. Overall, Br has r dimensions and 2r + 1levels, each with 2r vertices. The �rst and last r+1 levels form a butter yBFr. The number of edges in Br is 2r+2+22r and the maximum degree is4. Br is nonhamiltonian. 11

It is straightforward to obtain an upper bound for the skewness of Br,based on the similar bound for BFr . Because of the back-to-back con-tainment of the two BFr subgraphs, however, this reduces the number ofdiagonal crossing edges which can be redrawn as non-crossing arcs. No-tably, the crossing diagonals at all levels except for the �rst and last cannotbe avoided. In the �rst and last levels, one diagonal from each level can beredrawn as an exterior arc to avoid crossings. Also, since Br contains twocopies of Br�1, the four diagonals redrawn as exterior arcs in Br�1 mustbe redrawn as straight edges in Br. The number of crossings involving theset of diagonals in each of the �rst and last levels is 2r. Therefore, thefollowing recurrence is obtained: �(Br) � 2 � �(Br�1) + 2r+1 + 2. Theinitial conditions are �(B2) � 10, �(B3) � 38, and �(B4) � 110. Solvingthe recurrence, we obtain the following result:Theorem 16. �(Br) � r2r+1 � 2r � 2.An upper bound for the number of crossings in Br can be obtaineddirectly from Figure 7 by counting the crossings between edges of eachpair of adjacent levels. The number of crossings between the leftmost andrightmost pairs of adjacent levels is 2(2r�1)2 + 2 � 2r(2r�1 � 1). We mustalso add in the 4(2r�1 � 1) outer diagonal crossings of the four copiesof Br�1 which were not counted in the number of crossings for Br�1when redrawn as exterior arcs. Finally, we can subtract the crossings in-volving two of the exterior diagonals of Br, since two of these edges canbe redrawn as exterior arcs. Therefore, the total number of crossings is2 � 2r(2r�1 � 1) + 4(2r�1 � 1) + 2(2r�1)2 � 2(2r�1 + 2r�1 � 1). This leadsto the recurrence �(Br) � 2 � �(Br�1) + 3 � 22r�1 � 2r+1 � 2. The initialconditions are �(B2) � 14, �(B3) � 106, �(B4) � 562, and �(B5) � 2594.Solving the recurrence, we obtain the following result:Theorem 17. �(Br) � 3 � 4r � 5 � 2r � 2r2r + 2.Br is nonplanar for r � 2. Even B2 seems to require three planar layers.Hence, the following resultTheorem 18. �(Br) � 3.Proof: Place all horizontal edges in layer 1. Place all diagonal edgesdrawn in one direction in layer 2 and the remaining diagonal edges drawnin the opposite direction in layer 3. 212

7 Open ProblemsThere are three immediate open problems stemming from the research.First, �nding tighter upper bounds for the topological invariants is in or-der. We conjecture that the upper bounds given for the 2-dimensional torusare in fact as tight as possible, but those for the remaining networks canprobably be improved. These would involve a deeper analysis of the struc-ture of each network with regard to the containment of homeomorphs offorbidden planar subgraphs.Another problem is to �nd lower bounds for the given parameters. Re-sults of this type are known only for a few other networks, e.g. hypercube,cube-connected cycles [9, 10, 11].Finally, it would be interesting to investigate other parallel networkmodels with regard to these topological invariants, i.e., de Bruijn, pancake,star, shu�e-exchange, generalized hypercube, k-ary hypercube (k > 2).References[1] D. Barth and A. Raspaud. Two edge-disjoint hamiltonian cycles inthe butter y graph. Info. Proc. Letters 51 (1994) 175-179.[2] V. Bene�s. Permutation groups, complexes, and rearrangeable multi-stage connecting networks. Bell System Technical Journal 43 (1964)1619-1640.[3] M.R. Garey and D.S. Johnson. Computers and Intractability: AGuide to the Theory of NP-Completeness, W.H. Freeman & Co., NY(1979).[4] F. Harary, J.P. Hayes, and H.-J. Wu. A survey of the theory of hy-percube graphs. Comput. Math. Appl. 15 (4) (1988) 277-289.[5] J. Keller. Regular layouts of butter y networks. Integration 17 (1994)253-263.[6] F.T. Leighton. Introduction to Parallel Algorithms and Architectures:Arrays, Trees, Hypercubes, Morgan-Kaufman, San Mateo, CA (1992).[7] F.T. Leighton. New lower bound techniques for VLSI. Math. SystemsTheory 17 (1984) 47-70.[8] P.C. Liu and R.C. Geldmacher, On the deletion of nonplanar edgesof a graph, in Proc. 10th. Southeast Conf. Comb., Graph Theory, andComput., Boca Raton, FL, pp. 727-738, 1977.13

[9] T. Madej. Bounds for the crossing number of the n-cube. J. GraphTheory 15 (1991) 81-97.[10] O. S�ykora and I. Vr�to. On crossing numbers of hypercubes and cubeconnected cycles. BIT 33 (1993) 232-237.[11] O. S�ykora and I. Vr�to. On VLSI layouts of the star graph and relatednetworks. Integration 17 (1994) 83-93.[12] S.A. Wong. Hamilton cycles and paths in butter y graphs. Networks26 (1995) 145-150.

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