Improved Bounds on the Crossing Number of Butterﬂy Network

We draw the r - dimensional butterﬂy network with 14 4 r + O ( r 2 r ) crossings which improves the previous estimate given by Cimikowski (1996). We also give a lower bound which matches the upper bound obtained in this paper


Introduction
The crossing number cr(G) of a graph G is the minimum number of crossings of its edges among all drawings of G in the plane.It is an important measure of the non-planarity of a graph [12], with applications in discrete and computational geometry [11].The crossing number of a graph corresponding to the VLSI circuit has strong influence on the area layout as well as on the number of wire contact cuts that should be minimized.Discoveries of Leighton in the early 1980's made this, an area of high importance in the theoretical computer science [1,6].The chip area required for the realization of an electrical network is closely related to the crossing number of the underlying graph.In information visualization, one of the most important measures of the quality of a graph drawing is the number of crossings [9].The problem is NP-hard [5].
Cimikowski [4] has given an upper bound for the crossing number of the butterfly network as where r denotes the dimension of the network.In 2008, Manuel et al. [7] announced a drawing (D) of this butterfly network for which we obtained [10] an improved bound as In this paper we propose a new drawing (φ) of the butterfly network in the plane with number of crossings.We also give a lower bound for the crossing number of the butterfly network.

Butterfly Network
The set V of vertices of an r-dimensional butterfly correspond to pairs [w, i], where i, 0 ≤ i ≤ r is the dimension or level of the vertex and w is an r-bit binary number that denotes the row of the vertex.Two vertices [w, i] and [w , i ] are linked by an edge if and only if i = i + 1 and either: 1. w and w are identical, or 2. w and w differ in precisely the i th bit.
The r-dimensional butterfly network denoted by BF r has (r + 1)2 r vertices and r2 r+1 edges.It has r + 1 levels and there are 2 r vertices in each level.Each vertex on level 0 and level r is of degree 2. All the other vertices are of degree 4.

Proposed New Drawing
The new drawing φ of the butterfly network is constructed from the one (D) obtained by Manuel et al. [7] using some drawing pattern.It is easily seen from Figure 1 (b) and (c) that the D(BF 2 ) can be drawn as φ(BF 2 ).We describe next the procedure for the drawing of φ(BF 3 ).Consider imaginary lines through the level 0 and level 2 vertices of φ(BF 2 ).We call these lines as 0-line (horizontal line) and 2-line (vertical line) respectively.The 0-line divides the drawing of φ(BF 2 ) into two symmetrical parts say the upper half UH and the lower half LH.See Figure 2 (a).Reflect the lower half about the 0-line so that no two vertices overlap and the level 2 vertices of LH continue to lie on the 2-line.See Figure 2 (b).Call this diagram as φ (BF 2 ).Take a copy of φ (BF 2 ), call it φ (BF 2 ) and flip it vertically and place it below φ (BF 2 ).Now join the level 0 vertices of φ (BF 2 ) with the corresponding vertices of φ (BF 2 ) by means of 4-cycles which we call as binding 4-cycles.The resulting diagram is called φ(BF 3 ) and is shown in Figure 2 (c).The binding 4-cycles are shown by means of broken lines.The vertices of this graph are labeled as follows: Vertices of φ (BF 2 ) and φ (BF 2 ) are of the form . The new vertices in the binding 4-cycles are of the form [u 1 u 2 u 3 , 0] where u 1 , u 2 , u 3 ∈ {0, 1}.The procedure is recursive.

Lower Bound
We first solve the crossing number problem for BF 3 .We begin with the following results.Proof: Let t be the smallest non-negative integer such that the removal of some t edges from the graph G results in a planar graph G t .The graph G t is a connected spanning subgraph of the graph of G with 32 vertices and 64 − t edges.By Theorem 5, n − ε + f = 2.Here n = 32 and ε = 64 − t.Hence G t has 34 − t regions.Also by Theorem 6, since G t has girth four, 4(34 − t) ≤ 2(64 − r).Hence t ≥ 4 and therefore cr(G) ≥ 4. 2 Lemma 2 and Theorem 4 imply the following result.
We next proceed to obtain a lower bound for the crossing number which is fairly close to the upper bound obtained in Theorem 4.
The following terminologies related to butterfly network and some of the graph parameters are required in order to determine the lower bound for the crossing number of BF r for large values of r.
In a butterfly network the vertices on level 0 are called the inputs of the network, and the vertices on level r are called the outputs.An n -input butterfly network has n vertices in level 0. We say BF r is an 2 r -input butterfly network as there are 2 r vertices in level 0.
The bisection width of a graph G denoted by bw(G), is the minimum number of edges whose removal divides G into two parts having at most 2 |V | /3 vertices each.
The best upper bound we obtained in this paper is of order O(4 r ).A lower bound of matching order can be simply obtained by combining the known facts.
Theorem 8 Let G be an r-dimensional butterfly network.Then It is known [8] that for an n-vertex graph with the degree sequence It is also shown in [2] that for an 2 r -input butterfly network, the bisection width is given by,

Conclusion
Theorem 1 provides a 3 2 4 r +O(r2 r ) bound for the crossing number of an r-dimensional butterfly network.The drawing proposed in this paper drastically reduces the upper bound to 1 4 4 r + O(r2 r ).We have also obtained a matching lower bound as 1 59 4 r + O(r2 r ).It would be interesting to explore such tight bounds for other interconnection networks too.

Fig. 1 :
Fig. 1: (a) Original drawing of BF2 (b) Drawing D proposed by Manuel et al.(c) Drawing φ which we propose in this paper.