Strong Rainbow Edge Coloring of Some Interconnection Networks

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ScienceDirect Procedia Computer Science 57 (2015) 338 – 347

3rd International Conference on Recent Trends in Computing 2015

Strong Rainbow Edge Coloring of Some Interconnection Networks I. Annammal Arputhamarya,* , M. Helda Mercyb a b

Sathyabama University, Chennai – 600119, India.

Panimalar Engineering College, Chennai - 600123, India.

Abstract A rainbow edge coloring of a connected graph is a coloring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are colored the same. Rainbow coloring has received much attention recently in the field of interconnection networks. Computing the rainbow connection number of a graph is NP- hard and it finds its applications in the secure transfer of classified information between agencies and in cellular network . This paper investigates the strong rainbow connection numbers of butterfly network, Benes network and torus network. © 2015 The TheAuthors. Authors.Published Published Elsevier B.V. © 2015 byby Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing 2015 2015 (ICRTC-2015). (ICRTC-2015)

Keywords: Rainbow coloring; rainbow connection number; butterfly network; Benes network and torus network.

1. Introduction Interconnection network is the heart of parallel architecture. It plays a central role in determining the overall performance of the multiprocessor systems. An interconnection network is used for exchanging data between two processors in a multistage network. It can be modelled as a graph in which vertices represent processors and edges represent communication channels. Butterfly network is the most popular interconnection network which can perform the Fast Fourier Transforms very efficiently. The butterfly network consists of a series of switch stages and interconnection patterns, which allows n inputs to be connected to n outputs[15]. The Benes network consists of back-to-back butterflies. As butterfly is known for FFT, Benes is known for permutation routing. The butterfly and Benes networks are important multistage interconnection networks, which possess attractive topologies for

* Corresponding author. Tel.: 8056297388; fax: +0-000-000-0000 . E-mail address:[email protected]

1877-0509 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing 2015 (ICRTC-2015) doi:10.1016/j.procs.2015.07.348

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communication networks [14] and optical networks[13]. Torus network is a network topology for connecting processing nodes in a parallel computer system. Torus network is commonly used in parallel architectures. It has desirable properties because of symmetricity through the whole network 2. An overview of the paper All graphs considered in this paper are finite, undirected and simple. Let G be a nontrivial connected graph on which an edge-coloring c : E(G)→{1,2, · · · ,n}, n ‫ א‬N, is defined, where adjacent edges may be colored the same. The concept of rainbow colouring was introduced by Chartrand et al.,in [3]. A path is rainbow if no two edges of it are colored the same. An edge-colored graph G is rainbow connected if every two distinct vertices are connected by at least one rainbow path. An edge-coloring under which G is rainbow connected is called a rainbow coloring. An edge colored graph G is strongly rainbow connected if for every pair of distinct vertices, there exists atleast one shortest rainbow path. Thus, the rainbow connection number of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected and the strong rainbow connection number is the minimum number of colors that makes G strongly rainbow connected denoted by src(G). For example, the rainbow connection number of a complete graph is 1, that of a path is its length, and that of a tree is its number of edges. A rainbow coloring using rc(G) colors is called a minimum rainbow coloring. The rainbow connection number is not only a natural combinatorial measure, but it also has applications to the secure transfer of classified information between agencies. The rainbow connections of graphs are very new concepts and there has been great interest in these concepts and a lot of results have been published [3]. Precise values of rainbow connection number for many special graphs like complete multipartite graphs, Peterson graph and wheel graph were also determined. It was shown in Chakraborty et al.,in [4], that computing the rainbow connection number of an arbitrary graph is NP-Hard. Graphs have an important application in modelling communications networks. Generally, vertices in the network represent terminals, processors and edges represent transmission channels through which the data flows. The communication networks can be represented using the various mathematical structures which also help us to compare the various representations based on the characteristics of the network. We discuss the application of rainbow coloring to well-known interconnection networks such as the butterfly network , Benes network and torus network .They find widespread use in shared-memory multiprocessor systems ( [10], [7]), telecommunication networks, time division multiple accessed (TDMA) systems for satellite communication [6] and newer applications such as switching fabrics in internet routers. 3. Rainbow coloring of butterfly network BF(n) Butterfly graph of order n, BF(n) is an n- partite graph. It has a very good symmetry in structure. This is an advantage in networks as it lowers network cost. Cao, Du, Hsu, and Wan [2] have shown that BF(n) is 2-connected and its diameter is equal to 2n[2]. Note that BF(1) is a 4-cycle[12]. The vertices of n- dimensional butterfly network BF(n) correspond to the set of pairs [w,q], where q is the level or dimension of the vertex (0 d q d n) and w is an n- bit binary number that denotes the row of the vertex. Two ' ' vertices [ w, q] and [ w , q ] are connected by an edge iff q ' q  1 and either (i) w and w´ are identical or (ii) w and w´ differ in precisely the q th bit. ' ' ' n We call edge (( w, q), ( w, q )) a straight edge and edge (( w, q), ( w , q )) a cross edge[13]. BF(n) has (n  1)2 n 1 n vertices and n( 2 ) edges[9, 11]. It has n +1 levels and there are 2 vertices in each level. Each vertex on level 0 and level n is of degree 2. All other vertices are of degree 4.

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Fig. 1. BF(2)

3.1. Theorem The strong rainbow connection number of butterfly network BF(n) is n(2n 1  1) . Proof:

n

An n- dimensional butterfly network has n +1 levels namely level 0, level1,…,level n and 2 rows. In n n k BF(n), there are 2 straight edges and 2 cross edges between levels k and k-1, 1 d k d n . Let eki and e i be n the straight and cross edge of BF(n) between the level k and level k-1, 1 d k d n , 1 d i d 2 . Let cki and ak be n 1 the colors used between the levels k and k-1, 1 d k d n , 1 d i d 2 . We define c(eki ) as the color of the edge n eki , 1 d k d n , 1 d i d 2 .The coloring algorithm for coloring the straight edges and cross edges are given below. The edges between the levels 1 and 0 are colored as follows: The straight edges between the levels 1 and 0 are colored with the colors

c1i

­°c(e1(i  j ) ) , 0 d j d 2n 1  1 ® °¯c(e1(i  j 1) ) , i j  1

n 1

The straight edges between the levels 1 and 0 require 2 distinct colors . All the cross edges between the levels 1 n 1 and 0 are colored with the color a1. Total number of colors required is 2  1 . The edges between the levels 2 and 1 are colored as follows: The straight edges between the levels 2 and 1 are colored with the colors

c2i

­°c(e2 (i  2 j ) ) , 0 d j d 2n  2  1 ® °¯c(e2 (i  2 j  2) ) , 2 j  1 d i d 2 j  2

n 1

The straight edges between the levels 2 and 1 require 2 distinct colors . All the cross edges between the levels 2 n 1 and 1 are colored with the color a2. Total number of colors required is 2  1 . The edges between the levels 3 and 2 are colored as follows: All the straight edges between the levels 3 and 2 are colored with the colors

I. Annammal Arputhamary and M. Helda Mercy / Procedia Computer Science 57 (2015) 338 – 347

c3i

­°c(e3(i  4 j ) ) , 0 d j d 2n  3  1 ® °¯c(e3(i  4 j  4 ) ) , 4 j  1 d i d 4 j  4

n 1

The straight edges between the levels 3 and 2 require 2 distinct colors . All the cross edges between the levels 3 n 1 and 2 are colored with the color a3. Total number of colors required is 2  1 . In general the straight edges between the levels k and k-1, 1 d k d n are colored with the colors nk ­ °c(ek (i  2 k 1 j ) ) , 0 d j d 2  1 cki ® c (e ) , 2k 1 j  1 d i d 2k 1 j  2k 1 ° ¯ k (i  2 k 1 j  2 k 1 ) and the cross edges between the levels k and k-1, 1 d k d n are colored with the color ak .

Total number of colors required to rainbow color the butterfly network BF (n) n(2n1  1) which is also the strong rainbow connection number because every two vertices are connected by at least one shortest rainbow path. Therefore the strong rainbow connection number src ( BF (n)) (2n1  1) . For illustration BF(2) is shown in Fig. 2.

Fig 2: BF(2)

4. Rainbow Coloring of Benes Network BB(n) The Benes network was introduced in [1]. An n- dimensional Benes network BB(n) has 2n +1 levels , each level consists of 2n nodes. This n- dimensional Benes network consists of an n- dimensional butterfly network BF(n) which is followed by an inverse butterfly network BF(n) of dimension n.

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Fig 3: BB(2)

For ease of use, we consider the n- dimensional butterfly network BF(n) as left BF(n) and inverse butterfly network BF(n) as right BF(n). For illustration BB(2) is shown in fig.3. The middle level of the Benes network is shared by these butterflies. BBn is non Hamiltonian. Benes network is used to construct routing network and networks similar to ATM switching system. 4.1. Theorem The strong rainbow connection number of Benes network BB(n) is n(2n + 1). Proof: ' k Let eki and e ki , 1 d k d n , 1 d i d 2n be the straight edges in left BF(n) and right BF(n) respectively. Let e i 'k ' n and e i , 1 d k d n , 1 d i d 2 be the cross edges in left BF(n) and right BF(n) respectively. Let c ki and cki , 1 d k d n , 1 d i d 2n1 be the colors used for the straight edges in right BF(n) and left BF(n) respectively. Let ak , 1 d k d n be the color used for the cross edges in both right BF(n) and left BF(n). By theorem 3.1, the number of colors required for left BF(n) is n(2n 1  1) . The straight edges in right BF(n) can be colored as follows: The straight edges between the levels n and n-1 are colored with the colors

c

' 1i

­°c(e'1( i  j ) ) , 0 d j d 2n 1  1 ® ' °¯c(e 1( i  j 1) ) , i j  1

n 1

distinct colors . Total number of colors required is The straight edges between the levels n and n-1 require 2 2 n 1 . The straight edges between the levels n-1 and n-2 are colored with the colors '

c 2i

­°c(e'2(i  2 j ) ) , 0 d j d 2n  2  1 ® ' °¯c(e 2(i  2 j  2) ) , 2 j  1 d i d 2 j  2

The straight edges between the levels n-1 and n-2 require 2n-1 distinct colors. Total number of colors required is 2 n 1 . All the straight edges between the levels n-2and n- 3 are colored with the colors

I. Annammal Arputhamary and M. Helda Mercy / Procedia Computer Science 57 (2015) 338 – 347

'

c 3i

­°c(e'3(i  4 j ) ) , 0 d j d 2n  3  1 ® ' °¯c(e 3(i  4 j  4) ) , 4 j  1 d i d 4 j  4

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n 1

distinct colors. Total number of colors required is The straight edges between the levels n-2 and n-3 require 2 2 n 1 . In general the straight edges between the levels n  k  1 and n  k , 1 d k d n are colored with the colors '

c ki

­°c(e'k (i  2 k 1 j ) ) , 0 d j d 2n  k  1 ® ' °¯c(e k (i  2 k 1 j  2 k 1 ) ) , 2k 1 j  1 d i d 2k 1 j  2k 1

n 1

Therefore the colors required for the straight edges in right BF(n) is n2 .Note that the colors used for the straight edges in right BF(n) are distinct from the colors used for the straight edges in left BF(n). But the cross edges in right BF(n) can be colored with the same set of n colors that has been used for the cross edges in left BF(n). Therefore the number of colors required for the Benes network BB(n) n(2n  1) .We can see that every two vertices are connected by at least one shortest rainbow path. Therefore the strong rainbow connection number src ( BB(n)) n(2n  1) . 5. Rainbow Coloring of Torus Network A two dimensional mesh M (m, n) of size m u n is the cartesian product Pm u Pn of a path of length m-1 and a path of length n-1 with the vertex set V {vij | 1 d i d m,1 d j d n} [5]. Two vertices vij and vkl are adjacent if i k and j  l 1 or if j l and i  k 1 . A torus T (m, n) is a two dimensional mesh enhanced with wrap around edges. Vertices in T (m, n) is the set V {vij | 1 d i d m,1 d j d n} . Two vertices vij and vkl are adjacent in T (m, n) if (i) i k and j  l 1

j l and i  k 1 (iii) i k and j  l n  1 (iv) j l and i  k n  1 . (ii)

The edges which satisfy the first condition are referred to as horizontal edges, those which satisfy the second condition are referred to as vertical edges, while those which satisfy the third condition are called as horizontal wrap around edges and exactly those which satisfy the fourth condition are called as vertical wrap around edges. A one dimensional torus is a cycle.

Fig. 4: T (3,3)

5.1 Proposition The strong rainbow connection number of torus network

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(i) T (3,3)

2 ªnº (ii) T (3, n) « »  1, n t 4 «2» ªmº (iii) T (m,3) « »  1, m t 4 «2» Proof: Proof for (i) is straight forward. Proof for (ii)

ªnº

If n is even, the horizontal edges in the 3 rows are assigned with the colors 1,2,…, « » ,1,2,…, «2»

ªnº ªnº «« 2 »»  1 respectively from left to right. The m horizontal wrap around edges are assigned with a color «« 2 »» . The ªnº vertical edges in the n columns and the n vertical wrap around edges are assigned with a color « »  1 . The «2» above coloring defines a strong rainbow coloring ,because any two vertices are connected by at least one shortest rainbow path. If

ªnº

n is odd, the horizontal edges in the 3 rows are assigned with the colors 1,2,…, « » ,1,2,…, «2»

ªnº ªnº «« 2 »»  2 respectively from left to right. The m horizontal wrap around edges are assigned with a color «« 2 »»  1 . ªnº The vertical edges in the n columns and the n vertical wrap around edges are assigned with a color « »  1 . «2» ªnº Hence src (T (3, n)) « »  1, n t 4 . «2» Proof for (iii) follows from (ii). 5.2. Theorem For m > 3 and n > 3, the strong rainbow connection number of the torus network (a) src (T (m, n)) ª m  n º , if m and n are of different pairity «« 2 »» (b) src (T (m, n)) m  n , if both m and n are even 2 m § (c) src (T (m, n)) ¨  n ·¸  1, if both m and n are odd © 2 ¹ Proof: The torus network T (m, n) has m rows and n columns. To prove (a), we consider the following cases. Case(i) m is odd and n is even If m is odd and n is even, the edges can be colored as follows:

T (m, n) is

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ªnº

ªnº

(i)The horizontal edges in the m rows are assigned with the colors 1,2,…, « » ,1,2,…, « »  1 respectively from «2» «2» left to right.

ªnº

(ii)The m horizontal wrap around edges are assigned with a color « » . «2» (iii)The vertical edges in the n columns are

assigned

with

the

colors

ªm  nº ªnº ªm  nº ªnº ªnº ªnº « 2 »  1, « 2 »  2,..., « 2 », « 2 »  1, « 2 »  2,..., « 2 »  2 from top to bottom respectively. « » « » « » « » « » « »

ªm  nº  1. « 2 »»

(iv)The n vertical wrap around edges are assigned with the color «

We can see that any two vertices are connected by at least one shortest rainbow path. Hence the above coloring

ªm  nº » colors « 2 »

algorithm gives a strong rainbow edge coloring using « Case(ii) m is even and n is odd

ªm  nº » colors. « 2 »

If m is even and n is odd, the edges can be colored using the following coloring algorithm with « Coloring algorithm:

ªnº

ªnº

(i)The horizontal edges in the m rows are assigned with the colors 1,2,…, « » ,1,2,…, « »  2 respectively from «2» «2» left to right.

ªnº

(ii)The m horizontal wrap around edges are assigned with a color « »  1 . «2» (iii)The vertical edges in the n columns are

assigned

with

the

colors

ªm  nº ªnº ªm  nº ªnº ªnº ªnº « 2 »  1, « 2 »  2,..., « 2 », « 2 »  1, « 2 »  2,..., « 2 »  1 from top to bottom respectively. « » « » « » « » « » « »

ªm  nº . « 2 »» ªm  nº The above coloring algorithm gives a strong rainbow edge coloring using « » colors. « 2 » (iv)The n vertical wrap around edges are assigned with the color «

Now to prove (b) Coloring algorithm: (i)The horizontal edges in the m rows are assigned with the colors 1,2,…,

n n ,1,2,…,  1 respectively from left to 2 2

right. (ii)The m horizontal wrap around edges are assigned with a color (iii)The

vertical

edges

in

the

n

columns

n . 2 are

assigned

n n mn n n §m n· ,  1,  2,..., ¨  1,  2,..., ¸  1 from top to bottom respectively. 2 2 2 2 2 © 2 ¹

with

the

colors

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(iv)The n vertical wrap around edges are assigned with the color colors required for a strong rainbow edge coloring is To prove (c) Coloring algorithm:

mn colors. 2

mn . Therefore the minimum number of 2

ªnº

ªnº

(i)The horizontal edges in the m rows are assigned with the colors 1,2,…, « » ,1,2,…, « »  2 respectively from «2» «2» left to right.

ªnº

(ii)The m horizontal wrap around edges are assigned with a color « »  1 . «2» (iii)The vertical edges in the n columns are

assigned

with

the

colors

ªm  nº ªnº ªnº ªm  nº ªm  nº ªnº ªnº « 2 »  1, « 2 »  2,..., « 2 », « 2 »  1, « 2 »  1, «« 2 »»  2,..., «« 2 »»  1 from top to bottom respectively. « » » « » « « » « »

ªm  nº . The coloring algorithm defined above « 2 »»

(iv)The n vertical wrap around edges are assigned with the color «

ªm  nº +1 colors. « 2 »»

gives a strong rainbow edge coloring using « 6. Conclusion

In this paper the strong rainbow connection numbers of butterfly network BF(n) , Benes network BB(n) and torus network T (m, n) have been found in polynomial time. It would be interesting to find the strong rainbow connection numbers of interconnection networks such as star and pancake networks. 7. References 1. V.E Benes, Permutation groups, complexes and rearrangeable connecting network, The Bell system technical Journal 43(1964), p.1619-1640. 2. F. Cao. D. Z. Du, D. F. Hsu and P. Wan, Fault-tolerant routing in butterfly networks, Technical Report TR 95 – 073 , Department of Computer Science, University of Minnesota. (1995). 3. G.Chartrand, G.L. Johns,K.A. Mckeon and P.Zhang, Rainbow connection in graphs, Math. Bohemica 133(2008),p.85-98. 4. S.Chakraborty ,E.Fischer, A.Matsliah, R.Yuster, Hardness and algorithms for rainbow connection, Journal of Combinatorics Optimization, (2009),p.1-18. 5. L.H. Hsu and C.K.Lin, Graph Theory and interconnection networks, CRC Press, Inc. Boca Raton, FL, USA ©2008 6. S. Keshav. An Engineering Approach to Compter Networking. Addison-Wesley Publishers, 1997 7. F. T. Leighton. Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. Morgan Kaufmann Publishers, San Mateo, CA, 1992. 8. X. Li and and Y. Sun, Rainbow connections of graphs, Springer (2012). 9. P. Manuel, M.I. Abd-El-Barr, I. Rajasingh, B. Rajan, An Efficient representation of Benes Network and its Applications, Journal of Discrete Algorithms, 6(2008), p.11 – 19. 10. D. Nassimi, S. Sahni, “A Self-Routing Benes Network and Parallel Permutation Algorithms,” IEEE Transactions on Computers, 30(5),p.332-340 (1981) 11. B. Rajan, I. Rajasingh, P. Venugopal, Minimum Metric Dimension of Oriented Butterfly Network, Proc. of the fifth Asian Mathematical Conference, Malaysia, 2009. 12. S.Rujun, L.Changhong ,Y.Tianxing, On (d, 2)-dominating numbers of butterfly networks,Taiwanese Journal of Mathematics, 6(2002), no. 4, p. 515-521.

I. Annammal Arputhamary and M. Helda Mercy / Procedia Computer Science 57 (2015) 338 – 347 13. .K.M.Sivalingam and S. Subramaniam, Optical WDM Networks, Principles and Practice, Kluwer Academic publishers, 2000. 14. P. Vijaya Jyothi, B. Maheswari and I. Kelkar, 2-Domination Number of Butterfly Graphs, Chamchuri Journal of Mathematics, 1(2009) Number 1, p.73–79. 15. .A. William, A. Shanthakumari, Minimum cycle covers of butterfly and benes network, International journal mathematics and soft computing , 2(2012) No.1,p. 93 – 98

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