CONSTRUCTING EDGE-DISJOINT SPANNING TREES IN AUGMENTED CUBES
arXiv:1706.05215v1 [math.CO] 16 Jun 2017
S. A. Mane Center for Advanced Studies in Mathematics, Department of Mathematics, Savitribai Phule Pune University, Pune-411007, India.
[email protected] Abstract. Let T1 , T2 , ......., Tk be spanning trees in a graph G. If for any pair of vertices {u, v} of G, the paths between u and v in every Ti ( 1 ≤ i ≤ k) do not contain common edges then T1 , T2 , ......., Tk are called edge-disjoint spanning trees in G. The design of multiple edge-disjoint spanning trees has applications to the reliable communication protocols. The n−dimensional augmented cube, denoted as AQn , a variation of the hypercube, possesses some properties superior to those of the hypercube. For AQn (n ≥ 3), construction of n − 1 edge-disjoint spanning trees is given the result is optimal with respect to the number of edge-disjoint spanning trees.
Keywords: edge-disjoint spanning trees, Augmented cubes 1. Introduction A graph G is a triple consisting of a vertex set V (G), an edge set E(G), and a relation that associates with each edge two vertices called its endpoints[12]. The topology of the network is modeled in the form of a graph whose vertices correspond to nodes, while edges represent direct physical connections between nodes. This paper deals with the well-established problem of handling the maximum possible number of communication requests without using a single physical link more than once, known as the edge-disjoint spanning trees Problem. The hypercubes Qn are one of the most versatile and efficient interconnection networks discovered for parallel computation. Many variants of the hypercube have been proposed. The augmented cube, proposed by Choudam and Sunitha[2], is one of such variations. An n−dimensional augmented cube AQn can be formed as an extension of Qn by adding some links. For any positive integer n, AQn is a vertex transitive, (2n−1)-regular and (2n − 1)-connected graph with 2n vertices. AQn retains all favorable properties of Qn since Qn ⊂ AQn . Moreover, AQn possesses some embedding properties that Qn does not. The main merit of augmented cubes is that their diameters are about half of those of the corresponding hypercubes. A tree T is called a spanning tree of a graph G if V (T ) = V (G). Two spanning trees T1 and T2 in G are edge-disjoint if E(T1 ) ∩ E(T2 ) = φ. The edge-disjoint spanning trees(EDSTs for short) problem has received a great deal of attention in recent years because of its numerous applications on interconnection networks such as fault-tolerant broad casting and secure message distribution[3, 4, 5, 6, 7, 8, 9, 10, 13]. Barden, Hadas, Davis and Williams[1] proved there exist n EDSTs in a hypercube of dimension 2n and provided a construction for obtaining the maximum number of EDSTs in a hyperube. Wang, Shen and Fan[11] proved existence of n − 1 EDSTs in an augmented cube AQn and they asked, “how to derive an effective algorithm that constructs edgedisjoint spanning trees based on our algorithm, in an augmented cube?” Motivated by this 1
CONSTRUCTING EDGE-DISJOINT SPANNING TREES IN AUGMENTED CUBES
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question, we provide the construction for obtaining the maximum number of EDSTs (n − 1 EDSTs ) in augmented cube AQn ( n ≥ 3 ).
2. Preliminaries The definition of the n−dimensional augmented cube is stated as the following. Let n ≥ 1 be an integer. The n-dimensional augmented cube, denoted by AQn , is a graph with 2n vertices, and each vertex u can be distinctly labeled by an n-bit binary string, u = u1 u2 ....un . AQ1 is the graph K2 with vertex set {0, 1}. For n ≥ 2, AQn can be recursively constructed by two copies of AQn−1 , denoted by AQ0n−1 and AQ1n−1 , and by adding 2n edges between AQ0n−1 and AQ1n−1 as follows: Let V (AQ0n−1 ) = {0u2 ....un : ui ∈ {0, 1}, 2 ≤ i ≤ n} and V (AQ1n−1 ) = {1v2 ....vn : vi ∈ {0, 1}, 2 ≤ i ≤ n}. A vertex u = 0u2 ....un of AQ0n−1 is joined to a vertex v = 1v2 ....vn of AQ1n−1 if and only if for every i, 2 ≤ i ≤ n either 1. ui = vi ; in this case an edge hu, vi is called a hypercube edge and we say v = uh , or 2. ui = vi ; in this case an edge hu, vi is called a complement edge and we say v = uc . Let Enh = {hu, uh i : u ∈ V (AQ0n−1 )} and Enc = {hu, uc i : u ∈ V (AQ0n−1 )}. See Fig.1.
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For undefined terminology and notations see [12].
3. Construction of edge-disjoint spanning trees in augmented cubes Our proof is by induction. As AQn is (2n − 1)-regular, |E(AQn )| = (2n − 1)(2n−1 ) = n2n − 2n−1 . When we construct any spanning tree on 2n vertices of AQn we need exactly 2n −1 edges hence we can construct at most n − 1 EDSTs. Still, for n ≥ 3, 2n−1 + n − 1(< 2n − 1) number of edges remain uncovered by these n − 1 EDSTs, but by our method, we are able to construct the tree on 2n−1 + n vertices containing uncovered edges. Theorem 3.1. Let n ≥ 3 be an integer. There exist n − 1 edge-disjoint spanning trees in augmented cube AQn . Proof. First if n = 3, we construct two EDSTs T1 and T2 as follows. See Fig.2.
CONSTRUCTING EDGE-DISJOINT SPANNING TREES IN AUGMENTED CUBES
000q
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Fig.2(a). Spanning tree T1 in AQ3
Edges uncovered by T1 and T2 again form a tree say T3 on 7 vertices. See Fig.3. 000 r
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Fig.3. Tree T3 in AQ3 Let AQn+1 be decomposed into two augmented cubes say AQ0n and AQ1n with vertex set say {u0i , vi0 : 1 ≤ i ≤ 2n−1 } and {u1i , vi1 : 1 ≤ i ≤ 2n−1 } respectively. Denote by 0 T10 , T20 , ......., T(n−1) the EDSTs in AQ0n . Let the identical corresponding EDSTs in AQ1n 1 be denoted by T11 , T21 , ......., T(n−1) . Let the vertices of the tree say Tn0 which is made up of uncovered edges of n − 1 EDSTs in AQ0n be denoted by u01 , u02 , ....u02n−1 , v10 , v20 , ....vn0 and the identical corresponding vertices of the corresponding tree say Tn1 in AQ1n be denoted by u11 , u12 , ....u12n−1 , v11 , v21 , ....vn1 . We want to construct n EDSTs in AQn+1 , of which first n − 2 EDSTs are constructed from Ti0 and Ti1 by adding a hypercube edge hvi0 , vi1 i ∈ E(AQn+1 ) to connect two internal vertices vi0 ∈ V (Ti0 ) and vi1 ∈ V (Ti1 ), for 1 ≤ i ≤ n − 2. See Fig.4.
T10 T20
0 Tn−2
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Fig.4.
T11 T21
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0 c The (n − 1)th EDST is constructed from T(n−1) by adding all complement edges En+1 = c 0 {hu, u i : u ∈ V (AQn )}.See Fig.5.
q101
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CONSTRUCTING EDGE-DISJOINT SPANNING TREES IN AUGMENTED CUBES
q 10 )c (v
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AQ1n
1 The nth EDST is constructed from T(n−1) by adding to it hypercube edges hvi0 , vi1 i ∈ 1 E(AQn+1 ) to connect internal vertices vi0 ∈ V (AQ0n ) and vi1 ∈ V (T(n−1) ) n ≤ i ≤ 2n−1 . 1 via hypercube edges The 2n−1 + (n − 1) vertices in AQ0n which are not connected to T(n−1) 1 1 0 0 0 0 n−1 0 , by are ui (1 ≤ i ≤ 2 ) and v1 , v2 , ....vn−1 , but the edge hvn , vn i is included in T(n−1) 0 1 0 adding to T(n−1) the edges of the tree Tn through the vertex vn connect all remaining ver0 1 tices v10 , v20 , ....vn−1 and u0i (1 ≤ i ≤ 2n−1 ) to T(n−1) . See Fig.6.
vn1♣
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Still, we have uncovered edges of AQn+1 namely edges of tree Tn1 , the hypercube edge 0 1 i and hypercube edges hu0 , u1 i (1 ≤ i ≤ 2n−1 ). We can easily observe that these hvn−1 , vn−1 i i uncovered hypercube edges along with the tree Tn1 again form a new tree on (2n−1 + n) + (2n−1 + 1) = 2n + n + 1 vertices. See Fig.7.
1 vn−1 ♣
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u12n−1 Fig.7
Tn1
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Acknowledgment: The author gratefully acknowledges the Department of Science and Technology, New Delhi, India for the award of Women Scientist Scheme for research in Basic/Applied Sciences. References [1] B. Barden, R. Libeskind-Hadas, J. Davis, W. Williams, On edge-disjoint spanning trees in hypercubes, Inform. Process. Lett. 70(1)(1999)13 − 16. [2] S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40(2)(2002)71 − 84. [3] P. Fragopoulou, S.G. Akl, Edge-disjoint spanning trees on the star network with applications to fault tolerance, IEEE Trans. Comput. 45(2)(1996)174 − 185. [4] S.L. Johnsson, C.T. Ho, Optimal broadcasting and personalized communication in hypercubes, IEEE Transactions on Computing 38(9)(1989)1249 − 1268. [5] R. Libeskind-Hadas, D. Mazzoni, R. Rajagopalan, Tree-based multicasting in wormhole-routed irregular topologies, in: Proc. Merged 12th Internat. Parallel Processing Symp. and the 9th Symp. on Parallel and Distributed Processing, April 1998, pp.244 − 249. [6] S.-Y. Hsieh, C.-J. Tu, Constructing edge-disjoint spanning trees in locally twisted cubes, Theoret. Comput. Sci. 410(810)(2009)926 − 932. [7] J.-C. Lin, J.-S. Yang, C.-C. Hsu, J.-M. Chang, Independent spanning trees vs. edge-disjoint spanning trees in locally twisted cubes, Inform. Process. Lett. 110(10)(2010)414 − 419. [8] L. Ni, P. McKinley, A survey of wormhole routing techniques in direct networks. IEEE Comput. 26(2)(1993)62 − 76. [9] S.M. Tang, Y.L. Wang, Y.H. Leu, Optimal independent spanning trees on hypercubes, Journal of Information Science and Engineering 20(2004)143 − 155. [10] H. Wang, D. Blough, Tree-based multicast in wormholerouted torus networks, in: Proc. Internat. Conf. on Parallel and Distributed Processing Techniques and Applications, July 1998. [11] Yan Wang, Hong Shen, Jianxi Fan, Edge-independent spanning trees in augmented cubes. Theoret. Comput. Sci. 670(2017), 23 − 32. [12] Douglas. B. West, Introduction to Graph theory, Second Edition, Prentice-Hall of India, New Delhi, 2002. [13] J.S. Yang, S.M. Tang, J.M. Chang, Y.L. Wang, Parallel construction of optimal independent spanning trees on hypercubes, Parallel Computing 33(2007)73 − 79.
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