Minimum Tree Spanner Problem for Butterfly and Benes Networks Bharati Rajan, Indra Rajasingh, Department of Mathematics, Loyola College, Chennai 600 0, India Amutha A Department of Mathematics, Loyola College, Chennai 600 0, India E-mail: amutha_a@yahoo.co.in Paul Manuel Department of Information Science, Kuwait University, Kuwait 060 ABSTRACT The minimum tree spanner problem requires selecting a spanning tree of a fixed interconnection network that minimizes the cost of transmission between each pair of processors over the tree edges. In [8] we developed a technique to solve this problem for all parallel architectures including hypercube, CCC, wrapped butterfly, torus, star graphs which are classified under Cayley graphs. We introduced a new class of graphs called Diametrically Uniform Graphs and we provided a simple, efficient parallel algorithm to decide whether or not a parallel architecture is diametrically uniform. In this paper we consider the class of Butterfly and Benes networks, which are diametrically uniform and we solve the minimum tree spanner problem for this class. Keywords: Parallel architectures, Butterfly and Benes networks, t-spanner, tree t-spanner, minimum tree spanner.. Introduction The design of optimal communication and transportation networks, which satisfy a given set of requirements, has been studied extensively in the literature. We want to find a communication network among the vertices, where the communication delay is measured in terms of the length of a shortest path between the vertices. A desirable communication network is naturally one that minimizes the diameter. To keep the routing protocols simple, often the communication network is restricted to be a spanning tree. When communication takes place only between a specified collection of pairs of vertices, it is natural to minimize the maximum communication delay between vertices that need to communicate. In other words, it becomes necessary to find a spanning tree that connects all given vertices and satisfies their communication requirements with minimum communication links along the spanning tree. A similar situation arises in the field of parallel architecture. When algorithms are executed on a parallel computer, processors are often required to exchange information. It is well known that the overhead associated with this interprocessor communication is the major drawback of parallel computers in which processors are linked by an interconnection network. It is important to find ways to efficiently exchange messages through communication links. A common approach to implement communication algorithms on interconnection networks is to embed spanning trees with special properties on those networks []. The minimum tree spanner problem requires selecting a spanning tree of a network that minimizes the cost of transmission between any two sites over
the tree edges. A spanner ζ (T, G) of a spanning tree T of G is defined as ζ(t, G) = max {d T (u, v): (u, v) E(G)} where d T (u, v) denotes the distance between u and v in T. A minimum spanner ζ(g) of G is defined as ζ(g) = min {ζ(t, G): T is a spanning tree of G} A spanning tree T is called a minimum tree spanner, if ζ(t, G) = ζ(g). Equivalently T is a minimum tree spanner if ζ(t, G) ζ(t, G), for all spanning trees T of G. In other words, the minimum tree spanner problem of a graph G is to find a minimum tree spanner of G []. A polynomial time algorithm is available to solve this problem for digraphs [] and directed path graphs []. All the NPcomplete results on the tree t-spanner admissible problem [,, ] hold good for the minimum tree spanner problem. It is also shown that the minimum tree spanner problem is NP-complete for planar graphs []. In the literature the term t-spanner or tree t-spanner refers to a spanning sub graph or a spanning tree. We deviate from the standard notations for convenience. Throughout the paper by the term spanner we mean only the stretch factor and not the spanning subgraph or spanning tree. For each vertex u of a graph G, the maximum distance d(u, v) to any other vertex v of G is called its eccentricity and is denoted by ecc(u). In a graph G, the maximum value of eccentricity of vertices of G is called the diameter of G and is denoted by λ. Let G be a graph with diameter λ. A vertex v of G is said to be diametrically opposite to a vertex u of G, if d G (u, v) = λ. A graph G is said to be a diametrically uniform graph if every vertex of G has at least one diametrically opposite vertex. The set of diametrically opposite vertices of a vertex x in G is denoted by D(x). A graph G is diametrically uniform with λ = if and only if G is complete. Here we consider the diametrically uniform graphs with λ >. The parameter t of a tree t-spanner of a graph is always bounded by λ where λ is the diameter of the graph. The (, λ)- problem [6] is to find graphs with the maximal number of vertices with given constraints on the maximum degree and the diameter λ. A similar question arises in the context of minimum spanner. This problem is to find graphs with given constraints on the minimum spanner ζ(g) = λ or λ. We answer that the minimum spanner of a diametrically uniform graph is as large as twice of its diameter. In [8] we established sufficient conditions for diametrically uniform graphs to have the minimum spanner at least We identified several examples of diametrically uniform graphs and we studied the properties of those graphs. We also derived conditions under which the minimum spanner of diametrically uniform graphs is λ or λ. We devote the next section to recall the definitions and properties of the class of diametrically uniform graphs. We also list [8] a few important results on minimum spanner of diametrically uniform graphs.. Definitions and Properties of Diametrically Uniform Graphs Most of the well-known parallel architectures are diametrically uniform graphs. For example, hypercube, wrapped butterfly, torus and cycle are diametrically uniform graphs. An even Petersen graph P(n, ) is a diametrically uniform graph (Figure (a)) whereas an odd Petersen graph P(n+, ) is not diametrically uniform (Figure (b)).
0 8 0 8 6 6 0 0 8 8 (a) P(,) (b) P(,) Figure : Petersen Graphs Apart from these well-known graph families, one can construct any number of diametrically uniform graphs. See Figure. 6 6 such that (x*, y*) is an edge of G, then ζ(g) Theorem : Let G be a diametrically uniform graph with diameter λ >. If D(x) U D(y) is connected for every edge (x, y) of E(G), then ζ(g) Now we identify a few conditions under which the minimum spanner of a diametrically uniform graph is at most A subgraph H of a bipartite graph G would be a k-spanner of G if and only if it is a (k ) - spanner of G []. Thus we have the following result. Theorem : If G is a bipartite graph then ζ(g) is odd. In particular ζ(g) 8 6 0 Theorem 6: Let G be a diametrically uniform graph. If there exists a vertex x of G and a bfs tree BFS(x) rooted at x such that all the vertices of D(x) are in a subtree rooted at some vertex y (y x) in BFS(x), then ζ(g) Figure : A Diametrically Uniform Graph We list some of the results proved in [8]. The following theorem provides a necessary condition for tree (λ ) - spanner admissible graphs. Theorem : Let λ be the diameter of a graph G. If ζ(g) λ, then G is diametrically uniform. We now state [8] a few sufficient conditions for ζ(g) Theorem : Let G be graph such that G has a chordless cycle of length k+. Then ζ(g) k. Theorem : Let G be a diametrically uniform graph with diameter λ >. Given an edge (x, y) in E(G), if for every vertex x* of D(x) there exists a vertex y* of D(y) The following corollaries are direct applications of Theorem 6. Corollary : Let G be a diametrically uniform graph. If for some vertex x of G, D(x) is an independent set, then ζ(g) Corollary : Let G be a diametrically uniform graph. If D(x) is a singleton for some vertex x of G, then ζ(g). Tree Spanner Problem for Butterfly Networks The set of nodes V of an r-dimensional butterfly correspond to pairs [w, i], where i is the dimension or level of a node (0 i r) and w is an r-bit binary number that denotes the row of the node. Two nodes < w, i > and < w, i > are linked by an edge if and only if i = i + and either:. w and w are identical, or
. w and w differ in precisely the i th bit. The edges in the network are undirected. An r-dimensional butterfly is denoted by BF(r).. Tree Spanner Problem for Benes Networks An r-dimensional Benes network has r+ levels, each level with r nodes. The level zero to level r vertices in the network form an r-dimensional butterfly. The middle level of the Benes network is shared by these butterflies [0]. As butterfly is known for FFT, Benes is known for permutation routing (normal network). An r-dimensional Benes is denoted by B(r). Figure : A -dimensional Butterfly Network The proofs of the following Propositions are straightforward. Proposition : The r-dimensional Butterfly BF(r) is diametrically uniform. Proposition : Let G be BF(). Then D(x) U D(y) is connected for every edge (x, y) in E(G). Proposition : Let G be BF(r). Then for every edge (x, y) in E(G), there exists a vertex x* of D(x) and a vertex y* of D(y) such that (x*, y*) is an edge of G. Theorem : Let G be BF(). Then ζ(g) This follows from Propositions, and Theorem. Theorem 8: Let G be BF(r). Then ζ(g) This follows from Propositions, and Theorem. Theorem : The minimum spanner problem for Butterfly networks is polynomially solvable. Figure : A -dimensional Benes Network Proposition : The r-dimensional Benes network B(r) is diametrically uniform. Proposition : Let G be B(). Then D(x) U D(y) is connected for every edge (x, y) in E(G). Proposition 6: Let G be B(r). Then for every edge (x, y) in E(G), there exists a vertex x* of D(x) and a vertex y* of D(y) such that (x*, y*) is an edge of G. Theorem 0: Let G be B(). Then ζ(g) This follows from Propositions, and Theorem. Theorem : Let G be B(r). Then ζ(g) This follows from Propositions, 6 and Theorem. Theorem : The minimum spanner problem for Benes networks is polynomially solvable.
. Conclusion Using the techniques developed in [8], we have solved the minimum tree spanner problem for Butterfly and Benes networks. The problem is worth considering for other architectures like Pancake and shuffle exchange graphs. References: [] L. Cai and D.G. Corneil Tree Spanners, SIAM J. Discrete Math., 8, (), pp. -8. [] L. Cai and D.G. Corneil, Isomorphic tree spanner problems, Algorithmica,, (), pp. 8-. [] S.P. Fekete and J. Kremer, Tree Spanners in Planar Graphs, th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 8) Smolenice-Castle/Slovakia, June 8-0, 8. [] Fragopoulou, P. and S.G. Akl, Optimal communication algorithms on star graphs using spanning tree constructions, J. Parallel Distrib. Computing, Vol.,, pp.-. [] M.C. Heydemann, J.G. Peters and D. Sotteau, Spanners of hypercubederived networks, SIAM J. Discrete Math., (6), pp. -. [6] M.C. Heydemann, Cayley graphs and interconnection networks, In Graph Symmetry, eds. G. Hahn and G. Sabidussi, Kluwer Academic Publishers, The Netherlands, (), pp. 6-. [] M.S. Madanlal, G. Venkatesan and C. Pandu Rangan, Tree -spanners on interval, Permutation and regular bipartite graphs, Information Processing Letters, (6), pp. -0. [8] Paul Manuel, Bharati Rajan, Indra Rajasingh and Amutha Alaguvel, Tree Spanners, Cayley Graphs and Diametrically Uniform Graphs LNCS 880, (00), pp -. [] G. Venkatesan, U. Rotics, M.S.Madanlal, J.A. Makowski, C. Pandu Rangan, Restrictions of minimum spanner problem, Information and Computation, 6, (), pp. -6. [0] J. Xu, Topological Structure and Analysis of Interconnection Networks, Kluwer Academic Publishers, 00