Graphs on Alphabets as Models for Large Interconnection Networks

Gbmez, J., M.A. Fiol and J.L.A. Yebra, Graphs on alphabets as models foi large interconnection networks, Discrete Applied Mathematics 37138 (1992) 227-243. Graphs on alphabets are constructed by labelling the vertices with words on a given alphabet, and specifying a rule that relates pairs of different words to define the edges. They have proved to be quite suitable to model large interconnection networks since their structure usually provides efficient routing algorithms. The aim of this paper is to present several infinite families of such graphs with a large number of vertices for given values of their diameter and maximum degree. The advent of very large scale integrated (VLSI) circuit technology has enabled the construction of very complex interconnection networks. By most accounts, the next generation of supercomputers will achieve its gains by increasing the number of processing elements (PE), rather than by using faster processors. * In these computers the time of interprocessor data communication may-and usually shall-overcome the actual computation time. Hence the interest is studying the design of interconnection networks which should consider several factors, as for instance layout, transmission delay and traffic density, reliability or fault tolerance, existence of efficient routing algorithms, cost, etc. For a survey about interconnection networks, see for instance the paper of Feng [ 151. In this paper the concern is with the network topology. More precisely, we focus on two constraints inherent in such networks: the transmission delay should be small and each PE can be connected by links to just a few others. 0166-218X/92/$05.00 (Cl 1992-Elsevier Science Publishm B.V. All rights reserved 228 J. Gdrttez et al. It is well known that interconnection networks can be modeled by graphs. In our case, the vertices of the graph represent the PEs or nodes of the network and the edges represent the links between them. The distance between two vertices then represents the delay encountered in shortest path communication between the corresponding nodes, while the diameter of the graph measures the maximum possible delay. The degree of a vertex is the number of vertices it is connected to. For an account of results about graphs and interconnection networks, we refer the reader to the survey of Bermond, Bond, Paoli and Peyrat [3]. The graphs thus obtained can be directed or undirected, depending upon the links from each PE are used just for input or output, or for both. We are concerned here with undirected graphs only. Our aim is to propose several families of large graphs (i.e., graphs with a large number of vertices for given values of its degree and diameter) that have efficient routing algorithms. This last requirement puts aside several interesting constructions of large graphs described by Bermond, Norme and Quisquater in [4]. In their classification the graphs presented here are graphs on alphabets, since each vertex of the graph may be thought of as a word on a given alphabet. The organization of the paper is as follows. In the next section we state the basic concepts of graph theory related to the topology of networks. Then the best already known families of large graphs on alphabets are recalled. Sections 3,4 and 5 contain our new proposals and Section 6 sums up the best families of such graphs known to date. 2. Basic concepts and known results A graph G = ( V, E) consists of a set V of vertices and a set E of edges that join the vertices of I’. The number of vertices N= 1 VI is the order of the graph. If (x,y) is an edge of E, it is said that x and y are adjacent, and it is usually written -y-y. The degree of a vertex 6(x) is the number of vertices adjacent to x, and its maximum value over I/ is the degree of G, A =d(G)=max(S(x): XE V}. If S(x)=d for all XE V, it is said that the graph is regular of degree A. The distance between two vertices x and y, d(x,y), is the length of a shortest path between x and y, and its maximum value D= max(d(x, y): x,y~ V) is the diameter of the graph. For the definitions not given here we refer the reader to [8]. The order of a graph with maximum degree d ~3 and diameter D is easily seen to be bounded by Nrl+d+d(d-l)+.*.+d(d-l)D-‘= &l-1+2 A-2 . (2-l) The right-hand side is called the Moore bound, and it is known that when D# 1 it can only be attained for D = 2 and A = 3,7 or possibly 57, see Bannai and Ito [2] or Damerell [9]. Hence the interest in finding graphs which have a large number of vertices, as close as possible to the Moore bound. Large interconnection networks 229 Besides de Bruijn and Kautz digraphs, one of the first historic examples of large graphs on alphabets must be Akers graphs, also known as odd graphs, see [ 11. They are defined only for A = D + 1, with each vertex represented by a 24 1 length sequence of A 1 O’s and A l’s, and where each vertex is adjacent to all vertices that have just a common 1 with it. Akers graphs are regular of degree A, diameter D=A 1 and order N=(24,-i). In most cases such graphs have less order than the graphs of the following sections. In fact Akers graphs belong to a more genera1 family of graphs on alphabets known as Kneser graphs [25]. These graphs have as vertices the r-subsets of a l-set, tr2r+ 1, and two vertices are adjacent if the corresponding sets are disjoint. Another example of large graphs on alphabets is the family of trivalent (i.e., A = 3) graphs proposed by Leland and Solomon in [26]. They are defined on the vertex set V=Xk, where X= Z2 is the set of integers modulo 2, by the adjacency rules X2X3.. . X,X,, x1x2... Xk-IXk%kxl . ..xk-zxk-1. Xl . ..Xk_&-$k (.& =xi + 1 mod 2). It is shown that the diameter is upper bounded by L3k/2J so that their order satisfies N= 2k 1 22D’3. The first known infinite families of large (A, D) graphs, for even A and any D, were derived in a trivial way from the well-known families of de Bruijn digraphs [lo] and Kautz digraphs [23,24]. It suffices to leave out the orientation of the arcs and remove the possible loops and parallel edges, that is, to take their underlying graphs. Thus, de Bruijn graphs UB(d,k) have vertex set X”, 1x1 =d, adjacency conditions -&+-xkxk+!, xk+, EX, x1x2... xk-lxkxoxj . . . xk-$k-1, XOEX (2.2a) (2.2b) degree A = 2d (if k L 3) and diameter D =Lk. Therefore, their number of vertices in terms of A and D is A D N= z l 0 (2.3) These graphs were generalized by Delorme and Farhi in [ 131. They are also used by Jerrum and Skyum [22] to obtain the largest known graphs for fixed degree and very large diameter by substituting a graph with small average distance for each vertex of the de Bruijn graphs. The Kautz graph bi<(d, k) is the subgraph of the de Bruijn graph UB(d + 1, k) obtained by considering only the vertices represented by words whose consecutive letters (elements of X) are different: xi,1 #Xi, Isirk-1. For k23 and dr2, Kautz