Connectivity Models for Optoelectronic Computing Systems

Rent's rule and related concepts of connectivity such as dimensionality, line-length distributions, and separators have found great use in fundamental studies of di erent interconnection media, including superconductors and optics, as well as the study of optoelectronic computing systems. In this paper generalizations for systems for which the Rent exponent is not constan tthroughout the interconnection hierarchy are pro vided. The origin of Ren t's rule is stressed as resulting from the embedding of a high-dimensional information ow graph to twoor three-dimensional physical space. The applicability of these traditionally solid-wire-based concepts to free-space optically interconnected systems is discussed. 1 Connectivity ,Dimensionality ,and Rent's Rule The importance of wiring models has long been recognized and they have been used not only for design purposes but also for the fundamental study of interconnections and communication in computing. A central and ubiquitous concept appearing in such contexts is the connectivity of a circuit graph or computer netw ork. Connectedness has always been a central concept in mathematical graph theory [1{3], whose extensions play a central role in graph layout [4]. The purpose of these concepts is to quantify the communication requirements in computer circuits. This paper aims to discuss several concepts related to the connectivity of circuits (such as Rent's rule, dimensionality, line-length distributions, and separators), provide certain extensions, and brie y discuss some of their applications. We rst discuss generalizations for systems for which the Rent exponent is not constant throughout the interconnection hierarchy, and present a number of related results. Special emphasis is given to the role of discontinuities and the origin of Rent's rule. The applicability of these concepts to free-space optically in terconnected systems and the role of Rent's rule in fundamental studies of different interconnection media, including superconductors and optics, are brie y reviewed. Graph layout deals with the problem of how to situate the nodes and edges of an abstract graph in physical space. Optimal graph layout [5] is in general an NP-complete problem [6]. However, if a hierarc hical decompositionof a graph is pro vided, this graph can be laid out following relatively simple algorithms. A J. Rolim et al. (Eds.): IPDPS 2000 Workshops, LNCS 1800, pp. 1072-1088, 2000.  Springer-Verlag Berlin Heidelberg 2000 hierarchical decomposition of a graph consisting of N nodes and the associated decomposition function k( N) are obtained as follows: First, we remove k(N) edges in order to disconnect the graph into 2 subgraphs, each of approximately N=2 nodes. Roughly speaking, we try to do this by removing as few edges as possible. We repeat this procedure for the subgraphs thus created. The subgraphs will in general require di ering numbers of edges to be removed from them to be disconnected into subsubgraphs of ' N=2 nodes each. We denote the largest of these numbers as k(N=2). Continuing in this manner until the subgraphs consist of a single node each, we obtain the function k( N), the (worst case) number of edges removed during decomposition of subgraphs of N nodes. Once such a decomposition is found, it is possible to lay out the graph in the intuitively obvious manner by working upwards [7, 4, 6]. Whereas one can always nd some such decomposition, nding the decomposition that leads to a layout with some optimal property (such as minimal area) is not a trivial problem. We will assume that we agree on a particular decomposition obtained by some heuristic method. Now, let us de ne the connectivity p( N) and dimensionality n( N) associated with the hierarchical level of a decomposition involving subgraphs of N nodes by [8]