Connectivity Models for Optoelectronic Computing Systems

Abstract

Abstract. Rent's rule and related concepts of connectivity such a s d imensionality, line-length distributions, and separators have found great use in fundamental studies of di erent i n terconnection media, including superconductors and optics, as well as the study of optoelectronic computing systems. In this paper generalizations for systems for which t h e 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 two-or 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 RuleThe 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 c o n texts 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 o f circuits such a s R e n t'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 i s n o t constant throughout the interconnection hierarchy, and present a n umb e r o f r elated 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 interconnected systems and the role of Rent's rule in fundamental studies of different i n terconnection 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

Information flow and interconnections in computing: extensions and applications of Rent's rule

“…(5) by working up the hierarchy [49]. (Readers wishing to skip at first reading this reverse derivation given below may move directly to the paragraph following Eq.…”

“…In the remainder of this section, we discuss two further results regarding the calculation of the average and total connection lengths of a layout [49]. Readers wishing to skip these at a first reading may directly go to the next section.…”

“…(5) may be used directly for the range 1 <Ñ < 1000, and with a shift of origin for the range 1000 <Ñ < 100 × 1000. Reproduced with permission from [45,49].…”

“…The author finds the following viewpoint especially illuminating [49]. At the microscopic level, all information processing involves the distributed manipulation and backand-forth transfer of pieces of information.…”

“…The dimensionality of graphs corresponding to different levels for a hypothetical system with four levels. Reproduced with permission from [49].…”