Universality considerations in VLSI circuits

Abstract

The problem of embedding the interconnection pattern of a circuit into a two-dimensional surface of minimal area is discussed. Since even for some natural patterns graphs containing <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> connections may require Ω( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) area, in order to achieve compact embeddings restricted classes of graphs have to be considered. For example, arbitrary trees (of bounded degree) can be embedded in linear area without edges crossing over. Planar graphs can be embedded efficiently only if crossovers are allowed in the embedding.