Hypercrossing Number: A New and E ective Cost Function for Cell Placement Optimization

Traditionally, the algorithms that optimize placement of cells in a VLSI layout rely on a cost function that includes a distance metric. The choice of a cost function may limit the choices in devising new, more e cient and more e ective algorithms. The metric we propose relies on the interval graph of net nodes, induced by each speci c placement of cells into a row. Nominally, the metric associated with this graph is the chromatic number ( ). The new cost function, proposed in this paper, is the hypercrossing number (HC), de ned by the cardinality of the edge-set of the interval graph. This metric relates to the edge crossing number (EC) in the corresponding bipartite graph, depends on cell and net order only, and is independent of the target technology. Currently, there is no explicit algorithm to place cells such that the hypercrossing number is minimal, but edge crossing minimization algorithms are available. We demonstrate, experimentally and with statistical signi cance, the following results for the linear placement problem: (1) cost functions based on distance metric and technology-speci c wire length measured after routing have a positive linear correlation with the hypercrossing number (with a coe cient of correlation nearly 100%); (2) total wiring area, measured after routing, is highly correlated with the product of the hypercrossing and chromatic number; (3) the edge crossing number in the corresponding bipartite graph model is highly correlated with the hypercrossing number; (4) relatively simple heuristic algorithms that minimize the edge crossing number in bipartite graphs also e ectively minimize the hypercrossing number, producing linear placements of quality that can be better then or comparable to state-of-the-art placement algorithms { but at a much lower cost of computation.