Exact wirelength of hypercubes on a grid

• Published in: Discrete Applied Mathematics Vol. 157; no. 7; pp. 1486 - 1495
• Main Authors: Manuel, Paul, Rajasingh, Indra, Rajan, Bharati, Mercy, Helda
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 06.04.2009; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Congestion; Designs and configurations; Dilation; Embedding; Exact sciences and technology; Fixed interconnection parallel architecture; Grids; Hypercubes; Mathematical analysis; Mathematics; Operator theory; Sciences and techniques of general use; Theoretical computing; VLSI layout; Wirelength; Grids; VLSI layout; Fixed interconnection parallel architecture; Dilation; Hypercubes; Embedding; Congestion; Wirelength; Partition; Addition; Grid pattern; Computer theory; VLSI; Grid; Parallel architectures; Optimization; Design; Simulation; Layout; Interconnection; Hypercube; Combinatorics
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2008.09.013

Grid embeddings are used not only to study the simulation capabilities of a parallel architecture but also to design its VLSI layout. In addition to dilation and congestion, wirelength is an important measure of an embedding. There are very few papers in the literature which provide the exact wirelength of grid embedding. As far as the most versatile architecture hypercube is concerned, only approximate estimates of the wirelength of grid embedding are available. In this paper, we give an exact formula of minimum wirelength of hypercube layout into grids and thereby we solve completely the wirelength problem of hypercubes into grids. We introduce a new technique to estimate the wirelength of a grid embedding. This new technique is based on a Congestion Lemma and a Partition Lemma which we study in this paper.