### Abstract

Foldin2 or topological compaction of array-based VLSI layouts is an important optimization step that is carried out after logic synthesis. In this paper, a new approach to two-dimensional multiple folding of array-based VLSI layouts is presented. From the specification of the problem a pair of intersection graphs is created. We show that any pair of interval graphs that contain the intersection graphs as spanning subgraphs corresponds to a set of feasible foldings. Next, a complete and exact characterization of the folding problem is presented. In particular, it is shown that the set of all feasible foldings associated with a given pair of interval graphs corresponds to the set of independent colorings of a pair of compatibility graphs. The compatibility graphs are derived from a pair of interval graphs that contain the intersection graphs as spanning subgraphs. Thus, minimizing the area of a layout is tantamount to finding a pair of compatibility graphs such that the product of their chromatic numbers is minimum. As important as minimizing the area of a layout is, the ability to rapidly generate compact layouts over a wide range of aspect ratios is often equally, if not more, important. The interval graph-based formulation of the folding problem permits a controlled and systematic generation of compact layouts with varying aspect ratios. Efficient and provably correct algorithms to generate compact layouts that have a given number of rows or a given number of columns within their minimum and maximum possible values are given. The basic theory and methods are extended to include I/O and other types of constraints. Finally, the results of experiments that were carried out on a large number of benchmark problems are given. These results are compared with those obtained by previously reported methods.

Original language | English (US) |
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Pages (from-to) | 1201-1222 |

Number of pages | 22 |

Journal | IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems |

Volume | 13 |

Issue number | 10 |

DOIs | |

State | Published - Oct 1994 |

Externally published | Yes |

### ASJC Scopus subject areas

- Software
- Computer Graphics and Computer-Aided Design
- Electrical and Electronic Engineering