Graph-theory-based simplex algorithm for VLSI layout spacing problems with multiple variable constraints

An efficient algorithm is provided for solving a class of linear programming problems containing a large set of distance constraints of the form x_i-x_j⩾k and a small set of multivariable constraints of forms other than x_i-x_j⩾k. This class of linear programming formulation is applicable to very large scale integration (VLSI) layout spacing problems, including hierarchy-preserving hierarchical layout compaction, layout compaction with symmetric constraints, layout compaction with attractive and repulsive constraints, performance-driven layout compaction, etc. The longest path algorithm is efficient for solving spacing problems containing only distance constraints. However, it fails to solve problems that involve multiple-variable constraints. The linear programming formulation of a spacing problem requires use of the simplex method, which involves many matrix operations. This can be very time consuming when handling huge constraints systems derived from VLSI layouts. Herein it is found that most of the matrix operations can be replaced with fewer and faster graph operations, creating a more efficient graph-theory-based algorithm. Theoretical analysis shows that the proposed algorithm reduces the computation complexity of the simplex method