On VLSI interconnect optimization and linear ordering problem

Some well-known VLSI interconnect optimizations problems for timing, power and cross-coupling noise immunity share a property that enables mapping them into a specialized Linear Ordering Problem (LOP). Unlike the general LOP problem which is NP-complete, this paper proves that the specialized one has a closed-form solution. Let f (x, y) :R2 →R be symmetric, non-negative, defined for x ≥ 0 and y ≥ 0, and let f (x, y) be twice differentiable, satisfying ∂2f (x, y)/∂x∂y < 0. Let π be a permutation of {1, . . . , n}. The specialized LOP comprises n objects, each associated with a real value parameter ri , 1 ≤ i ≤ n, and a cost f (ri, rj ) associated to any two objects if |π(i)−π(j)| = 1,1 ≤ i, j ≤ n, and f (ri, rj )= 0 otherwise. We show that the permutation π which minimizes ∑n−1 i=1 f (rπ−1(i), rπ−1(i+1)), called “symmetric hill”, is determined upfront by the relations between the parameter values ri .