GORDIAN

The idea

GORDIAN keeps PROUD's spring mathematics but disciplines the collapse problem with constraints instead of pure recursion. Cells are assigned (by partitioning) to regions; each region contributes a linear constraint fixing the center of mass of its cells to the region's center. Minimizing the global quadratic objective subject to these constraints is still a linear-algebra problem (the constraints eliminate variables), and crucially the optimization stays global: every cell still feels every spring, unlike pure divide-and-conquer where regions go blind to each other. As partitioning refines regions, constraints multiply, and the continuous placement gradually disciplines itself into spread-out form; a final step converts region populations into a legal arrangement via slicing-style area assignment, with linear wirelength re-optimization (GORDIAN-L) as a refinement.

Why it is the reference point

The center-of-mass constraint is the cleanest early answer to analytical placement's central question, how to reconcile smooth wirelength with non-overlap, and the solve-partition-constrain-resolve loop became the canonical architecture: BonnPlace and a generation of quadratic placers are recognizably GORDIAN-shaped. Its separation of concerns (global numerics for wirelength, combinatorial assignment for area) is the same division modern placers keep, with density penalties replacing hard constraints.

Where you meet it today

In every lecture as the quadratic placement algorithm, and in the architecture of successors: partitioning-constrained quadratic placement remained industrially competitive well into the 2000s and its skeleton is visible inside FastPlace, SimPL-class placers, and the warm-start phases of current tools.