### Faster minimization of linear wirelength for global placement

C.J. Alpert, T.F. Chan, A.B. Kahng, I.L. Markov, P. Mulet
1998 IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
A linear wirelength objective more e ectively captures timing, congestion, and other global placement considerations than a squared wirelength objective. The GORDIAN-L cell placement tool 16 minimizes linear wirelength by rst approximating the linear wirelength objective b y a modi ed squared wirelength objective, then executing the following loop 1 minimize the current objective to yield some approximate solution, and 2 use the resulting solution to construct a more accurate objective u n til
more » ... he solution converges. In this paper, we rst show that the GORDIAN-L loop can be viewed as a special case of a new algorithm that generalizes a 1937 iteration due to Weiszfeld 19 . Specically, w e formulate the Weiszfeld iteration using a regularization parameter to control the tradeo between convergence and solution accuracy; the GORDIAN-L iteration is equivalent to setting this regularization parameter to zero. Other novel numerical methods described in the paper, the Primal Newton iteration and the Primal-Dual Newton iteration, further improve upon the linearly convergent W eiszfeld iteration. Our Primal-Dual Newton iteration stably attains quadratic convergence, making it a superior choice for implementing a placer such a s G O R D I A N -L , o r f o r a n y linear wirelength optimization. P n j=1 aij, i . e . , qii is the degree of vertex vi. De nition: The n-dimensional placement vector x = xi corresponds to the physical locations of cells v1; : : : ; v n on the real line, i.e., xi is the coordinate of vertex vi. GORDIAN 9 uses a squared wirelength objective: