Faster minimization of linear wirelength for global placement

Abstract

A linear wirelength objective more effectively captures timing, congestion, and other global placement considerations than a squared wirelength objective. The GORDIAN-L cell placement tool [19] minimizes linear wirelength by first approximating the linear wirelength objective by a modified 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-until the solution converges. This paper shows how to apply a generalization [5], [6] of a 1937 algorithm due to Weiszfeld [22] to placement with a linear wirelength objective and that the main GORDIAN-L loop is actually a special case of this algorithm. We then propose applying a regularization parameter to the generalized Weiszfeld algorithm to control the tradeoff between convergence and solution accuracy; the GORDIAN-L iteration is equivalent to setting this regularization parameter to zero. We also apply novel numerical methods, such as the primal-Newton and primal-dual Newton iterations, to optimize the linear wirelength objective. Finally, we show both theoretically and empirically that the primal-dual Newton iteration stably attains quadratic convergence, while the generalized Weiszfeld iteration is linear convergent. Hence, primal-dual Newton is a superior choice for implementing a placer such as GORDIAN-L, or for any linear wirelength optimization.Index Terms-Analytic placement, GORDIAN-L, interconnect delay, linear placement, linear wirelength, Newton, primal dual, quadratic placement, Weiszfeld. include the design of efficient computational algorithms for large scale scientific computing (e.g., multigrid and domain decomposition algorithms, iterative methods, Krylov subspace methods, and parallel algorithms), VLSI circuit placement, and PDE methods for image processing.

Efficient optimization by modifying the objective function: applications to timing-driven VLSI layout

“…Since the exact wiring of nets is unknown until routing occurs, wirelength estimations are used. For instance, one can solve the problem (1) where is the cell location vector and each nonnegative number weights the importance of keeping cells and in close proximity (when cells and are not connected, ). The matrix represents generic linear placement constraints via the linear system of equations .…”

“…W E CONSIDER minimization of convex and continuous objective functions that are differentiable almost everywhere, except for points where directional derivatives disagree, e.g., functions involving absolute values. Examples in VLSI analytic placement include wirelength [1], [28], and delay [13], [14], [29], [27], both of which depend on the absolute value of cell-to-cell distances. Examples abound in other applications, e.g., multifacility location [6], [11], [20] and de-noising in image processing.…”

“…Classic-minimization algorithms, e.g., Newton methods and variants [23], [16], [35], assume differentiability and are inapplicable if optima occur at or near points of nondifferentiability. 1 Therefore, problem-specific algorithms have been developed. In several works [1], [11], [20], nondifferentiability has been addressed by function regularization, i.e., removing nondifferentiabilities without significantly changing the set of minimizers.…”

“…1 Therefore, problem-specific algorithms have been developed. In several works [1], [11], [20], nondifferentiability has been addressed by function regularization, i.e., removing nondifferentiabilities without significantly changing the set of minimizers. The main benefit of a successful regularization is that Newton-type methods become applicable.…”

“…This paper proposes new generalized approaches to construct regularizations for given objectives including arbitrary convex piecewise-linear functions that are widely used in synthesis and analysis of electrical networks [19]. The described methods extend those applied to analytical placement in VLSI layout in [1], where the regularized objective lead to a new interpretation of the well-known heuristic GORDIAN-L [28] and suggested a faster algorithm. Comparable regularizations of linear wirelength and path-delay based objectives cannot be produced by previous approaches.…”