Toward Near-Optimal Floorplanning with Complex Design Constraints for Modern SoCs

1 Introduction

• Modern evaluations require integrated handling of coupled design constraints alongside wirelength and area optimization.These constraints are fundamental to design validity and engineering feasibility.Source: §1 · p. 1
• Existing analytical methods primarily address overlap removal and wirelength, while soft penalties for coupled constraints can trap solvers in poor local optima.Subsequent legalization may then struggle to achieve strict feasibility without severe quality degradation.Source: §1 · pp. 1–2
• 90% lower HPWL and 16% lower area on average than state-of-the-art methods, with all constraints satisfied and the fastest runtime.Wirelength and area are also 2% and 1% larger than ground-truth references, respectively.Source: §1 · p. 2
• The paper introduces an end-to-end analytical framework combining SDP, ALM, and SOCP to handle complex design constraints.Its contributions include two-step SDP initialization, differentiable objectives and constraints in incremental ALM, and SOCP legalization.Source: §1 · p. 2

2 Constrained Floorplanning Problem

• Each block is represented by center coordinates and dimensions, and floorplanning determines the shape and placement of all blocks.The optimization variables collect each block’s x and y coordinates, width, and height.Source: §2 · p. 2
• The objectives are weighted half-perimeter wirelength and the area of the enclosing bounding box.The supplied passage identifies the metrics but does not provide further formula details.Source: §2 · p. 2
• The formulation includes hard- and soft-block requirements, non-overlap, fixed outlines and positions, multi-instance dimensions, grouping, and boundary constraints.Grouping requires a spanning tree whose adjacent blocks share a non-zero boundary; boundary-group blocks must touch their assigned die edges.Source: §2 · p. 2
• Because the constraints make the optimization highly non-convex and computationally intractable to solve directly, the paper proposes a three-stage analytical framework.The framework is introduced as the response to constraints (1b)–(1j).Source: §2 · p. 2, Figure 3 · p. 3

3 Constraint-Driven Floorplanning Framework

• Two-step SDP initialization first anchors global structure, then adds pruned non-overlap and MST-based grouping constraints to form a robust seed.Pruning unnecessary constraints reduces complexity.Source: §3 · p. 3
• Incremental ALM starts from the seed and jointly suppresses overlap, grouping, boundary, and MIB constraints using balanced penalty initialization, locality-constrained updates, and feasibility projection.The global-floorplanning objective and constraints are differentiable.Source: §3 · p. 3
• Legalization builds horizontal and vertical constraint graphs and solves a unified SOCP with non-overlap, grouping, soft-block, and boundary constraints.The pipeline converts the near-feasible global result into a fully legal floorplan.Source: §3 · p. 3

3.1 Initialization via Two-Step Semidefinite Programming

• The two-step SDP is designed to balance computational cost and geometric quality in a highly non-convex landscape with poor local minima.It prunes unnecessary constraints and incorporates practical design requirements to provide a strong seed for rectangle-based global floorplanning.Source: §3.1 · p. 3

3.1.1 Basic SDP Formulation.

• The basic SDP abstracts each block as a circle and relaxes the non-convex Gram-matrix relation to a positive-semidefinite lifted matrix.The remaining non-convexity is handled through rank control during iterative refinement.Source: §3.1.1 · p. 3

3.1.2 Step I: Trial Placement with Anchor Constraints.

• Step I solves a relaxed SDP using wirelength and fixed-anchor constraints while deferring complex overlap constraints.Preplaced and boundary blocks are unified as anchor nodes with fixed coordinates; the resulting solution captures global structure and respects those locations.Source: §3.1.2 · pp. 3–4

3.1.3 Step II: Constraint-Driven SDP Initialization.

• The Step I placement guides construction of the constraints used to improve geometric quality and prepare for global floorplanning.Figure 4 contrasts the anchor-only first step with the second step’s additional non-overlap and grouping constraints.Source: §3.1.3 · p. 4, Figure 4 · p. 4
• Step II refines the Step I solution by adding non-overlap constraints only for identified overlapping pairs and grouping constraints on minimum-spanning-tree edges.Rank control alternately optimizes W and Z, increasing the penalty coefficient until an approximate rank-2 solution yields the initial placement.Source: §3.1.3 · p. 4

3.2 Global Floorplanning via Multi-Constraint Augmented Lagrangian Method

• The differentiable global-floorplanning problem jointly minimizes wirelength and bounding-box area while modeling overlap, grouping, boundary, and multi-instantiation constraints.It starts from an initialization-stage solution.Source: §3.2 · p. 4

3.2.1 Floorplan Area.

• The smoothed bounding-box area surrogate makes area part of the differentiable global-floorplanning objective.Log-sum-exp approximates coordinate extrema.Source: §3.2.1 · p. 4

3.2.2 Multi-Instantiation Block Constraints.

• MIB groups are driven toward identical aspect ratios using a group-wise mean and a variance-like inconsistency penalty.A zero penalty indicates perfect shape consistency.Source: §3.2.2 · p. 4

3.2.3 Overlap Constraint.

• Pairwise overlap is modeled with smooth penalties that decay to zero when modules are sufficiently separated.The two-dimensional overlap term is the product of the one-dimensional terms, and normalization by block area avoids bias toward large modules.Source: §3.2.3 · p. 5

3.2.4 Grouping Constraints.

• Grouping penalties use minimum-spanning-tree edges to preserve group connectivity while limiting constraint coupling.The edge penalty uses an abutment potential and a small tolerance for slight gaps.Source: §3.2.4 · p. 5

3.2.5 Boundary Constraint.

• Boundary penalties measure normalized slack for each constrained side and become zero when a constrained module exactly touches that side.Normalization by current box span reduces scale sensitivity across floorplan sizes.Source: §3.2.5 · p. 5

3.2.6 Incremental Augmented Lagrangian Optimization.

• An initial-solution-aware incremental ALM progressively tightens constraints while preserving the initialization’s global topology.Locality-constrained updates restrict early step magnitudes, and balanced penalty initialization prevents constraint penalties from abruptly disrupting the floorplan.Source: §3.2.6 · p. 6

3.3 Legalization via Second-Order Cone Programming

• SOCP legalization addresses residual overlaps and minor constraint violations in the global solution while minimizing bounding-box area.The formulation is intended to satisfy all constraints in the resulting floorplan.Source: §3.3 · p. 6

3.3.1 Constraint Graph and Non-Overlap Constraints.

• Horizontal and vertical constraint graphs encode non-overlap requirements for legal placement.The graph is constructed from the global floorplanning solution.Source: §3.3.1 · p. 6, Figure 6 · p. 7

3.3.2 Soft Block Constraints.

• Soft-block area and aspect-ratio requirements are reformulated as a second-order cone constraint with linear inequalities.The convex formulation allows dimensions to adjust while maintaining mathematical tractability.Source: §3.3.2 · p. 6

3.3.3 Grouping Constraints.

• For each group’s minimum-spanning-tree edge, the primary abutment direction comes from the global solution, with zero gap in that direction and minimum overlap orthogonally.The orthogonal overlap is enforced by a positive tolerance.Source: §3.3.3 · p. 7

3.3.4 Boundary Constraints.

• Designated blocks are constrained to chip boundaries through linear equalities on their edges.The constraint graph identifies edge types and their associated constraints.Source: §3.3.4 · p. 7

3.3.5 Handling Remaining Constraints.

• Hard-block dimensions and preplaced-block geometry are fixed, while blocks in each MIB group share width and height variables.Treating fixed geometry as constants reduces the effective dimension; shared MIB variables further reduce variable count.Source: §3.3.5 · p. 7

3.3.6 SOCP Formulation.

• The SOCP minimizes W + H as a convex proxy for area and yields a fully legal floorplan satisfying all constraints.The passage states that the convex problem can be solved to a global optimum in polynomial time.Source: §3.3.6 · p. 7

4.1 Experimental Setup

• The method was evaluated on FloorSet-lite with fixed IO positions and IO bounding boxes as fixed outlines; block aspect ratios ranged from 1/3 to 3.The comparison included PARSAC and two ablated variants; each method had 100 parallel trials, with the best result reported.Source: §4.1 · p. 7, Table 1 · p. 7

4.2 Comparison with SOTA and Ground-Truth

• 90% lower HPWL and 16% lower area on average versus PARSAC; the method satisfies all constraints across 11 benchmarks and is fastest overall, with about 72% lower average runtime.Ground-truth references are near-optimal: their theoretical optima are 98% and 99% of the method's achieved HPWL and area, respectively.Source: §4.2 · pp. 7–8, Table 2 · p. 8
• PARSAC begins to show constraint violations beyond intel_lite60, reaching up to 6 on larger cases, while the method's quality gap widens as benchmark scale increases.Figure 7 qualitatively compares the method's strictly legal intel_lite60 floorplan with its ground-truth reference.Source: §4.2 · pp. 7–8, Figure 7 · p. 8

4.3 Ablation Study

• Against Ours w/o Init & Incre, the full method reduces HPWL by about 4.0% and improves area by 3.0%; the center-based start without refinement uses about 50% of full-flow runtime but degrades quality.This ablation compares the full method with a variant lacking both SDP initialization and incremental optimization.Source: §4.3 · p. 8
• The incremental stage adds a further 2.0% reduction in both HPWL and area versus Ours w/o Incre, although it increases runtime.The complete flow remains faster than PARSAC while improving solution quality and robustly satisfying constraints.Source: §4.3 · p. 8

• The framework unifies SDP-based initialization, incremental augmented Lagrangian optimization, and SOCP-based legalization in an end-to-end solution for constrained floorplanning.The method optimizes wirelength and area while handling complex coupled design constraints.Source: §5 · p. 8