layoutfold

Layoutfold is a computational technique for transforming a two-dimensional layout into a folded, multi-layer representation while preserving prescribed spatial relationships. It is used in fields such as integrated circuit design, microfluidics, and dense user interface layouts where compact, layered deployments are desirable.

Definition and model: Given a grid-like arrangement of elements with adjacency constraints, a layoutfold is a sequence of fold operations along grid lines that maps cells to layers in a stack. The mapping must avoid overlaps and typically aims to minimize layers, folded area, or interconnect length. Folds are reflections, translations, or rotations restricted to the grid.

Algorithms and complexity: Layoutfold problems are usually NP-hard. Practical solutions rely on heuristics and exact methods for small instances. Common approaches include constraint programming, simulated annealing, and greedy folding with conflict resolution. Evaluation criteria include clash avoidance, balanced layering, preservation of adjacencies, and efficiency of subsequent routing or assembly steps.

Applications: In 3D integrated circuits, folding a planar layout reduces footprint. In microfluidics, layoutfold helps compact networks of channels. In printing and electronics, it supports densification while maintaining manufacturability. In software, a virtual layoutfold can guide responsive user interfaces that reflow content into stacked panels while preserving relationships between elements.

History and status: Layoutfold appears in theoretical discussions and some tool documentation as a folding-based layout optimization concept. It is not universally standardized, and implementations vary by domain, constraints, and goals. See also layout optimization, graph embedding, folding theory.