We introduce staged self-assembly of Wang tiles, where tiles can be added dynamically in sequence and where intermediate constructions can be stored for later mixing. This model and its various constraints and performance measures are motivated by a practical nanofabrication scenario through protein-based bioengineering. Staging allows us to break through the traditional lower bounds in tile self-assembly by encoding the shape in the staging algorithm instead of the tiles. All of our results are based on the practical assumption that only a constant number of glues, and thus only a constant number of tiles, can be engineered. Under this assumption, traditional tile self-assembly cannot even manufacture an n 9 n square; in contrast, we show how staged assembly in theory enables manufacture of arbitrary shapes in a variety of precise formulations of the model.

We introduce staged self-assembly of Wang tiles, where tiles can be added dynamically in sequence and where intermediate constructions can be stored for later mixing. This model and its various constraints and performance measures are motivated by a practical nanofabrication scenario through protein-based bioengineering. Staging allows us to break through the traditional lower bounds in tile self-assembly by encoding the shape in the staging algorithm instead of the tiles. All of our results are based on the 123Nat Comput (2008) 7:347-370 DOI 10.1007 practical assumption that only a constant number of glues, and thus only a constant number of tiles, can be engineered. Under this assumption, traditional tile self-assembly cannot even manufacture an n 9 n square; in contrast, we show how staged assembly in theory enables manufacture of arbitrary shapes in a variety of precise formulations of the model.

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Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (adjacent to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces. These constraints are described by combinatorial pseudo-triangulations, first defined and studied in this paper. Also of interest are our 32 R. Haas et al. / Computational Geometry 31 (2005) two proof techniques, one based on Henneberg inductive constructions from combinatorial rigidity theory, the other on a generalization of Tutte's barycentric embeddings to directed graphs.

The goal of this paper is to show that the concept of the shortest path inside a polygonal region contributes to the design of e cient algorithms for certain geometric optimization problems involving simple polygons: computing optimum separators, maximum area or perimeter inscribed triangles, a minimum area circumscribed concave quadrilateral, or a maximum area contained triangle. The structure for our algorithms is as follows: a) decompose the initial problem into a low-degree polynomial number of optimization problems; b) solve each individual subproblem in constant time using standard methods of calculus, basic methods of numerical analysis, or linear programming. These same optimization techniques can be applied to splinegons (curved polygons). To do this, we rst develop a decomposition technique for curved polygons which we substitute for triangulation in creating equally e cient curved versions of the algorithms for the shortest-path tree, ray-shooting and two-point shortest path problems. The maximum-are or perimeter inscribed triangle problem, the minimum area circumscribed concave quadrilateral problem and maximum area contained triangle problem have applications to robotics and stock-cutting. The results of this paper will appear also in 33].