Learn More
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(More)
We extend the results of straight-edged computational geometry into the curved world by defining a pair of new geometric objects, thesplinegon and thesplinehedron, as curved generalizations of the polygon and polyhedron. We identify three distinct techniques for extending polygon algorithms to splinegons: the carrier polygon approach, the bounding polygon(More)
A splinegon is a polygon whose edges have been replaced by “well-behaved” curves. We show how to decompose a simple splinegon into a union of monotone pieces and into a union of differences of unions of convex pieces. We also show how to use a fast triangulation algorithm to test whether two given simple splinegons intersect. We conclude with examples of(More)
It is well known that, given two simple n-sided polygons, it may not be possible to triangulate the two polygons in a compatible fashion, if one's choice of triangulation vertices is restricted to polygon corners. Is it always possible to produce compatible triangulations if additional vertices inside the polygon are allowed? We give a positive answer and(More)
The goal of this paper is to show that the concept of the shortest path inside a polygonal region contributes to the design of eecient 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(More)
Let P = fp 1 ; : : : ; p n g and Q = fq 1 ; : : : ; q n g be two point sets lying in the interior of rectangles in the plane. We show how to construct a piecewise linear homeomorphism of size O(n 2) between the rectangles which maps p i to q i for each i. This bound is optimal in the worst case; i.e., there exist point sets for which any piecewise linear(More)
As proposed by Liu [8] the simplicial depth of a point with respect to a data set in is the fraction of closed sim-plices given by ! # " of the data points containing. We propose an alternative definition for simplicial depth which remains valid over a continuous probability field and fixes some problems in the finite sample case. A data depth measures how(More)
Let S and T be two sets of points with total cardinality n. The minimum-cost many-to-many matching problem matches each point in S to at least one point in T and each point in T to at least one point in S, such that sum of the matching costs is minimized. Here we examine the special case where both S and T lie on the line and the cost of matching s ∈ S to t(More)
Data depth is a statistical analysis method that assigns a numeric value to a point based on its centrality relative to a data set. Examples include the half-space depth (also known as Tukey depth), convex-hull peeling depth and L1 depth. Data depth has significant potential as a data analysis tool. The lack of efficient computational tools for depth based(More)