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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 bioengi-neering. Staging allows us to break through… (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)

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)

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)

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)

We study the configuration space of rectangulations and convex subdivisions of n points in the plane. It is shown that a sequence of O(n log n) elementary flip and rotate operations can transform any rectangulation to any other rectangulation on the same set of n points. This bound is the best possible for some point sets, while Θ(n) operations are… (More)