In this paper we study the separability in the plane by two criteria: double-wedge separability and Θ-separability. We give an O(N logN)-time optimal algorithm for computing all the vertices of separating double wedges of two disjoint sets of objects (points, segments, polygons and circles) and an O((N/Θ0) logN)-time algorithm for computing a nearly-straight minimal Θ-polygonal chain separating...

A tiling of the plane by polygons is unilateral if each edge of the tiling is a side of at most one polygon of the tiling. A tiling is equitransitive if for any two congruent tiles in the tiling, there exists a symmetry of the tiling mapping one to the other. It is known that a unilateral and equitransitive (UE) tiling can be made with any finite number of congruence classes of squares. This ar...

We investigate the dynamics of tiling dynamical systems and their deformations. If two tiling systems have identical combinatorics, then the tiling spaces are homeomorphic, but their dynamical properties may differ. There is a natural map I from the parameter space of possible shapes of tiles to H of a model tiling space, with values in R. Two tiling spaces that have the same image under I are ...

Simple cell-cell interactions can give rise to complex cellular patterns. For example, neurons of the same type can interact to create a complex patchwork of non-overlapping dendrite arbors, a pattern known as dendrite tiling. Dendrite tiling often involves mutual repulsion between neighboring neurons. While dendrite tiling is found across nervous systems, the nematode Caenorhabditis elegans ha...

Tiling is a crucial loop transformation for generating high performance code on modern architectures. Efficient generation of multi-level tiled code is essential for maximizing data reuse in systems with deep memory hierarchies. Tiled loops with parametric tile sizes (not compile-time constants) facilitate runtime feedback and dynamic optimizations used in iterative compilation and automatic tu...

Starting with a substitution tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles have fractal boundary. We show that each of the new tilings is mutually locally derivable to the original tiling. Thus, at the tiling space level, the new substitution rules are expressing...