Umbrella Trees for Nesting Problem

Abstract

The problem of irregular-shaped pieces representation in order to minimize the nesting time is considered. A representation in the form of umbrella tree with a square crown consisting of tricolor nodes and bicolor leaves equidistant from the root is proposed. The construction of an umbrella tree from a image based on the transformation of Cartesian pixel coordinates into reference coordinates of tree leaves is shown. The translation and rotation of pieces functions on umbrella trees with quadratic estimates of computational complexity are determined. Examples of rounding coordinates heuristics and the experimental values of the threshold recoloring leaves during rotating a piece are given. The quadratic problem of intersection localization of the slab contents and the piece is reduced to the evaluating of color mutual exclusivity predicate of the pairs of nodes of the slab tree and the piece tree with linearrhythmic estimation of time in the applied cases. A pair of nodes is mutually exclusive if at least one of the nodes is white, and it isn't mutually exclusive if both are black. In other cases a pair of nodes is considered mutually exclusive if all four pairs of child nodes are mutually exclusive. The price of reducing the computational complexity of intersection localization is an increase of the required memory size by 33 % compared to the raster representation. Umbrella trees make it possible to use photo and video frames of pieces and slabs without converting them into a vector representation.