Computational geometry

We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

Triangulating a simple polygon in linear time

Computational geometry.

Origami describes rules for creating folded structures from patterns on a flat sheet, but does not prescribe how patterns can be designed to fit target shapes. Here, starting from the simplest periodic origami pattern that yields one-degree-of-freedom collapsible structures-we show that scale-independent elementary geometric constructions and constrained optimization algorithms can be used to determine spatially modulated patterns that yield approximations to given surfaces of constant or varying curvature. Paper models confirm the feasibility of our calculations. We also assess the difficulty of realizing these geometric structures by quantifying the energetic barrier that separates the metastable flat and folded states. Moreover, we characterize the trade-off between the accuracy to which the pattern conforms to the target surface, and the effort associated with creating finer folds. Our approach enables the tailoring of origami patterns to drape complex surfaces independent of absolute scale, as well as the quantification of the energetic and material cost of doing so.

Programming curvature using origami tessellations

Voting based extreme learning machine

Biedl et al.1 presented an algorithm for unfolding orthostacks into one piece without overlap by using arbitrary cuts along the surface. They conjectured that orthostacks could be unfolded using cuts that lie in a plane orthogonal to a coordinate axis and containing a vertex of the orthostack. We prove the existence of a vertex unfolding using only such cuts.

/pdf/grid-vertex-unfolding-orthostacks-4y7iqaht4t.pdf

Grid vertex-unfolding orthostacks

We consider the problem of computing an order type invariant labeling for a given set of n points. In 2D, such a labeling can be constructed in O(hn 2 ) time, where h is the size of the smallest convex layer. In 3D the time complexity is O(n 3 log n) if the point set is in general position 1 . This is useful to test if two point sets have the same order type within the same time bounds. It can also be used as preprocessing for any order type invariant algorithm, such as triangulation/tetrahedralization, or polygonization.

/pdf/order-type-invariant-labeling-and-comparison-of-point-sets-4u3lf3xuid.pdf

Order type invariant labeling and comparison of point sets

In this paper, we analyze the time complexity of finding regular polygons in a set of n points. We use two different approaches to find regular polygons, depending on their number of edges. Those with o(n 0.068 ) edges are found by sweeping a line through the set of points, while the larger polygons are found by random sampling. We can find all the polygons with high probability in O(n 2.068+� ) expected time for every positive �. This compares well to the O(n 2.136+� ) deterministic algorithm of Brass [1]. Our method can also be used to find incomplete regular polygons, where up to a constant fraction of the vertices are missing.

/pdf/where-to-build-a-temple-and-where-to-dig-to-find-one-3i1zad1mry.pdf

Where to build a temple, and where to dig to find one

A V-shape is an innite polygonal region bounded by two pairs of parallel rays emanating from two vertices (see Figure 1) We describe a randomized algorithm that, given n points and an integer k 0, nds the minimum-width V-shape enclosing all but k of the points with probability 1 1=n c for any c > 0, with expected running time O(cn 2 (k + 1) 4 logn(logn log logn + k))

How to Cover Most of a Point Set with a V-Shape of Minimum Width

The B (cid:15) -tree [Brodal and Fagerberg 2003] is a simple I/O-eﬃcient external-memory-model data structure that supports updates orders of magnitude faster than B-tree with a query performance comparable to the B-tree: for any positive constant (cid:15) < 1 insertions and deletions take O ( 1 B 1 − (cid:15) log B N ) time (rather than O (log B N ) time for the classic B-tree), queries take O (log B N ) time and range queries returning k items take O (log B N + kB ) time. Although the B (cid:15) -tree has an optimal update/query tradeoﬀ, the runtimes are amortized. Another structure, the write-optimized skip list, introduced by Bender et al. [PODS 2017], has the same performance as the B (cid:15) -tree but with runtimes that are randomized rather than amortized. In this paper, we present a variant of the B (cid:15) -tree with deterministic worst-case running times that are identical to the original’s amortized running times.

John Iacono

Papers

Grid vertex-unfolding orthostacks

Order type invariant labeling and comparison of point sets

Where to build a temple, and where to dig to find one

How to Cover Most of a Point Set with a V-Shape of Minimum Width

External-memory dictionaries with worst-case update cost