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

The Tukey depth (Proceedings of the International Congress of Mathematicians, vol. 2, pp. 523---531, 1975) of a point p with respect to a finite set S of points is the minimum number of elements of S contained in any closed halfspace that contains p. Algorithms for computing the Tukey depth of a point in various dimensions are considered. The running times of these algorithms depend on the value of the output, making them suited to situations, such as outlier removal, where the value of the output is typically small.

/pdf/output-sensitive-algorithms-for-tukey-depth-and-related-tnmo5vfo1a.pdf

Output-sensitive algorithms for Tukey depth and related problems

Given a fixed distribution of point location queries among the triangles in a triangulation of the plane, a data structure is presented that achieves, within constant multiplicative factors, the entropy bound on the expected point location query time. The data structure is a simple variation of Kirkpatrick's classic planar point location structure [D.G. Kirkpatrick, SIAM J. Comput. 12 (1) (1983) 28-35], and has linear construction costs and space requirements.

Expected asymptotically optimal planar point location

We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the lp norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p=1, p=2, and p=∞. For p=2, we reduce the problem to standard convolution, while for p=∞ and p=1, we reduce the problem to (min,+) convolution and (median,+) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X + Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X + Y matrix. All of our algorithms run in o(n2) time, whereas the obvious algorithms for these problems run in Θ (n2) time.

/pdf/necklaces-convolutions-and-x-y-29hpni4gow.pdf

Necklaces, convolutions, and X + Y

This paper presents a simple dictionary structure designed for a hierarchical memory The proposed data structure is cache oblivious and locality preserving A cache-oblivious data structure has memory performance optimized for all levels of the memory hierarchy even though it has no memory-hierarchy-specific parameterization A locality-preserving dictionary maintains elements of similar key values stored close together for fast access to ranges of data with consecutive keysThe data structure presented here is a simplification of the cache-oblivious B-tree of Bender, Demaine, and Farach-Colton Like the cache-oblivious B-tree, this structure supports search operations using only O(logBN) block operations at a level of the memory hierarchy with block size B Insertion and deletion operations use O(logBN + log2N/B) amortized block transfers Finally, the data structure returns all k data items in a given search range using O(logBN + k/B) block operationsThis data structure was implemented and its performance was evaluated on a simulated memory hierarchy This paper presents the results of this simulation for various combinations of block and memory sizes

A locality-preserving cache-oblivious dynamic dictionary

We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the p norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p = 1, p even, and p = \infty. For p even, we reduce the problem to standard convolution, while for p = \infty and p = 1, we reduce the problem to (min, +) convolution and (median, +) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X + Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X + Y matrix. All of our algorithms run in o(n^2) time, whereas the obvious algorithms for these problems run in \Theta(n^2) time.

John Iacono

Papers

Output-sensitive algorithms for Tukey depth and related problems

Expected asymptotically optimal planar point location

Necklaces, convolutions, and X + Y

A locality-preserving cache-oblivious dynamic dictionary

Necklaces, Convolutions, and X+Y