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

Given a set R of red points and a set B of blue points, the nearest-neighbour decision rule classifies a new point q as red (respectively, blue) if the closest point to q in R ⋃ B comes from R (respectively, B). This rule implicitly partitions space into a red set and a blue set that are separated by a red-blue decision boundary. In this paper we develop output-sensitive algorithms for computing this decision boundary for point sets on the line and in ℝ2. Both algorithms run in time O(n log k), where k is the number of points that contribute to the decision boundary. This running time is the best possible when parameterizing with respect to n and k.

/pdf/output-sensitive-algorithms-for-computing-nearest-neighbour-1108c6e06w.pdf

Output-Sensitive Algorithms for Computing Nearest-Neighbour Decision Boundaries

A locality-preserving cache-oblivious dynamic dictionary

We present a novel connection between binary search trees (BSTs) and points in the plane satisfying a simple property. Using this correspondence, we achieve the following results:1. A surprisingly clean restatement in geometric terms of many results and conjectures relating to BSTs and dynamic optimality.2. A new lower bound for searching in the BST model, which subsumes the previous two known bounds of Wilber [FOCS'86].3. The first proposal for dynamic optimality not based on splay trees. A natural greedy but offline algorithm was presented by Lucas [1988], and independently by Munro [2000], and was conjectured to be an (additive) approximation of the best binary search tree. We show that there exists an equal-cost online algorithm, transforming the conjecture of Lucas and Munro into the conjecture that the greedy algorithm is dynamically optimal.

/pdf/the-geometry-of-binary-search-trees-2v4dv49bvo.pdf

The geometry of binary search trees

Splay trees are a self adjusting form of search tree that supports access operations in O(log n) amortized time. Splay trees also have several amazing distribution sensitive properties, the strongest two of which are the working set theorem and the dynamic finger theorem. However, these two theorems are shown to poorly bound the performance of splay trees on some simple access sequences. The unified conjecture is presented, which subsumes the working set theorem and dynamic finger theorem, and accurately bounds the performance of splay trees over some classes of sequences where the existing theorems' bounds are not tight. While the unified conjecture for splay trees is unproven, a new data structure, the unified structure, is presented where the unified conjecture does hold. This structure also has a worst case of O(log n) per operation, in contrast to the O(n) worst case runtime of splay trees. A second data structure, the working set structure, is introduced. The working set structure has the same performance attributed to splay trees through the working set theorem, except the runtime is worst case per operation rather than amortized.

Alternatives to splay trees with O(log n) worst-case access times

Given a set R of red points and a set B of blue points, the nearest-neighbour decision rule classifies a new point q as red (respectively, blue) if the closest point to q in R ∪ B comes from R (respectively, B). This rule implicitly partitions space into a red set and a blue set that are separated by a red-blue decision boundary. In this paper we develop output-sensitive algorithms for computing this decision boundary for point sets on the line and in ℝ2. Both algorithms run in time O(n log k), where k is the number of points that contribute to the decision boundary. This running time is the best possible when parameterizing with respect to n and k.

John Iacono

Papers

Output-Sensitive Algorithms for Computing Nearest-Neighbour Decision Boundaries

A locality-preserving cache-oblivious dynamic dictionary

The geometry of binary search trees

Alternatives to splay trees with O(log n) worst-case access times

Output-Sensitive Algorithms for Computing Nearest-Neighbour Decision Boundaries