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

Encoding 2D range maximum queries

Over the last decade, there have been several data structures that, given a planar subdivision and a probability distribution over the plane, provide a way for answering point location queries that is fine-tuned for the distribution. All these methods suffer from the requirement that the query distribution must be known in advance. We present a new data structure for point location queries in planar triangulations. Our structure is asymptotically as fast as the optimal structures, but it requires no prior information about the queries. This is a 2-D analogue of the jump from Knuth's optimum binary search trees (discovered in 1971) to the splay trees of Sleator and Tarjan in 1985. While the former need to know the query distribution, the latter are statically optimal. This means that we can adapt to the query sequence and achieve the same asymptotic performance as an optimum static structure, without needing any additional information.

A static optimality transformation with applications to planar point location

In the past ten years, there have been a number of data structures that, given a distribution of planar point location queries, produce a planar point location data structure that is tuned for the provided distribution. These structures all suffer from the requirement that the query distribution be provided in advance. For the problem of point location in a triangulation, a data structure is presented that performs asymptotically as well as these structures, but does not require the distribution to be provided in advance. This result is the 2-d analogue of the jump from the optimum binary search trees of Knuth in 1971 which required that the distribution be provided, to the splay trees of Sleator and Tarjan in 1985 where in the static optimality theorem it was proven that splay trees had the same asymptotic performance of optimum search trees without being provided the probability distribution.

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We consider the \emph{two-dimensional range maximum query (2D-RMQ)} problem: given an array $A$ of ordered values, to pre-process it so that we can find the position of the smallest element in the sub-matrix defined by a (user-specified) range of rows and range of columns. We focus on determining the \emph{effective} entropy of 2D-RMQ, i.e., how many bits are needed to encode $A$ so that 2D-RMQ queries can be answered \emph{without} access to $A$. We give tight upper and lower bounds on the expected effective entropy for the case when $A$ contains independent identically-distributed random values, and new upper and lower bounds for arbitrary $A$, for the case when $A$ contains few rows. The latter results improve upon previous upper and lower bounds by Brodal et al. (ESA 2010). In some cases we also give data structures whose space usage is close to the effective entropy and answer 2D-RMQ queries rapidly.

Encoding 2-D Range Maximum Queries

The complexity of the visibility region formed by a point light source after k diffuse reflections in a simple n-sided polygon is O(n9), which is the first result polynomial in n, with no dependence on k. This bound is an exponential improvement over the previous bound of O(n$^{\rm 2\lceil ({\it k}+1)/2 \rceil +1}$) due to Prasad et al.[8].

John Iacono

Papers

Encoding 2D range maximum queries

A static optimality transformation with applications to planar point location

A static optimality transformation with applications to planar point location

Encoding 2-D Range Maximum Queries

The complexity of diffuse reflections in a simple polygon