John Iacono

Bio: John Iacono is an academic researcher from Université libre de Bruxelles. The author has contributed to research in topics: Data structure & Amortized analysis. The author has an hindex of 24, co-authored 170 publications receiving 2130 citations. Previous affiliations of John Iacono include New York University & Aarhus University.

Partitioning a Regular n-gon into n+1 Convex Congruent Pieces is Impossible, for Sufficiently Large n.

Abstract: The interest in polygon decomposition emanates from the theoretical importance of the problem on one hand and the many applications that it has on the other. The decomposition problem has been extensively studied in the literature and yet variations of the problem remain open [3]. The existence of a huge literature on this problem can be informally explained by the fact that there are numerous ways in which we can decompose a polygon and there are many types of polygons to decompose. Decomposition can be defined as partitioning a polygon into components according to a set of rules. In other words, each kind of decomposition has a set of constraints either on the type of the pieces, the number of pieces, the length of the cuts, the areas of the partitions. In this paper, we discuss the following problem posed by Joseph O’Rourke at the Fall Workshop on Computational Geometry in 2004: Is it possible to partition a regular n-gon into n+1 congruent pieces? It is obviously possible for the equilateral triangle and the square as shown below. However, is it ever possible for n ≥ 5?

The Complexity of Order Type Isomorphism

Abstract: The order type of a point set in $R^d$ maps each $(d{+}1)$-tuple of points to its orientation (e.g., clockwise or counterclockwise in $R^2$). Two point sets $X$ and $Y$ have the same order type if there exists a mapping $f$ from $X$ to $Y$ for which every $(d{+}1)$-tuple $(a_1,a_2,\ldots,a_{d+1})$ of $X$ and the corresponding tuple $(f(a_1),f(a_2),\ldots,f(a_{d+1}))$ in $Y$ have the same orientation. In this paper we investigate the complexity of determining whether two point sets have the same order type. We provide an $O(n^d)$ algorithm for this task, thereby improving upon the $O(n^{\lfloor{3d/2}\rfloor})$ algorithm of Goodman and Pollack (1983). The algorithm uses only order type queries and also works for abstract order types (or acyclic oriented matroids). Our algorithm is optimal, both in the abstract setting and for realizable points sets if the algorithm only uses order type queries.

Sublinear Explicit Incremental Planar Voronoi Diagrams

Abstract: A data structure is presented that explicitly maintains the graph of a Voronoi diagram of $N$ point sites in the plane or the dual graph of a convex hull of points in three dimensions while allowing insertions of new sites/points Our structure supports insertions in $\tilde O (N^{3/4})$ expected amortized time, where $\tilde O$ suppresses polylogarithmic terms This is the first result to achieve sublinear time insertions; previously it was shown by Allen et al that $\Theta(\sqrt{N})$ amortized combinatorial changes per insertion could occur in the Voronoi diagram but a sublinear-time algorithm was only presented for the special case of points in convex position

Approximability of (Simultaneous) Class Cover for Boxes

Abstract: Bereg et al. (2012) introduced the Boxes Class Cover problem, which has its roots in classification and clustering applications: Given a set of n points in the plane, each colored red or blue, find the smallest cardinality set of axis-aligned boxes whose union covers the red points without covering any blue point. In this paper we give an alternative proof of APX-hardness for this problem, which also yields an explicit lower bound on its approximability. Our proof also directly applies when restricted to sets of points in general position and to the case where so-called half-strips are considered instead of boxes, which is a new result. We also introduce a symmetric variant of this problem, which we call Simultaneous Boxes Class Cover and can be stated as follows: Given a set S of n points in the plane, each colored red or blue, find the smallest cardinality set of axis-aligned boxes which together cover S such that all boxes cover only points of the same color and no box covering a red point intersects a box covering a blue point. We show that this problem is also APX-hard and give a polynomial-time constant-factor approximation algorithm.

Spanning Properties of Theta–Theta-6

Abstract: We show that, unlike the Yao–Yao graph $$YY_6$$, the Theta–Theta graph $${\varTheta }{\varTheta }_6$$ defined by six cones is a spanner for sets of points in convex position. We also show that, for sets of points in non-convex position, the spanning ratio of $${\varTheta }{\varTheta }_6$$ is unbounded.

Triangulating a simple polygon in linear time

Abstract: 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.

Programming curvature using origami tessellations

Abstract: 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.