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

It is shown how to enhance any data structure in the pointer model to make it confluently persistent, with efficient query and update times and limited space overhead Updates are performed in O(log n) amortized time, and following a pointer takes O(log c log n) time where c is the in-degree of a node in the data structure In particular, this proves that confluent persistence can be achieved at a logarithmic cost in the bounded in-degree model used widely in previous work This is a O(n/log n)-factor improvement over the previous known transform to make a data structure confluently persistent

https://epubs.siam.org/doi/pdf/10.1137/1.9781611973099.50

Confluent persistence revisited

Let $P$ and $Q$ be finite point sets of the same cardinality in $\mathbb{R}^2$, each labelled from $1$ to $n$. Two noncrossing geometric graphs $G_P$ and $G_Q$ spanning $P$ and $Q$, respectively, are called compatible if for every face $f$ in $G_P$, there exists a corresponding face in $G_Q$ with the same clockwise ordering of the vertices on its boundary as in $f$. In particular, $G_P$ and $G_Q$ must be straight-line embeddings of the same connected $n$-vertex graph. Deciding whether two labelled point sets admit compatible geometric paths is known to be NP-complete. We give polynomial-time algorithms to find compatible paths or report that none exist in three scenarios: $O(n)$ time for points in convex position; $O(n^2)$ time for two simple polygons, where the paths are restricted to remain inside the closed polygons; and $O(n^2 \log n)$ time for points in general position if the paths are restricted to be monotone.

/pdf/compatible-paths-on-labelled-point-sets-2wzdoopz0y.pdf

Compatible Paths on Labelled Point Sets

Polygon decomposition problems are well studied in the literature [6], yet many variants of these problems remain open. In this paper, we are interested in partitioning a polygon into mirror congruent pieces. Symmetry detection algorithms solve problems of the same flavor by detecting all kinds of isometries in a polygon, a set of points, a set of line segments and some classes of polyhedra [2]. Two open problems with unknown complexity were posed in [2]: the minimum symmetric decomposition (MSD) problem and the minimal symmetric partition (MSP) problem. Given a set D in R (d ∈ 2, 3), the goal is to find a set of symmetric (nondisjoint for MSD and disjoint for MSP) subsets {D1, D2, . . . , Dk} of D such that the union of the Di is D and k is minimum. The following problem is a decision version of MSP where k = 2: Problem 1 Given a polygon P with n vertices, compute a partition of P into two (properly or mirror) congruent polygons P1 and P2, or indicate such a partition does not exist.

/pdf/partitioning-a-polygon-into-two-mirror-congruent-pieces-4s4tacxkrz.pdf

Partitioning a Polygon into Two Mirror Congruent Pieces

In this paper, we analyze the time complexity of finding regular polygons in a set of n points. We combine two different approaches to find regular polygons, depending on their number of edges. Our result depends on the parameter alpha, which has been used to bound the maximum number of isosceles triangles that can be formed by n points. This bound has been expressed as O(n^{2+2alpha+epsilon}), and the current best value for alpha is ~0.068. 
Our algorithm finds polygons with O(n^alpha) edges by sweeping a line through the set of points, while larger polygons are found by random sampling. We can find all regular polygons with high probability in O(n^{2+alpha+epsilon}) expected time for every positive epsilon. This compares well to the O(n^{2+2alpha+epsilon}) deterministic algorithm of Brass.

Detecting all regular polygons in a point set

The modular subset sum problem consists of deciding, given a modulus $m$, a multiset $S$ of $n$ integers in $0..m$, and a target integer $t$, whether there exists a subset of $S$ with elements summing to $t \pmod{m}$, and to report such a set if it exists. We give a simple $O(m \log m)$-time with high probability (w.h.p.) algorithm for the modular subset sum problem. This builds on and improves on a previous $\tilde{O}(m)$ w.h.p. algorithm from Axiotis, Backurs, Jin, Tzamos, and Wu (SODA 19). Our method utilizes the ADT of the dynamic strings structure of Gawrychowski et. al (SODA 18). However, as this structure is rather complicated we present a much simpler alternative which we call the Data Dependent Tree. As an application, we consider the computational version of a fundamental theorem in zero-sum Ramsey theory. The Erdős-Ginzburg-Ziv Theorem states that a multiset of $2n - 1$ integers always contains a subset of cardinality exactly $n$ whose values sum to a multiple of $n$. We give an algorithm for finding such a subset in time $O(n \log n)$ w.h.p. which improves on an $O(n^2)$ algorithm due to Del Lungo, Marini, and Mori (Disc. Math. 09).

John Iacono

Papers

Confluent persistence revisited

Compatible Paths on Labelled Point Sets

Partitioning a Polygon into Two Mirror Congruent Pieces

Detecting all regular polygons in a point set

Modular Subset Sum, Dynamic Strings, and Zero-Sum Sets.