Showing papers by "John Iacono published in 2006"

Necklaces, convolutions, and X + Y

Abstract: 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.

An O(n log n)-time algorithm for the restriction scaffold assignment problem.

Abstract: The restriction scaffold assignment problem takes as input two finite point sets S and T (with S containing more points than T ) and establishes a correspondence between points in S and points in T , such that each point in S maps to exactly one point in T and each point in T maps to at least one point in S. An algorithm is presented that finds a minimum-cost solution for this problem in O(n log n) time, provided that the points in S and T are restricted to lie on a line and the cost function delta is the L(1) metric. This algorithm runs in linear time, if S and T are presorted. This improves the previously best-known O(n (2))-time algorithm for this problem.

Data structures for halfplane proximity queries and incremental voronoi diagrams

Abstract: We consider preprocessing a set S of n points in the plane that are in convex position into a data structure supporting queries of the following form: given a point q and a directed line l in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q subject to being to the left of line l. We present two data structures for this problem. The first data structure uses O(n1+e) space and preprocessing time, and answers queries in O(21/e log n) time. The second data structure uses O(n log3n) space and polynomial preprocessing time, and answers queries in O(log n) time. These are the first solutions to the problem with O(log n) query time and o(n2) space. In the process of developing the second data structure, we develop a new representation of nearest-point and farthest-point Voronoi diagrams of points in convex position. This representation supports insertion of new points in counterclockwise order using only O(log n) amortized pointer changes, subject to supporting O(log n)-time point-location queries, even though every such update may make Θ(n) combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n) pointer changes per operation while keeping O(log n)-time point-location queries.

The complexity of diffuse reflections in a simple polygon

Abstract: 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].

Data structures for halfplane proximity queries and incremental voronoi diagrams

Abstract: We consider preprocessing a set S of n points in the plane that are in convex position into a data structure supporting queries of the following form: given a point q and a directed line in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q subject to being to the left of line . We present two data structures for this problem. The first data structure uses O(n 1+e ) space and preprocessing time, and answers queries in O(2 1/e logn) time. The second data structure uses O(n log 3 n) space and polynomial preprocessing time, and answers queries in O(log n) time. These are the first solutions to the problem with O(logn) query time and o(n 2 ) space. In the process of developing the second data structure, we develop a new representation of nearest-point and farthest-point Voronoi diagrams of points in convex position. This representation supports insertion of new points in counterclockwise order using only O(log n.) amortized pointer changes, subject to supporting O (log n)-time point-location queries, even though every such update may make ⊖(n) combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n) pointer changes per operation while keeping O(log n)-time point-location queries.

Curves in the Sand: Algorithmic Drawing

Abstract: 1 IntroductionEthnomathematics is the study of mathematics in the worksof art of various cultures [3, 4, 10, 14]. The concepts in thispaper are inspired by the visual art of sand drawings that hasdeveloped independently in different forms in diverse cul-tures. Generally speaking, the artist draws a set of dots onsome flat surface (usually in the sand or in powder on thefloor) and then draws one continuous curve that surroundsthe dots and crosses itself repeatedly. Although not univer-sally the case, we focus on drawings in which there is exactlyone dot per bounded face (and no dots in the outside face).In particular, sand drawings made by the Tshokwe people inthe West Central Bantu area of Africa are called sona.Sona drawings have been considered in the field of topol-ogy under an equivalent guise as generic planar closedcurves (immersions of the unit circle into the plane). Sev-eral topological invariants about such curves are proved byArnol

Where to build a temple, and where to dig to find one

Abstract: In this paper, we analyze the time complexity of finding regular polygons in a set of n points. We use two different approaches to find regular polygons, depending on their number of edges. Those with o(n 0.068 ) edges are found by sweeping a line through the set of points, while the larger polygons are found by random sampling. We can find all the polygons with high probability in O(n 2.068+� ) expected time for every positive �. This compares well to the O(n 2.136+� ) deterministic algorithm of Brass [1]. Our method can also be used to find incomplete regular polygons, where up to a constant fraction of the vertices are missing.

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?