An Improved Implementation and Inalysis of the Diaz and O'Rourke Algorithm for Finding the Simpson Point of a Convex Polygon

This paper focuses on the well-known Diaz and O'Rourke [M. Diaz and J. O'Rourke, Algorithms for computing the center of area of a convex polygon, Visual Comput. 10 (1994), 432–442.] iterative search algorithm to find the Simpson Point of a market, described by a convex polygon. In their paper, they observed that their algorithm did not appear to converge pointwise, and therefore, modified it to do so. We first present an enhancement of their algorithm that improves its time complexity from O(log2?) to O(n log 1/?). This is then followed by a proof of pointwise convergence and derivation of explicit bounds on convergence rates of our algorithm. It is also shown that with an appropriate interpretation, our convergence results extend to all similar iterative search algorithms to find the Simpson Point – a class that includes the original unmodified Diaz–O'Rourke algorithm. Finally, we explore how our algorithm and its convergence guarantees might be modified to find the Simpson Point when the demand distribution is non-uniform

Minimum Convex Partitions and Maximum Empty Polytopes

Let $S$ be a set of $n$ points in $\mathbb{R}^d$. A Steiner convex partition is a tiling of ${\rm conv}(S)$ with empty convex bodies. For every integer $d$, we show that $S$ admits a Steiner convex partition with at most $\lceil (n-1)/d\rceil$ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension $d\geq 3$. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any $n$ points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is $\omega(1/n)$. Here we give a $(1-\varepsilon)$-approximation algorithm for computing the maximum volume of an empty convex body amidst $n$ given points in the $d$-dimensional unit box $[0,1]^d$.Comment: 16 pages, 4 figures; revised write-up with some running times improve

Approximating the Maximum Overlap of Polygons under Translation

Let $P$ and $Q$ be two simple polygons in the plane of total complexity $n$, each of which can be decomposed into at most $k$ convex parts. We present an $(1-\varepsilon)$-approximation algorithm, for finding the translation of $Q$, which maximizes its area of overlap with $P$. Our algorithm runs in $O(c n)$ time, where $c$ is a constant that depends only on $k$ and $\varepsilon$. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time

Combined Coverage Area Reporting and Geographical Routing in Wireless Sensor-Actuator Networks for Cooperating with Unmanned Aerial Vehicles

In wireless sensor network (WSN) applications with multiple gateways, it is key to route location dependent subscriptions efficiently at two levels in the system. At the gateway level, data sinks must not waste the energy of the WSN by injecting subscriptions that are not relevant for the nodes in their coverage area and at WSN level, energy-efficient delivery of subscriptions to target areas is required. In this paper, we propose a mechanism in which (1) the WSN provides an accurate and up-to-date coverage area description to gateways and (2) the wireless sensor network re-uses the collected coverage area information to enable efficient geographical routing of location dependent subscriptions and other messages. The latter has a focus on routing of messages injected from sink nodes to nodes in the region of interest. Our proposed mechanisms are evaluated in simulation

Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference