On the time complexity for circumscribing a convex polygon

A recent article "Circumscribing a Convex Polygon by a Polygon of Fewer Sides with Minimal Area Addition" by Dori and Ben-Bassat, Comput. Vision Graph. Image Process. 24, 1983, 131-159, raised several interesting questions including the time complexity of their algorithm. In this paper, the time complexity on circumscribing an n-gon by an m-gon, where m n, is analyzed to be O(n 1g n).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25655/1/0000207.pd

On the number of sides necessary for polygonal approximation of black-and-white figures in a plane

A bound on the number of extreme points or sides necessary to approximate a convex planar figure by an enclosing polygon is described. This number is found to be proportional to the fourth root of the figure's area divided by the square of a maximum Euclidean distance approximation parameter.An extension of this bound, preserving its fourth root quality, is made to general planar figures. This is done by decomposing the general figure into nearly convex sets defined by inflection points, cusps, and multiple windings.A procedure for performing actual encoding of this type is described. Comparisons of parsimony are made with contemporary figure encoding schemes

Stock Cutting Of Complicated Designs by Computing Minimal Nested Polygons

This paper studies the following problem in stock cutting: when it is required to cut out complicated designs from parent material, it is cumbersome to cut out the exact design or shape, especially if the cutting process involves optimization. In such cases, it is desired that, as a first step, the machine cut out a relatively simpler approximation of the original design, in order to facilitate the optimization techniques that are then used to cut out the actua1 design. This paper studies this problem of approximating complicated designs or shapes. The problem is defined formally first and then it is shown that this problem is equivalent to the Minima1 Nested Polygon problem in geometry. Some properties of the problem are then shown and it is demonstrated that the problem is related to the Minimal Turns Path problem in geometry. With these results, an efficient approximate algorithm is obtained for the origina1 stock cutting problem. Numerica1 examples are provided to illustrate the working of the algorithm in different cases

Optimization of Spatial Joins Using Filters

When viewing present-day technical applications that rely on the use of database systems, one notices that new techniques must be integrated in database management systems to be able to support these applications efficiently. This paper discusses one of these techniques in the context of supporting a Geographic Information System. It is known that the use of filters on geometric objects has a significant impact on the processing of 2-way spatial join queries. For this purpose, filters require approximations of objects. Queries can be optimized by filtering data not with just one but with several filters. Existing join methods are based on a combination of filters and a spatial index. The index is used to reduce the cost of the filter step and to minimize the cost of retrieving geometric objects from disk. In this paper we examine n-way spatial joins. Complex n-way spatial join queries require solving several 2-way joins of intermediate results. In this case, not only the profit gained from using both filters and spatial indices but also the additional cost due to using these techniques are examined. For 2-way joins of base relations these costs are considered part of physical database design. We focus on the criteria for mutually comparing filters and not on those for spatial indices. Important aspects of a multi-step filter-based n-way spatial join method are described together with performance experiments. The winning join method uses several filters with approximations that are constructed by rotating two parallel lines around the object

Optimal clustering of a pair of irregular objects

Cutting and packing problems arise in many fields of applications and theory. When dealing with irregular objects, an important subproblem is the identification of the optimal clustering of two objects. Within this paper we consider a container (rectangle, circle, convex polygon) of variable sizes and two irregular objects bounded by circular arcs and/or line segments, that can be continuously translated and rotated. In addition minimal allowable distances between objects and between each object and the frontier of a container, may be imposed. The objects should be arranged within a container such that a given objective will reach its minimal value. We consider a polynomial function as the objective, which depends on the variable parameters associated with the objects and the container. The paper presents a universal mathematical model and a solution strategy which are based on the concept of phi-functions and provide new benchmark instances of finding the containing region that has either minimal area, perimeter or homothetic coefficient of a given container, as well as finding the convex polygonal hull (or its approximation) of a pair of objects

Multi-Step Processing of Spatial Joins

Spatial joins are one of the most important operations for combining spatial objects of several relations. In this paper, spatial join processing is studied in detail for extended spatial objects in twodimensional data space. We present an approach for spatial join processing that is based on three steps. First, a spatial join is performed on the minimum bounding rectangles of the objects returning a set of candidates. Various approaches for accelerating this step of join processing have been examined at the last year’s conference [BKS 93a]. In this paper, we focus on the problem how to compute the answers from the set of candidates which is handled by the following two steps. First of all, sophisticated approximations are used to identify answers as well as to filter out false hits from the set of candidates. For this purpose, we investigate various types of conservative and progressive approximations. In the last step, the exact geometry of the remaining candidates has to be tested against the join predicate. The time required for computing spatial join predicates can essentially be reduced when objects are adequately organized in main memory. In our approach, objects are first decomposed into simple components which are exclusively organized by a main-memory resident spatial data structure. Overall, we present a complete approach of spatial join processing on complex spatial objects. The performance of the individual steps of our approach is evaluated with data sets from real cartographic applications. The results show that our approach reduces the total execution time of the spatial join by factors

Identifying Alternate Optimal Solutions to the Design Approximation Problem in Stock Cutting

The design approximation problem is a well known problem in stock cutting, where, in order to facilitate the optimization techniques used in the cutting process, it is required to approximate complex designs by simpler ones. Although there are algorithms available to solve this problem, they all suffer from an undesirable feature that they only produce one optimal solution to the problem, and do not identify the complete set of all optimal solutions. The focus of this paper is to study this hitherto unexplored aspect of the problem: specifically, the case is considered in which both the design and the parent material are convex shapes, and some essential properties of all optimal solutions to the design approximation problem are ascertained. These properties are then used to devise two efficient schemes to identify the set of all optimal solutions to the problem. Finally, the recovery of a desired optimal approximation from the identified sets of optimal solutions, is discussed

The gerrymandering of educational boundaries and the segregation of American schools : a geospatial analysis

Despite steady and substantial decreases in residential racial/ethnic segregation since the 1960s, public school segregation is increasing steadily. As a result of these trends, schools, which have historically been less segregated than their surrounding neighborhoods, are now becoming more segregated than neighborhoods, underscoring the need for research on the ways in which educational institutions are facilitating segregation. Adopting a “student exchange” framework from the literature on electoral gerrymandering, this study provides initial empirical evidence examining how gerrymandered educational boundaries exacerbate or ameliorate patterns of residential segregation by “zoning in” certain students and “zoning out” others. Using a large, nationally-representative sample of 9,717 school attendance zones and 9,796 school districts, this study employs geospatial analytic techniques to investigate the effects of school attendance zone and school district gerrymandering on the racial/ethnic diversity of schools and districts. The effect of gerrymandering on diversity is assessed by comparing the characteristics of students residing in current boundaries to those residing in the “natural”, compact zone or district that would be expected in the absence of gerrymandering, operationalized as the equal land area circle of Angel and Parent (2011) and convex Voronoi polygons. Analyses reveal that, on average, both school attendance zones and school districts are gerrymandered to “zone out” more racially/ethnically dissimilar students in favor of more racially/ethnically similar students. As a result, schools and districts are significantly more racially and ethnically homogeneous than they would be in the absence of gerrymandering. While gerrymandering serves to segregate students of all races and ethnicities, it particularly serves to exclude blacks and Hispanics from predominantly white schools and districts, reinforcing the historical divisions between these groups. Indeed, estimates suggest that, on average, school attendance zones and school districts are 15% and 14% less black-white diverse, respectively, than would be expected if their boundaries were not gerrymandered. Findings suggest that the gerrymandering of boundaries adds another pernicious layer of segregation to public education institutions, which are already highly segregated by residency. The finding that the gerrymandering of school attendance zones and school districts serves to segregate underscores the importance of educational boundaries as a contemporary mechanism of segregation. However, findings also warrant some optimism. Because attendance zone and district boundaries are modifiable and subject to policy intervention, state standards for boundary compactness and rezoning efforts designed to create more equitable boundaries present cost-effective opportunities to achieve meaningful gains in integration. While changing school district boundaries is less politically feasible than changing school attendance zones, when such windows of opportunity arise, they have the potential to reduce school finance inequities and equalize educational opportunity while also increasing racial/ethnic equity.Educational Administratio