On finding widest empty curved corridors

Open archive-ElsevierAn α-siphon of width w is the locus of points in the plane that are at the same distance w from a 1-corner polygonal chain C such that α is the interior angle of C. Given a set P of n points in the plane and a fixed angle α, we want to compute the widest empty α-siphon that splits P into two non-empty sets.We present an efficient O(n log3 n)-time algorithm for computing the widest oriented α-siphon through P such that the orientation of a half-line of C is known.We also propose an O(n3 log2 n)-time algorithm for the widest arbitrarily-oriented version and an (nlog n)-time algorithm for the widest arbitrarily-oriented α-siphon anchored at a given point

The siphon problem

An α-siphon is the locus of points in the plane that are at the same distance ǫ from a polygonal chain consisting of two half-lines emanating from a common point such that α is the interior angle of the half-lines. Given a set S of n points in the plane and a fixed angle α, we want to compute an α-siphon of largest width ǫ such that no points of S lies in its interior. We present an efficient O(n2)-time algorithm for computing an orthogonal siphon. The approach can be handled to solve the problem of the oriented α-siphon for which the orientation of a half-line is known. We also propose an O(n3 log n)-time algorithm for the arbitrarily oriented version.Ministerio de Ciencia y TecnologiaFondo Europeo de Desarrollo RegionalGeneralitat de Cataluny

Approximating Geometric Knapsack via L-packings

We study the two-dimensional geometric knapsack problem (2DK) in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is (2 + \epsilon) [Jansen and Zhang, SODA 2004]. In this paper, we break the 2 approximation barrier, achieving a polynomial-time (17/9 + \epsilon) < 1.89 approximation, which improves to (558/325 + \epsilon) < 1.72 in the cardinality case. Essentially all prior work on 2DK approximation packs items inside a constant number of rectangular containers, where items inside each container are packed using a simple greedy strategy. We deviate for the first time from this setting: we show that there exists a large profit solution where items are packed inside a constant number of containers plus one L-shaped region at the boundary of the knapsack which contains items that are high and narrow and items that are wide and thin. As a second major and the main algorithmic contribution of this paper, we present a PTAS for this case. We believe that this will turn out to be useful in future work in geometric packing problems. We also consider the variant of the problem with rotations (2DKR), where items can be rotated by 90 degrees. Also, in this case, the best-known polynomial-time approximation factor (even for the cardinality case) is (2 + \epsilon) [Jansen and Zhang, SODA 2004]. Exploiting part of the machinery developed for 2DK plus a few additional ideas, we obtain a polynomial-time (3/2 + \epsilon)-approximation for 2DKR, which improves to (4/3 + \epsilon) in the cardinality case.Comment: 64pages, full version of FOCS 2017 pape

Building sustainable learning environments that are ‘fit for the future’ with reference to Egypt

Perhaps there is no building type that has a more significant impact on our lives than the Kindergarten to high School (K-12). We continue to carry the memories of our early learning environments through the residue of our lives. It is the quality of those learning environments that play a crucial role in enhancing or hampering our learning experience. Learning spaces are complex spaces where the collective skills, knowledge, and practices of a culture are taught, shaped, encouraged, and transmitted. Comfortable/safe and creative learning spaces can inspire and motivate users, while ugly/unsafe spaces can oppress. Based on these two attitudes, the aims of this paper are to; firstly, developing Sustainable learning environments (SLE) in the Middle-East countries with reference to Egypt. Secondly, to reviewing and extending the planning and design of the internal, external and landscaping features of a proposed eco-class to collectively pass to the learners for enhancing the quality of learning space and thus education. After the Egyptian Revolution on the 25th of January, 2011 and the hopes and dreams this brings with it, for a major transformation in all life sectors, the Egyptian government needs to recognise the right of children and young people to learn in an environment which is safe, healthy and achieves the highest quality possible. We must all be committed to improving the quality, attractiveness and health of the learning and communal spaces in our schools. Environmental factors have significant effects on pupil and teacher wellbeing. In contrast, poor school and classroom design can affect concentration, creativity and general well-being; in addition, poor quality lighting, ventilation, acoustics and furniture all have a negative effect on student achievement and health. Nowadays, Egypt endure deterioration of education quality as a result of deficient learning spaces, high number of pupils in class, insufficient governmental expenditure and funding, and lack of proper research in education developmental strategies. Therefore, new learning spaces should be able to increase flexibility in order to support hands-on and outside-class learning activities. Furthermore, they intend to encourage extra-curricula activities beyond conventional learning times. Currently, these integral learning-components are crucial for socio-cultural sustainability and positive initiatives towards minimizing recent educational underachievement. Undoubtedly, comfortable, safe and creative learning spaces can inspire and motivate users, while ugly/unsafe spaces can depress. Therefore, well-designed learning spaces are able to support creative, productive and efficient learning processes on one hand. On the other hand, ecological design measures became increasingly major keystone for modern sustainable learning-spaces. Thus, learning-spaces’ design process, form, components, materials, features, and energy-saving technologies can generate well-educated, environmental-literate, energy-conscious, and innovative future-generations. (Continued

Minimax and Maximin Fitting of Geometric Objects to Sets of Points

This thesis addresses several problems in the facility location sub-area of computational geometry. Let S be a set of n points in the plane. We derive algorithms for approximating S by a step function curve of size k \u3c n, i.e., by an x-monotone orthogonal polyline ℜ with k \u3c n horizontal segments. We use the vertical distance to measure the quality of the approximation, i.e., the maximum distance from a point in S to the horizontal segment directly above or below it. We consider two types of problems: min-ε, where the goal is to minimize the error for a given number of horizontal segments k and min-#, where the goal is to minimize the number of segments for a given allowed error ε. After O(n) preprocessing time, we solve instances of the latter in O(min{k log n, n}) time per instance. We can then solve the former problem in O(min{n2, nk log n}) time. Both algorithms require O(n) space. The second contribution is a heuristic for the min-ε problem that computes a solution within a factor of 3 of the optimal error for k segments, or with at most the same error as the k-optimal but using 2k - 1 segments. Furthermore, experiments on real data show even better results than what is guaranteed by the theoretical bounds. Both approximations run in O(n log n) time and O(n) space. Then, we present an exact algorithm for the weighted version of this problem that runs in O(n2) time and generalize the heuristic to handle weights at the expense of an additional log n factor. At this point, a randomized algorithm that runs in O(n log2 n) expected time for the unweighted version is presented. It easily generalizes to the weighted case, though at the expense of an additional log n factor. Finally, we treat the maximin problem and present an O(n3 log n) solution to the problem of finding the furthest separating line through a set of weighted points. We conclude with solutions to the obnoxious wedge problem: an O(n2 log n) algorithm for the general case of a wedge with its apex on the boundary of the convex hull of S and an O(n2) algorithm for the case of the apex of a wedge coming from the input set S

Domestic spaces and cultural geography

This paper discusses the geographic dimension of domestic spaces and advocates for their integration on cultural geographers' research agenda. The argument moves from a general discussion of how domestic spaces can be defined to the presentation of a case study conducted by the author among the Inuinnait (formerly known as Copper Inuit) of the Canadian Arctic at the turn of the 2000s. This serves as a base to demonstrate why domestic spaces matter for cultural geographers. Within the book in it which the paper is published, it serves as an example of the diversity of topics cultural geographers tackle.Cet article traite des espaces domestiques, de leur géographicité et de leur importance pour la géographie culturelle

Querying for the Largest Empty Geometric Object in a Desired Location

We study new types of geometric query problems defined as follows: given a geometric set $P$, preprocess it such that given a query point $q$, the location of the largest circle that does not contain any member of $P$, but contains $q$ can be reported efficiently. The geometric sets we consider for $P$ are boundaries of convex and simple polygons, and point sets. While we primarily focus on circles as the desired shape, we also briefly discuss empty rectangles in the context of point sets.Comment: This version is a significant update of our earlier submission arXiv:1004.0558v1. Apart from new variants studied in Sections 3 and 4, the results have been improved in Section 5.Please note that the change in title and abstract indicate that we have expanded the scope of the problems we stud