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

Finding a widest empty 1-corner corridor

Given a set of n points in the plane, we consider the problem of computing a widest empty 1-corner corridor. We star giving a characterization of the 1-corner corridors that we call locally widest. Our approach to finding a widest empty 1-corner corridor consists of identifying a set of 1-corner corridors locally widest, that is guaranteed to contain a solution. We describe an algorithm that solves the problem in O(n4 log n) time and O(n) space.Ministerio de Ciencia y TecnologíaNational Science FoundationGeneralitat de Cataluny

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

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

Thermal Performance of a High-Rise Residential Building with Internal Courtyard in Tropical Climate

Natural ventilation is an effectual passive design approach to create a better indoor thermal condition as well as energy efficiency. The primary goal of building design is providing a healthy and comfortable indoor environment titled as sustainable architecture.  Literature suggests that the significant feature that alteration has to take place on for better energy performance is the envelope design. This paper aims to augment the Window to Wall Ratio (WWR), orientation and courtyard corridor size for improving the design of naturally ventilated courtyard high-rise residential buildings. Briefly, the findings indicate that contending with WWR, orientation and courtyard corridor size could increase the potential of improving its natural ventilation and thus, thermal performance

The function of pedestrian-oriented open space in Knoxville\u27s central business district

Man is constantly affected by his physical environment. Pro-viding pleasant physical elements in open spaces—such as parks, squares, plazas, small green spaces, and sidewalks--helps to produce a better environment in densely developed urban areas. The purpose of this study is to find how the existing open spaces in Knoxville\u27s central business district function by surveying their characteristics of continuity, direction, orientation, center, linkage, enclosure, interruption, circulation, and sequence. A check-list was made to make sure that each site was inspected under the same terms. The data which was obtained by observing the physical items under each characteristic is shown in figures and tables, which indicate the overall condition of a certain survey item. The analysis of the data indicate the strengths and weaknesses of each open space. The result--a description of the character of the whole system of pedestrian open spaces—showed that many marginally functional open spaces break the continuity of open spaces which function well and complement Knoxville\u27s central business district. These findings indicate which places need to be improved the most. Open spaces on Hill Avenue, Main Avenue, Summit Hill Drive, Wall Avenue, Market Square, and Market Street were found to have a relatively pleasant pedestrian environment

Sleeping dragons

In six months Wyce has learned almost nothing about the research compound that has become his prison. That there are mages amongst his captors can only mean that someone within the Magister\u27s Council endorses this facility, but while this fact disturbs him deeply as a student of magic himself, it leads him no closer to understanding. His reality has become a waking nightmare, and over time constant experimentation and abuse degrade even his natural curiosity. He fights an impending sense of complacency, internally outraged at his failure to act in his own interest, constantly reminded that because of his spell stutter - a magical disability - he is less of a threat to the soldiers that guard him than a well-educated child. When the woman in the cell next to his proposes an escape Wyce pledges his loyalty and assistance in spite of her refusal to share the details of her plan. Escaping seems to be the only thing that matters, but in the full course of time Wyce must realize that there are always other considerations. The chapters herein form the opening to a novel-in-progress that imagines a world where modern science forges magic instead of machines; where commercialism is rampant and apathy is a default state of being. The novel follows Wyce and Reina in their escape from the compound until their status as wanted criminals finally lands them in the hands of a young fortune hunter named Spades, whose decision not to hand the two over to authorities forces him into taking a stand on issues that he\u27s made a career out of avoiding. How to navigate the growing tide of modernity is a question that speaks to all of us, and the struggle of these three companions to do just that is a story for all of us