Pub Date : 2024-04-17DOI: 10.1016/j.comgeo.2024.102101
Christian Jung, Claudia Redenbach
Tessellations are an important tool to model the microstructure of cellular and polycrystalline materials. Classical tessellation models include the Voronoi diagram and the Laguerre tessellation whose cells are polyhedra. Due to the convexity of their cells, those models may be too restrictive to describe data that includes possibly anisotropic grains with curved boundaries. Several generalizations exist. The cells of the generalized balanced power diagram are induced by elliptic distances leading to more diverse structures. So far, methods for computing the generalized balanced power diagram are restricted to discretized versions in the form of label images. In this work, we derive an analytic representation of the vertices and edges of the generalized balanced power diagram in 2d. Based on that, we propose a novel algorithm to compute the whole diagram.
{"title":"An analytical representation of the 2d generalized balanced power diagram","authors":"Christian Jung, Claudia Redenbach","doi":"10.1016/j.comgeo.2024.102101","DOIUrl":"https://doi.org/10.1016/j.comgeo.2024.102101","url":null,"abstract":"<div><p>Tessellations are an important tool to model the microstructure of cellular and polycrystalline materials. Classical tessellation models include the Voronoi diagram and the Laguerre tessellation whose cells are polyhedra. Due to the convexity of their cells, those models may be too restrictive to describe data that includes possibly anisotropic grains with curved boundaries. Several generalizations exist. The cells of the generalized balanced power diagram are induced by elliptic distances leading to more diverse structures. So far, methods for computing the generalized balanced power diagram are restricted to discretized versions in the form of label images. In this work, we derive an analytic representation of the vertices and edges of the generalized balanced power diagram in 2d. Based on that, we propose a novel algorithm to compute the whole diagram.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0925772124000233/pdfft?md5=fa13c50875805de5231691dc670463bc&pid=1-s2.0-S0925772124000233-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140633213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-09DOI: 10.1016/j.comgeo.2024.102099
Marcus Brazil , Michael Hendriksen , Jae Lee , Michael S. Payne , Charl Ras , Doreen Thomas
The Euclidean k-Steiner tree problem asks for a minimum-cost network connecting n given points in the plane, allowing at most k additional nodes referred to as Steiner points. In the classical Steiner tree problem in which there is no restriction on the number of nodes, every Steiner point must be of degree 3. The k-Steiner problem differs in that Steiner points of degree 4 may be included in an optimal solution. This simple change leads to a number of complexities when attempting to create a generation algorithm for optimal k-Steiner trees, which has proven to be a powerful component of the flagship algorithm, namely GeoSteiner, for solving the classical Steiner tree problem. In the present paper we firstly extend the basic framework of GeoSteiner's generation algorithm to include degree-4 Steiner points. We then introduce a number of novel results restricting the structural and geometric properties of optimal k-Steiner trees, and then show how these properties may be used as topological pruning methods underpinning our generation algorithm. Finally, we present experimental data to show the effectiveness of our pruning methods in reducing the number of sub-optimal solution topologies.
{"title":"An exact algorithm for the Euclidean k-Steiner tree problem","authors":"Marcus Brazil , Michael Hendriksen , Jae Lee , Michael S. Payne , Charl Ras , Doreen Thomas","doi":"10.1016/j.comgeo.2024.102099","DOIUrl":"https://doi.org/10.1016/j.comgeo.2024.102099","url":null,"abstract":"<div><p>The Euclidean <em>k</em>-Steiner tree problem asks for a minimum-cost network connecting <em>n</em> given points in the plane, allowing at most <em>k</em> additional nodes referred to as <em>Steiner points</em>. In the classical Steiner tree problem in which there is no restriction on the number of nodes, every Steiner point must be of degree 3. The <em>k</em>-Steiner problem differs in that Steiner points of degree 4 may be included in an optimal solution. This simple change leads to a number of complexities when attempting to create a <em>generation algorithm</em> for optimal <em>k</em>-Steiner trees, which has proven to be a powerful component of the flagship algorithm, namely GeoSteiner, for solving the classical Steiner tree problem. In the present paper we firstly extend the basic framework of GeoSteiner's generation algorithm to include degree-4 Steiner points. We then introduce a number of novel results restricting the structural and geometric properties of optimal <em>k</em>-Steiner trees, and then show how these properties may be used as topological pruning methods underpinning our generation algorithm. Finally, we present experimental data to show the effectiveness of our pruning methods in reducing the number of sub-optimal solution topologies.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S092577212400021X/pdfft?md5=e2fe47e64b9ad0273f7021d80587df58&pid=1-s2.0-S092577212400021X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140555528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-05DOI: 10.1016/j.comgeo.2024.102100
Kentaro Hayakawa , Zeyuan He , Simon D. Guest
In this study, we lay the groundwork for a systematic investigation of the rigidity and flexibility of rigid origami by using the mathematical model referred to as the panel-point model. Rigid origami is commonly known as a type of panel-hinge structure where rigid polygonal panels are connected by rotational hinges, and its motion and stability are often investigated from the perspective of its consistency constraints representing the rigidity and connection conditions of panels. In the proposed methodology, vertex coordinates are directly treated as the variables to represent the rigid origami in the panel-point model, and these variables are constrained by the conditions for the out-of-plane and in-plane rigidity of panels. This model offers several advantages including: 1) the simplicity of polynomial consistency constraints; 2) the ease of incorporating displacement boundary conditions; and 3) the straightforwardness of numerical simulation and visualization. It is anticipated that the presented theories in this article are valuable to a broad audience, including mathematicians, engineers, and architects.
{"title":"Panel-point model for rigidity and flexibility analysis of rigid origami","authors":"Kentaro Hayakawa , Zeyuan He , Simon D. Guest","doi":"10.1016/j.comgeo.2024.102100","DOIUrl":"https://doi.org/10.1016/j.comgeo.2024.102100","url":null,"abstract":"<div><p>In this study, we lay the groundwork for a systematic investigation of the rigidity and flexibility of rigid origami by using the mathematical model referred to as the panel-point model. Rigid origami is commonly known as a type of panel-hinge structure where rigid polygonal panels are connected by rotational hinges, and its motion and stability are often investigated from the perspective of its consistency constraints representing the rigidity and connection conditions of panels. In the proposed methodology, vertex coordinates are directly treated as the variables to represent the rigid origami in the panel-point model, and these variables are constrained by the conditions for the out-of-plane and in-plane rigidity of panels. This model offers several advantages including: 1) the simplicity of polynomial consistency constraints; 2) the ease of incorporating displacement boundary conditions; and 3) the straightforwardness of numerical simulation and visualization. It is anticipated that the presented theories in this article are valuable to a broad audience, including mathematicians, engineers, and architects.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140555529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-06DOI: 10.1016/j.comgeo.2024.102088
Sang Won Bae , Sandip Banerjee , Arpita Baral , Priya Ranjan Sinha Mahapatra , Sang Duk Yoon
Given a set of n colored points with k colors in the plane, we study the problem of computing a maximum-width rainbow-bisecting empty annulus (of objects specifically axis-parallel square, axis-parallel rectangle and circle) problem. We call a region rainbow if it contains at least one point of each color. The maximum-width rainbow-bisecting empty annulus problem asks to find an annulus A of a particular shape with maximum possible width such that A does not contain any input points and it bisects the input point set into two parts, each of which is a rainbow. We compute a maximum-width rainbow-bisecting empty axis-parallel square, axis-parallel rectangular and circular annulus in time using space, in time using space and in time using space respectively.
给定平面上 k 种颜色的 n 个彩色点的集合,我们研究的是计算最大宽度彩虹等分线空环面(对象具体为轴平行的正方形、轴平行的矩形和圆形)问题。如果一个区域至少包含每种颜色的一个点,我们就称该区域为彩虹区域。最大宽度彩虹分叉空环问题要求找到一个特定形状的最大宽度环 A,使得 A 不包含任何输入点,并且将输入点集一分为二,每一部分都是彩虹。我们使用 O(n) 空间在 O(n3) 时间内、使用 O(nlogn) 空间在 O(k2n2logn) 时间内以及使用 O(n2) 空间在 O(n3) 时间内分别计算出了最大宽度的彩虹分叉空轴平行正方形、轴平行矩形和圆形环面。
{"title":"Maximum-width rainbow-bisecting empty annulus","authors":"Sang Won Bae , Sandip Banerjee , Arpita Baral , Priya Ranjan Sinha Mahapatra , Sang Duk Yoon","doi":"10.1016/j.comgeo.2024.102088","DOIUrl":"https://doi.org/10.1016/j.comgeo.2024.102088","url":null,"abstract":"<div><p>Given a set of <em>n</em> colored points with <em>k</em> colors in the plane, we study the problem of computing a maximum-width rainbow-bisecting empty annulus (of objects specifically axis-parallel square, axis-parallel rectangle and circle) problem. We call a region <em>rainbow</em> if it contains at least one point of each color. The maximum-width rainbow-bisecting empty annulus problem asks to find an annulus <em>A</em> of a particular shape with maximum possible width such that <em>A</em> does not contain any input points and it bisects the input point set into two parts, each of which is a <em>rainbow</em>. We compute a maximum-width rainbow-bisecting empty axis-parallel square, axis-parallel rectangular and circular annulus in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span> time using <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> space, in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> time using <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> space and in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span> time using <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> space respectively.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140113725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-04DOI: 10.1016/j.comgeo.2024.102090
Peyman Afshani , Rasmus Killmann , Kasper G. Larsen
In colored range counting (CRC), the input is a set of points where each point is assigned a “color” (or a “category”) and the goal is to store them in a data structure such that the number of distinct categories inside a given query range can be counted efficiently. CRC has strong motivations as it allows data structure to deal with categorical data.
However, colors (i.e., the categories) in the CRC problem do not have any internal structure, whereas this is not the case for many datasets in practice where hierarchical categories exist or where a single input belongs to multiple categories. Motivated by these, we consider variants of the problem where such structures can be represented. We define two variants of the problem called hierarchical range counting (HCC) and sub-category colored range counting (SCRC) and consider hierarchical structures that can either be a DAG or a tree. We show that the two problems on some special trees are in fact equivalent to other well-known problems in the literature. Based on these, we also give efficient data structures when the underlying hierarchy can be represented as a tree. We show a conditional lower bound for the general case when the existing hierarchy can be any DAG, through a reduction from the orthogonal vectors problem.
{"title":"Hierarchical categories in colored searching","authors":"Peyman Afshani , Rasmus Killmann , Kasper G. Larsen","doi":"10.1016/j.comgeo.2024.102090","DOIUrl":"10.1016/j.comgeo.2024.102090","url":null,"abstract":"<div><p>In colored range counting (CRC), the input is a set of points where each point is assigned a “color” (or a “category”) and the goal is to store them in a data structure such that the number of distinct categories inside a given query range can be counted efficiently. CRC has strong motivations as it allows data structure to deal with categorical data.</p><p>However, colors (i.e., the categories) in the CRC problem do not have any internal structure, whereas this is not the case for many datasets in practice where hierarchical categories exist or where a single input belongs to multiple categories. Motivated by these, we consider variants of the problem where such structures can be represented. We define two variants of the problem called hierarchical range counting (HCC) and sub-category colored range counting (SCRC) and consider hierarchical structures that can either be a DAG or a tree. We show that the two problems on some special trees are in fact equivalent to other well-known problems in the literature. Based on these, we also give efficient data structures when the underlying hierarchy can be represented as a tree. We show a conditional lower bound for the general case when the existing hierarchy can be any DAG, through a reduction from the orthogonal vectors problem.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0925772124000129/pdfft?md5=58168aae21edfa03ea4bb23171502329&pid=1-s2.0-S0925772124000129-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140056725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-01DOI: 10.1016/j.comgeo.2024.102089
Yuan Luo , Bradley J. Nelson
Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations performed on many similar filtered topological spaces which is based on updating associated matrix factorizations. Our approach improves the update scheme of Cohen-Steiner, Edelsbrunner, and Morozov for permutations by additionally handling addition and deletion of cells in a filtered topological space and by processing changes in a single batch. We show that the complexity of our scheme scales with the number of elementary changes to the filtration which as a result is often less expensive than the full persistent homology computation. Finally, we perform computational experiments demonstrating practical speedups in several situations including feature generation and optimization guided by persistent homology.
{"title":"Accelerating iterated persistent homology computations with warm starts","authors":"Yuan Luo , Bradley J. Nelson","doi":"10.1016/j.comgeo.2024.102089","DOIUrl":"10.1016/j.comgeo.2024.102089","url":null,"abstract":"<div><p>Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations performed on many similar filtered topological spaces which is based on updating associated matrix factorizations. Our approach improves the update scheme of Cohen-Steiner, Edelsbrunner, and Morozov for permutations by additionally handling addition and deletion of cells in a filtered topological space and by processing changes in a single batch. We show that the complexity of our scheme scales with the number of elementary changes to the filtration which as a result is often less expensive than the full persistent homology computation. Finally, we perform computational experiments demonstrating practical speedups in several situations including feature generation and optimization guided by persistent homology.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140056633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have emerged as a way to provide a more precise and expressive control over the solution, relying on persistent homology to quantify and reduce its roughness. All such existing techniques back-propagate gradients through the persistence diagram, which is a summary of the topological features of a function. Their downside is that they provide information only at the critical points of the function. We propose a method that instead builds on persistence-sensitive simplification and translates the required changes to the persistence diagram into changes on large subsets of the domain, including both critical and regular points. This approach enables a faster and more precise topological regularization, the benefits of which we illustrate with experimental evidence.
{"title":"Topological regularization via persistence-sensitive optimization","authors":"Arnur Nigmetov , Aditi Krishnapriyan , Nicole Sanderson , Dmitriy Morozov","doi":"10.1016/j.comgeo.2024.102086","DOIUrl":"10.1016/j.comgeo.2024.102086","url":null,"abstract":"<div><p>Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have emerged as a way to provide a more precise and expressive control over the solution, relying on persistent homology to quantify and reduce its roughness. All such existing techniques back-propagate gradients through the persistence diagram, which is a summary of the topological features of a function. Their downside is that they provide information only at the critical points of the function. We propose a method that instead builds on persistence-sensitive simplification and translates the required changes to the persistence diagram into changes on large subsets of the domain, including both critical and regular points. This approach enables a faster and more precise topological regularization, the benefits of which we illustrate with experimental evidence.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0925772124000087/pdfft?md5=6740a147d9e195f49dbdb29746bfe080&pid=1-s2.0-S0925772124000087-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140007733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-27DOI: 10.1016/j.comgeo.2024.102087
Daniel McGinnis
We define a to be a family of k sets such that when (indices are taken modulo k). We show that if is a family of compact, convex sets that does not contain a , then there are lines that pierce . Additionally, we give an example of a family of compact, convex sets that contains no and cannot be pierced by lines.
我们将 a 定义为这样的集合族,即当(指数取模)时,a 。我们证明,如果是一个不包含 a 的紧凑凸集族,那么就有直线穿透 。此外,我们还给出了一个紧凑凸集合族的例子,它不包含且不能被直线穿透。
{"title":"Piercing families of convex sets in the plane that avoid a certain subfamily with lines","authors":"Daniel McGinnis","doi":"10.1016/j.comgeo.2024.102087","DOIUrl":"10.1016/j.comgeo.2024.102087","url":null,"abstract":"<div><p>We define a <span><math><mi>C</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> to be a family of <em>k</em> sets <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> such that <span><math><mtext>conv</mtext><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∪</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>∩</mo><mtext>conv</mtext><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∪</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mo>∅</mo></math></span> when <span><math><mo>{</mo><mi>i</mi><mo>,</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>}</mo><mo>∩</mo><mo>{</mo><mi>j</mi><mo>,</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo>}</mo><mo>=</mo><mo>∅</mo></math></span> (indices are taken modulo <em>k</em>). We show that if <span><math><mi>F</mi></math></span> is a family of compact, convex sets that does not contain a <span><math><mi>C</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span>, then there are <span><math><mi>k</mi><mo>−</mo><mn>2</mn></math></span> lines that pierce <span><math><mi>F</mi></math></span>. Additionally, we give an example of a family of compact, convex sets that contains no <span><math><mi>C</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> and cannot be pierced by <span><math><mrow><mo>⌈</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>−</mo><mn>1</mn></math></span> lines.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140007731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-22DOI: 10.1016/j.comgeo.2023.102078
Judith Brecklinghaus, Ulrich Brenner, Oliver Kiss
We examine rectangle packing problems where only the areas of the rectangles to be packed are given while their aspect ratios may be chosen from a given interval . In particular, we ask for the smallest possible size of a rectangle R such that, under these constraints, any collection of rectangle areas of total size 1 can be packed into R. As for standard square packing problems, which are contained as a special case for , this question leads us to three different answers, depending on whether the aspect ratio of R is given or whether we may choose it either with or without knowing the areas . Generalizing known results for square packing problems, we provide upper and lower bounds for the size of R with respect to all three variants of the problem, which are tight at least for larger values of γ. Moreover, we show how to improve these bounds on the size of R if we restrict ourselves to instances where the largest element in is bounded.
我们研究的矩形打包问题只给出待打包矩形的面积 a1、...、an,而它们的长宽比可以从给定区间 [1γ,γ]中选择。对于作为 γ=1 的特例而包含的标准正方形堆积问题,这个问题有三种不同的答案,取决于 R 的长宽比是给定的,还是可以在知道或不知道面积 a1、...、an 的情况下选择。根据已知的正方形包装问题的结果,我们提供了与问题的所有三个变体有关的 R 大小的上界和下界,这些上界和下界至少对较大的 γ 值是严密的。此外,我们还展示了如果我们将自己限制在 a1、...,an 中最大元素有界的实例中,如何改进 R 大小的这些界值。
{"title":"Bounds on soft rectangle packing ratios","authors":"Judith Brecklinghaus, Ulrich Brenner, Oliver Kiss","doi":"10.1016/j.comgeo.2023.102078","DOIUrl":"10.1016/j.comgeo.2023.102078","url":null,"abstract":"<div><p><span>We examine rectangle packing problems where only the areas </span><span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span> of the rectangles to be packed are given while their aspect ratios may be chosen from a given interval </span><span><math><mo>[</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>γ</mi></mrow></mfrac><mo>,</mo><mi>γ</mi><mo>]</mo></math></span>. In particular, we ask for the smallest possible size of a rectangle <em>R</em> such that, under these constraints, any collection <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span> of rectangle areas of total size 1 can be packed into </span><em>R</em>. As for standard square packing problems, which are contained as a special case for <span><math><mi>γ</mi><mo>=</mo><mn>1</mn></math></span>, this question leads us to three different answers, depending on whether the aspect ratio of <em>R</em> is given or whether we may choose it either with or without knowing the areas <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span>. Generalizing known results for square packing problems, we provide upper and lower bounds for the size of </span><em>R</em> with respect to all three variants of the problem, which are tight at least for larger values of <em>γ</em>. Moreover, we show how to improve these bounds on the size of <em>R</em> if we restrict ourselves to instances where the largest element in <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is bounded.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139025451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}