{"title":"Dependable Spanners via Unreliable Edges","authors":"Sariel Har-Peled, Maria C. Lusardi","doi":"arxiv-2407.01466","DOIUrl":"https://doi.org/arxiv-2407.01466","url":null,"abstract":"Let $P$ be a set of $n$ points in $mathbb{R}^d$, and let $varepsilon,psi\u0000in (0,1)$ be parameters. Here, we consider the task of constructing a\u0000$(1+varepsilon)$-spanner for $P$, where every edge might fail (independently)\u0000with probability $1-psi$. For example, for $psi=0.1$, about $90%$ of the\u0000edges of the graph fail. Nevertheless, we show how to construct a spanner that\u0000survives such a catastrophe with near linear number of edges. The measure of reliability of the graph constructed is how many pairs of\u0000vertices lose $(1+varepsilon)$-connectivity. Surprisingly, despite the spanner\u0000constructed being of near linear size, the number of failed pairs is close to\u0000the number of failed pairs if the underlying graph was a clique. Specifically, we show how to construct such an exact dependable spanner in\u0000one dimension of size $O(tfrac{n}{psi} log n)$, which is optimal. Next, we\u0000build an $(1+varepsilon)$-spanners for a set $P subseteq mathbb{R}^d$ of $n$\u0000points, of size $O( C n log n )$, where $C approx 1/bigl(varepsilon^{d}\u0000psi^{4/3}bigr)$. Surprisingly, these new spanners also have the property that\u0000almost all pairs of vertices have a $leq 4$-hop paths between them realizing\u0000this short path.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Edge-Unfolding Polycubes with Orthogonally Convex Layers","authors":"Mirela Damian, Henk Meijer","doi":"arxiv-2407.01326","DOIUrl":"https://doi.org/arxiv-2407.01326","url":null,"abstract":"A polycube is an orthogonal polyhedron composed of unit cubes glued together\u0000along entire faces, homeomorphic to a sphere. A polycube layer is the section\u0000of the polycube that lies between two horizontal cross-sections of the polycube\u0000at unit distance from each other. An edge unfolding of a polycube involves\u0000cutting its surface along any of the constituent cube edges and flattening it\u0000into a single, non-overlapping planar piece. We show that any polycube with\u0000orthogonally convex layers can be edge unfolded.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Strictly Self-Assembling Discrete Self-Similar Fractals Using Quines","authors":"Daniel Hader, Matthew J. Patitz","doi":"arxiv-2406.19595","DOIUrl":"https://doi.org/arxiv-2406.19595","url":null,"abstract":"The abstract Tile-Assembly Model (aTAM) was initially introduced as a simple\u0000model for DNA-based self-assembly, where synthetic strands of DNA are used not\u0000as an information storage medium, but rather a material for nano-scale\u0000construction. Since then, it has been shown that the aTAM, and variant models\u0000thereof, exhibit rich computational dynamics, Turing completeness, and\u0000intrinsic universality, a geometric notion of simulation wherein one aTAM\u0000system is able to simulate every other aTAM system not just symbolically, but\u0000also geometrically. An intrinsically universal system is able to simulate all\u0000other systems within some class so that $mtimes m$ blocks of tiles behave in\u0000all ways like individual tiles in the system to be simulated. In this paper, we\u0000explore the notion of a quine in the aTAM with respect to intrinsic\u0000universality. Typically a quine refers to a program which does nothing but\u0000print its own description with respect to a Turing universal machine which may\u0000interpret that description. In this context, we replace the notion of machine\u0000with that of an aTAM system and the notion of Turing universality with that of\u0000intrinsic universality. Curiously, we find that doing so results in a\u0000counterexample to a long-standing conjecture in the theory of tile-assembly,\u0000namely that discrete self-similar fractals (DSSFs), fractal shapes generated\u0000via substitution tiling, cannot be strictly self-assembled. We find that by\u0000growing an aTAM quine, a tile system which intrinsically simulates itself, DSSF\u0000structure is naturally exhibited. This paper describes the construction of such\u0000a quine and even shows that essentially any desired fractal dimension between 1\u0000and 2 may be achieved.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ivor van der Hoog, André Nusser, Eva Rotenberg, Frank Staals
{"title":"Fully-Adaptive Dynamic Connectivity of Square Intersection Graphs","authors":"Ivor van der Hoog, André Nusser, Eva Rotenberg, Frank Staals","doi":"arxiv-2406.20065","DOIUrl":"https://doi.org/arxiv-2406.20065","url":null,"abstract":"A classical problem in computational geometry and graph algorithms is: given\u0000a dynamic set S of geometric shapes in the plane, efficiently maintain the\u0000connectivity of the intersection graph of S. Previous papers studied the\u0000setting where, before the updates, the data structure receives some parameter\u0000P. Then, updates could insert and delete disks as long as at all times the\u0000disks have a diameter that lies in a fixed range [1/P, 1]. The state-of-the-art\u0000for storing disks in a dynamic connectivity data structure is a data structure\u0000that uses O(Pn) space and that has amortized O(P log^4 n) expected amortized\u0000update time. Connectivity queries between disks are supported in O( log n /\u0000loglog n) time. The state-of-the-art for Euclidean disks immediately implies a\u0000data structure for connectivity between axis-aligned squares that have their\u0000diameter in the fixed range [1/P, 1], with an improved update time of O(P log^4\u0000n) amortized time. We restrict our attention to axis-aligned squares, and study fully-dynamic\u0000square intersection graph connectivity. Our result is fully-adaptive to the\u0000aspect ratio, spending time proportional to the current aspect ratio {psi}, as\u0000opposed to some previously given maximum P. Our focus on squares allows us to\u0000simplify and streamline the connectivity pipeline from previous work. When $n$\u0000is the number of squares and {psi} is the aspect ratio after insertion (or\u0000before deletion), our data structure answers connectivity queries in O(log n /\u0000loglog n) time. We can update connectivity information in O({psi} log^4 n +\u0000log^6 n) amortized time. We also improve space usage from O(P n log n) to O(n\u0000log^3 n log {psi}) -- while generalizing to a fully-adaptive aspect ratio --\u0000which yields a space usage that is near-linear in n for any polynomially\u0000bounded {psi}.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141530774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Robust Classification of Dynamic Bichromatic point Sets in R2","authors":"Erwin Glazenburg, Frank Staals, Marc van Kreveld","doi":"arxiv-2406.19161","DOIUrl":"https://doi.org/arxiv-2406.19161","url":null,"abstract":"Let $R cup B$ be a set of $n$ points in $mathbb{R}^2$, and let $k in\u00001..n$. Our goal is to compute a line that \"best\" separates the \"red\" points $R$\u0000from the \"blue\" points $B$ with at most $k$ outliers. We present an efficient\u0000semi-online dynamic data structure that can maintain whether such a separator\u0000exists. Furthermore, we present efficient exact and approximation algorithms\u0000that compute a linear separator that is guaranteed to misclassify at most $k$,\u0000points and minimizes the distance to the farthest outlier. Our exact algorithm\u0000runs in $O(nk + n log n)$ time, and our $(1+varepsilon)$-approximation\u0000algorithm runs in $O(varepsilon^{-1/2}((n + k^2) log n))$ time. Based on our\u0000$(1+varepsilon)$-approximation algorithm we then also obtain a semi-online\u0000data structure to maintain such a separator efficiently.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Outperforming the Best 1D Low-Discrepancy Constructions with a Greedy Algorithm","authors":"François Clément","doi":"arxiv-2406.18132","DOIUrl":"https://doi.org/arxiv-2406.18132","url":null,"abstract":"The design of uniformly spread sequences on $[0,1)$ has been extensively\u0000studied since the work of Weyl and van der Corput in the early $20^{text{th}}$\u0000century. The current best sequences are based on the Kronecker sequence with\u0000golden ratio and a permutation of the van der Corput sequence by Ostromoukhov.\u0000Despite extensive efforts, it is still unclear if it is possible to improve\u0000these constructions further. We show, using numerical experiments, that a\u0000radically different approach introduced by Kritzinger in seems to perform\u0000better than the existing methods. In particular, this construction is based on\u0000a emph{greedy} approach, and yet outperforms very delicate number-theoretic\u0000constructions. Furthermore, we are also able to provide the first numerical\u0000results in dimensions 2 and 3, and show that the sequence remains highly\u0000regular in this new setting.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Covering Simple Orthogonal Polygons with Rectangles","authors":"Aniket Basu Roy","doi":"arxiv-2406.16209","DOIUrl":"https://doi.org/arxiv-2406.16209","url":null,"abstract":"We study the problem of Covering Orthogonal Polygons with Rectangles. For\u0000polynomial-time algorithms, the best-known approximation factor is\u0000$O(sqrt{log n})$ when the input polygon may have holes [Kumar and Ramesh,\u0000STOC '99, SICOMP '03], and there is a $2$-factor approximation algorithm known\u0000when the polygon is hole-free [Franzblau, SIDMA '89]. Arguably, an easier\u0000problem is the Boundary Cover problem where we are interested in covering only\u0000the boundary of the polygon in contrast to the original problem where we are\u0000interested in covering the interior of the polygon, hence it is also referred\u0000as the Interior Cover problem. For the Boundary Cover problem, a $4$-factor\u0000approximation algorithm is known to exist and it is APX-hard when the polygon\u0000has holes [Berman and DasGupta, Algorithmica '94]. In this work, we investigate how effective is local search algorithm for the\u0000above covering problems on simple polygons. We prove that a simple local search\u0000algorithm yields a PTAS for the Boundary Cover problem when the polygon is\u0000simple. Our proof relies on the existence of planar supports on appropriate\u0000hypergraphs defined on the Boundary Cover problem instance. On the other hand,\u0000we construct instances where support graphs for the Interior Cover problem have\u0000arbitrarily large bicliques, thus implying that the same local search technique\u0000cannot yield a PTAS for this problem. We also show large locality gap for its\u0000dual problem, namely the Maximum Antirectangle problem.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"79 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Constraints Matrices and Convergence Proof of TPMS2STEP","authors":"Yaonaiming Zhao, Qiang Zou","doi":"arxiv-2407.03352","DOIUrl":"https://doi.org/arxiv-2407.03352","url":null,"abstract":"TPMS is consistently described in the functional representation (F-rep)\u0000format, while modern CAD/CAM/CAE tools are built upon the boundary\u0000representation (B-rep) format. To solve this issue, translating TPMS to STEP is\u0000needed, called TPMS2STEP. This paper provides constraint matrices and\u0000convergence proof of TPMS2STEP so that $C^2$ continuity and an error bound of\u0000$2epsilon$ on the deviation can be ensured during the translation.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"SplineGen: a generative model for B-spline approximation of unorganized points","authors":"Qiang Zou, Lizhen Zhu","doi":"arxiv-2406.09692","DOIUrl":"https://doi.org/arxiv-2406.09692","url":null,"abstract":"This paper presents a learning-based method to solve the traditional\u0000parameterization and knot placement problems in B-spline approximation.\u0000Different from conventional heuristic methods or recent AI-based methods, the\u0000proposed method does not assume ordered or fixed-size data points as input.\u0000There is also no need for manually setting the number of knots. It casts the\u0000parameterization and knot placement problems as a sequence-to-sequence\u0000translation problem, a generative process automatically determining the number\u0000of knots, their placement, parameter values, and their ordering. Once trained,\u0000SplineGen demonstrates a notable improvement over existing methods, with a one\u0000to two orders of magnitude increase in approximation accuracy on test data.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Seongjun Hong, Yongmin Kwon, Dongju Shin, Jangseop Park, Namwoo Kang
{"title":"DeepJEB: 3D Deep Learning-based Synthetic Jet Engine Bracket Dataset","authors":"Seongjun Hong, Yongmin Kwon, Dongju Shin, Jangseop Park, Namwoo Kang","doi":"arxiv-2406.09047","DOIUrl":"https://doi.org/arxiv-2406.09047","url":null,"abstract":"Recent advancements in artificial intelligence (AI) have significantly\u0000influenced various fields, including mechanical engineering. Nonetheless, the\u0000development of high-quality, diverse datasets for structural analysis still\u0000needs to be improved. Although traditional datasets, such as simulated jet\u0000engine bracket dataset, are useful, they are constrained by a small number of\u0000samples, which must be improved for developing robust data-driven surrogate\u0000models. This study presents the DeepJEB dataset, which has been created using\u0000deep generative models and automated engineering simulation pipelines, to\u0000overcome these challenges. Moreover, this study provides comprehensive 3D\u0000geometries and their corresponding structural analysis data. Key experiments validated the effectiveness of the DeepJEB dataset,\u0000demonstrating significant improvements in the prediction accuracy and\u0000reliability of surrogate models trained on this data. The enhanced dataset\u0000showed a broader design space and better generalization capabilities than\u0000traditional datasets. These findings highlight the potential of DeepJEB as a\u0000benchmark dataset for developing reliable surrogate models in structural\u0000engineering. The DeepJEB dataset supports advanced modeling techniques, such as\u0000graph neural networks (GNNs) and high-dimensional convolutional networks\u0000(CNNs), leveraging node-level field data for precise predictions. This dataset\u0000is set to drive innovation in engineering design applications, enabling more\u0000accurate and efficient structural performance predictions. The DeepJEB dataset\u0000is publicly accessible at: https://www.narnia.ai/dataset","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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