Discrete & Computational Geometry最新文献_第7页

Computing the Homology Functor on Semi-algebraic Maps and Diagrams 计算半代数映射和图的同调函数

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-02-14 DOI: 10.1007/s00454-024-00627-z

Saugata Basu, Negin Karisani

{"title":"Computing the Homology Functor on Semi-algebraic Maps and Diagrams","authors":"Saugata Basu, Negin Karisani","doi":"10.1007/s00454-024-00627-z","DOIUrl":"https://doi.org/10.1007/s00454-024-00627-z","url":null,"abstract":"Developing an algorithm for computing the Betti numbers of semi-algebraic sets with singly exponential complexity has been a holy grail in algorithmic semi-algebraic geometry and only partial results are known. In this paper we consider the more general problem of computing the image under the homology functor of a continuous semi-algebraic map (f:X rightarrow Y) between closed and bounded semi-algebraic sets. For every fixed (ell ge 0) we give an algorithm with singly exponential complexity that computes bases of the homology groups (text{ H}_i(X), text{ H}_i(Y)) (with rational coefficients) and a matrix with respect to these bases of the induced linear maps (text{ H}_i(f):text{ H}_i(X) rightarrow text{ H}_i(Y), 0 le i le ell ). We generalize this algorithm to more general (zigzag) diagrams of continuous semi-algebraic maps between closed and bounded semi-algebraic sets and give a singly exponential algorithm for computing the homology functors on such diagrams. This allows us to give an algorithm with singly exponential complexity for computing barcodes of semi-algebraic zigzag persistent homology in small dimensions.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"13 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139766220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Convexity, Elementary Methods, and Distances 凸性、初等方法和距离

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-02-03 DOI: 10.1007/s00454-023-00625-7

Oliver Roche-Newton, Dmitrii Zhelezov

引用次数: 0

Locally Finite Completions of Polyhedral Complexes 多面体复合物的局部有限补全

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-02-03 DOI: 10.1007/s00454-024-00629-x

Desmond Coles, Netanel Friedenberg

引用次数: 0

Peeling Sequences 剥离序列

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-02-02 DOI: 10.1007/s00454-023-00616-8

Adrian Dumitrescu, Géza Tóth

引用次数: 0

Computable Bounds for the Reach and r-Convexity of Subsets of $${{mathbb {R}}}^d$$ $${{mathbb {R}}^d$$ 子集的可达性和 r-凸性的可计算边界

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-01-27 DOI: 10.1007/s00454-023-00624-8

Ryan Cotsakis

引用次数: 0

The Cone of $$5times 5$$ Completely Positive Matrices $$5 times 5$ 完全正矩阵的圆锥体

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-01-24 DOI: 10.1007/s00454-023-00620-y

Max Pfeffer, José Alejandro Samper

引用次数: 0

Almost Congruent Triangles 几乎全等的三角形

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-01-11 DOI: 10.1007/s00454-023-00623-9

{"title":"Almost Congruent Triangles","authors":"","doi":"10.1007/s00454-023-00623-9","DOIUrl":"https://doi.org/10.1007/s00454-023-00623-9","url":null,"abstract":"<h3>Abstract</h3> Almost 50 years ago Erdős and Purdy asked the following question: Given n points in the plane, how many triangles can be approximate congruent to equilateral triangles? They pointed out that by dividing the points evenly into three small clusters built around the three vertices of a fixed equilateral triangle, one gets at least (leftlfloor frac{n}{3} rightrfloor cdot leftlfloor frac{n+1}{3} rightrfloor cdot leftlfloor frac{n+2}{3} rightrfloor ) such approximate copies. In this paper we provide a matching upper bound and thereby answer their question. More generally, for every triangle T we determine the maximum number of approximate congruent triangles to T in a point set of size n. Parts of our proof are based on hypergraph Turán theory: for each point set in the plane and a triangle T, we construct a 3-uniform hypergraph (mathcal {H}=mathcal {H}(T)) , which contains no hypergraph as a subgraph from a family of forbidden hypergraphs (mathcal {F}=mathcal {F}(T)) . Our upper bound on the number of edges of (mathcal {H}) will determine the maximum number of triangles that are approximate congruent to T.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"75 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139460336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Euclidean TSP in Narrow Strips 窄带欧氏 TSP

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-01-08 DOI: 10.1007/s00454-023-00609-7

Henk Alkema, Mark de Berg, Remco van der Hofstad, Sándor Kisfaludi-Bak

{"title":"Euclidean TSP in Narrow Strips","authors":"Henk Alkema, Mark de Berg, Remco van der Hofstad, Sándor Kisfaludi-Bak","doi":"10.1007/s00454-023-00609-7","DOIUrl":"https://doi.org/10.1007/s00454-023-00609-7","url":null,"abstract":"We investigate how the complexity of Euclidean TSP for point sets P inside the strip ((-infty ,+infty )times [0,delta ]) depends on the strip width (delta ). We obtain two main results.<ul>\u0000<li>\u0000For the case where the points have distinct integer x-coordinates, we prove that a shortest bitonic tour (which can be computed in (O(nlog ^2 n)) time using an existing algorithm) is guaranteed to be a shortest tour overall when (delta leqslant 2sqrt{2}), a bound which is best possible.\u0000</li>\u0000<li>\u0000We present an algorithm that is fixed-parameter tractable with respect to (delta ). Our algorithm has running time (2^{O(sqrt{delta })} n + O(delta ^2 n^2)) for sparse point sets, where each (1times delta ) rectangle inside the strip contains O(1) points. For random point sets, where the points are chosen uniformly at random from the rectangle ([0,n]times [0,delta ]), it has an expected running time of (2^{O(sqrt{delta })} n). These results generalise to point sets P inside a hypercylinder of width (delta ). In this case, the factors (2^{O(sqrt{delta })}) become (2^{O(delta ^{1-1/d})}).\u0000</li>\u0000</ul>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"72 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139408141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Counting Arcs in $${mathbb {F}}_q^2$$ 计算 $${mathbb {F}}_q^2$ 中的弧线

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-01-08 DOI: 10.1007/s00454-023-00622-w

Krishnendu Bhowmick, Oliver Roche-Newton

{"title":"Counting Arcs in $${mathbb {F}}_q^2$$","authors":"Krishnendu Bhowmick, Oliver Roche-Newton","doi":"10.1007/s00454-023-00622-w","DOIUrl":"https://doi.org/10.1007/s00454-023-00622-w","url":null,"abstract":"An arc in (mathbb F_q^2) is a set (P subset mathbb F_q^2) such that no three points of P are collinear. We use the method of hypergraph containers to prove several counting results for arcs. Let ({mathcal {A}}(q)) denote the family of all arcs in (mathbb F_q^2). Our main result is the bound $$begin{aligned} |{mathcal {A}}(q)| le 2^{(1+o(1))q}. end{aligned}$$This matches, up to the factor hidden in the o(1) notation, the trivial lower bound that comes from considering all subsets of an arc of size q. We also give upper bounds for the number of arcs of a fixed (large) size. Let (k ge q^{2/3}(log q)^3), and let ({mathcal {A}}(q,k)) denote the family of all arcs in (mathbb F_q^2) with cardinality k. We prove that $$begin{aligned} |{mathcal {A}}(q,k)| le left( {begin{array}{c}(1+o(1))q kend{array}}right) . end{aligned}$$This result improves a bound of Roche-Newton and Warren [12]. A nearly matching lower bound $$begin{aligned} |{mathcal {A}}(q,k)| ge left( {begin{array}{c}q kend{array}}right) end{aligned}$$follows by considering all subsets of size k of an arc of size q.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"27 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139415409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Topological Optimization with Big Steps 大步拓扑优化

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-01-05 DOI: 10.1007/s00454-023-00613-x

Arnur Nigmetov, Dmitriy Morozov

引用次数: 0