Discrete & Computational Geometry最新文献_第8页

Plurality in Spatial Voting Games with Constant $$beta $$ 具有恒定 $$beta $$ 的空间投票游戏中的多元性

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-01-03 DOI: 10.1007/s00454-023-00619-5

Arnold Filtser, Omrit Filtser

{"title":"Plurality in Spatial Voting Games with Constant $$beta $$","authors":"Arnold Filtser, Omrit Filtser","doi":"10.1007/s00454-023-00619-5","DOIUrl":"https://doi.org/10.1007/s00454-023-00619-5","url":null,"abstract":"Consider a set V of voters, represented by a multiset in a metric space (X, d). The voters have to reach a decision—a point in X. A choice (pin X) is called a (beta )-plurality point for V, if for any other choice (qin X) it holds that (|{vin Vmid beta cdot d(p,v)le d(q,v)}| ge frac{|V|}{2}). In other words, at least half of the voters “prefer” p over q, when an extra factor of (beta ) is taken in favor of p. For (beta =1), this is equivalent to Condorcet winner, which rarely exists. The concept of (beta )-plurality was suggested by Aronov, de Berg, Gudmundsson, and Horton [TALG 2021] as a relaxation of the Condorcet criterion. Let (beta ^*_{(X,d)}=sup {beta mid text{ every } text{ finite } text{ multiset } V{ in}X{ admitsa}beta text{-plurality } text{ point }}). The parameter (beta ^*) determines the amount of relaxation required in order to reach a stable decision. Aronov et al. showed that for the Euclidean plane (beta ^*_{({mathbb {R}}^2,Vert cdot Vert _2)}=frac{sqrt{3}}{2}), and more generally, for d-dimensional Euclidean space, (frac{1}{sqrt{d}}le beta ^*_{({mathbb {R}}^d,Vert cdot Vert _2)}le frac{sqrt{3}}{2}). In this paper, we show that (0.557le beta ^*_{({mathbb {R}}^d,Vert cdot Vert _2)}) for any dimension d (notice that (frac{1}{sqrt{d}}<0.557) for any (dge 4)). In addition, we prove that for every metric space (X, d) it holds that (sqrt{2}-1le beta ^*_{(X,d)}), and show that there exists a metric space for which (beta ^*_{(X,d)}le frac{1}{2}).","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"47 4 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139094710","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

Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs 在简单绘制的完整图形中寻找平面结构的扭曲方法

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-01-03 DOI: 10.1007/s00454-023-00610-0

Oswin Aichholzer, Alfredo García, Javier Tejel, Birgit Vogtenhuber, Alexandra Weinberger

{"title":"Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs","authors":"Oswin Aichholzer, Alfredo García, Javier Tejel, Birgit Vogtenhuber, Alexandra Weinberger","doi":"10.1007/s00454-023-00610-0","DOIUrl":"https://doi.org/10.1007/s00454-023-00610-0","url":null,"abstract":"Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). A simple drawing is c-monotone if there is a point O such that each ray emanating from O crosses each edge of the drawing at most once. We introduce a special kind of c-monotone drawings that we call generalized twisted drawings. A c-monotone drawing is generalized twisted if there is a ray emanating from O that crosses all the edges of the drawing. Via this class of drawings, we show that every simple drawing of the complete graph with n vertices contains (Omega (n^{frac{1}{2}})) pairwise disjoint edges and a plane cycle (and hence path) of length (Omega (frac{log n }{log log n})). Both results improve over best previously published lower bounds. On the way we show several structural results and properties of generalized twisted and c-monotone drawings, some of which we believe to be of independent interest. For example, we show that a drawing D is c-monotone if there exists a point O such that no edge of D is crossed more than once by any ray that emanates from O and passes through a vertex of D.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139094704","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

Dynamic Connectivity in Disk Graphs 磁盘图中的动态连接性

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-01-03 DOI: 10.1007/s00454-023-00621-x

Alexander Baumann, Haim Kaplan, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam Roditty, Paul Seiferth

{"title":"Dynamic Connectivity in Disk Graphs","authors":"Alexander Baumann, Haim Kaplan, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam Roditty, Paul Seiferth","doi":"10.1007/s00454-023-00621-x","DOIUrl":"https://doi.org/10.1007/s00454-023-00621-x","url":null,"abstract":"Let (S subseteq mathbb {R}^2) be a set of n sites in the plane, so that every site (s in S) has an associated radius (r_s > 0). Let (mathcal {D}(S)) be the disk intersection graph defined by S, i.e., the graph with vertex set S and an edge between two distinct sites (s, t in S) if and only if the disks with centers s, t and radii (r_s), (r_t) intersect. Our goal is to design data structures that maintain the connectivity structure of (mathcal {D}(S)) as sites are inserted and/or deleted in S. First, we consider unit disk graphs, i.e., we fix (r_s = 1), for all sites (s in S). For this case, we describe a data structure that has (O(log ^2 n)) amortized update time and (O(log n/log log n)) query time. Second, we look at disk graphs with bounded radius ratio (Psi ), i.e., for all (s in S), we have (1 le r_s le Psi ), for a parameter (Psi ) that is known in advance. Here, we not only investigate the fully dynamic case, but also the incremental and the decremental scenario, where only insertions or only deletions of sites are allowed. In the fully dynamic case, we achieve amortized expected update time (O(Psi log ^{4} n)) and query time (O(log n/log log n)). This improves the currently best update time by a factor of (Psi ). In the incremental case, we achieve logarithmic dependency on (Psi ), with a data structure that has (O(alpha (n))) amortized query time and (O(log Psi log ^{4} n)) amortized expected update time, where (alpha (n)) denotes the inverse Ackermann function. For the decremental setting, we first develop an efficient decremental disk revealing data structure: given two sets R and B of disks in the plane, we can delete disks from B, and upon each deletion, we receive a list of all disks in R that no longer intersect the union of B. Using this data structure, we get decremental data structures with a query time of (O(log n/log log n)) that supports deletions in (O(nlog Psi log ^{4} n)) overall expected time for disk graphs with bounded radius ratio (Psi ) and (O(nlog ^{5} n)) overall expected time for disk graphs with arbitrary radii, assuming that the deletion sequence is oblivious of the internal random choices of the data structures.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"16 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139094707","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

When is a Planar Rod Configuration Infinitesimally Rigid? 什么情况下平面杆配置具有无限刚性？

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-12-19 DOI: 10.1007/s00454-023-00617-7

Signe Lundqvist, Klara Stokes, Lars-Daniel Öhman

{"title":"When is a Planar Rod Configuration Infinitesimally Rigid?","authors":"Signe Lundqvist, Klara Stokes, Lars-Daniel Öhman","doi":"10.1007/s00454-023-00617-7","DOIUrl":"https://doi.org/10.1007/s00454-023-00617-7","url":null,"abstract":"We investigate the rigidity properties of rod configurations. Rod configurations are realizations of rank two incidence geometries as points (joints) and straight lines (rods) in the Euclidean plane, such that the lines move as rigid bodies, connected at the points. Note that not all incidence geometries have such realizations. We show that under the assumptions that the rod configuration exists and is sufficiently generic, its infinitesimal rigidity is equivalent to the infinitesimal rigidity of generic frameworks of the graph defined by replacing each rod by a cone over its point set. To put this into context, the molecular conjecture states that the infinitesimal rigidity of rod configurations realizing 2-regular hypergraphs is determined by the rigidity of generic body and hinge frameworks realizing the same hypergraph. This conjecture was proven by Jackson and Jordán in the plane, and by Katoh and Tanigawa in arbitrary dimension. Whiteley proved a version of the molecular conjecture for hypergraphs of arbitrary degree that have realizations as independent body and joint frameworks. Our result extends his result to hypergraphs that do not necessarily have realizations as independent body and joint frameworks, under the assumptions listed above.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138818652","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

Deep Cliques in Point Sets 点集合中的深度聚类

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-12-18 DOI: 10.1007/s00454-023-00612-y

Stefan Langerman, Marcelo Mydlarz, Emo Welzl

{"title":"Deep Cliques in Point Sets","authors":"Stefan Langerman, Marcelo Mydlarz, Emo Welzl","doi":"10.1007/s00454-023-00612-y","DOIUrl":"https://doi.org/10.1007/s00454-023-00612-y","url":null,"abstract":"Let (n in mathbb {N}) and (k in mathbb {N}_0). Given a set P of n points in the plane, a pair ({p,q}) of points in P is called k-deep, if there are at least k points from P strictly on each side of the line spanned by p and q. A k-deep clique is a subset of P with all its pairs k-deep. We show that if P is in general position (i.e., no three points on a line), there is a k-deep clique of size at least ( max {1,lfloor frac{n}{k+1} rfloor }); this is tight, for example in convex position. A k-deep clique in any set P of n points cannot have size exceeding (n-lceil frac{3k}{2} rceil ); this is tight for (k le frac{n}{3}). Moreover, for (k le lfloor frac{n}{2} rfloor - 1), a k-deep clique cannot have size exceeding (2sqrt{n(lfloor frac{n}{2} rfloor -k)}); this is tight within a constant factor. We also pay special attention to ((frac{n}{2}-1))-deep cliques (for n even), which are called halving cliques. These have been considered in the literature by Khovanova and Yang, 2012, and they play a role in the latter bound above. Every set P in general position with a halving clique Q of size m must have at least (lfloor frac{(m-1)(m+3)}{2}rfloor ) points. If Q is in convex position, the set P must have size at least (m(m-1)). This is tight, i.e., there are sets (Q_m) of m points in convex position which can be extended to a set of (m(m-1)) points where (Q_m) is a halving clique. Interestingly, this is not the case for all sets Q in convex position (even if parallel connecting lines among point pairs in Q are excluded).","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"16 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138741131","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

Restricted Birkhoff Polytopes and Ehrhart Period Collapse 受限伯克霍夫多边形和艾哈特周期坍缩

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-12-16 DOI: 10.1007/s00454-023-00611-z

Per Alexandersson, Sam Hopkins, Gjergji Zaimi

引用次数: 0

Orthogonal Dissection into Few Rectangles 正交分解成几个矩形

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-12-12 DOI: 10.1007/s00454-023-00614-w

David Eppstein

引用次数: 0

Improved Estimates on the Number of Unit Perimeter Triangles 单位周长三角形数量的改进估算

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-12-09 DOI: 10.1007/s00454-023-00615-9

Ritesh Goenka, Kenneth Moore, Ethan Patrick White

引用次数: 0

The Parameterized Complexity of Guarding Almost Convex Polygons 保护近凸多边形的参数化复杂度

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-12-05 DOI: 10.1007/s00454-023-00569-y

Akanksha Agrawal, Kristine V. K. Knudsen, Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi

{"title":"The Parameterized Complexity of Guarding Almost Convex Polygons","authors":"Akanksha Agrawal, Kristine V. K. Knudsen, Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi","doi":"10.1007/s00454-023-00569-y","DOIUrl":"https://doi.org/10.1007/s00454-023-00569-y","url":null,"abstract":"The Art Gallery problem is a fundamental visibility problem in Computational Geometry. The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point in C is visible to at least one guard. In the classic formulation of Art Gallery, G and C consist of all the points within P. Other well-known variants restrict G and C to consist either of all the points on the boundary of P or of all the vertices of P. Recently, three new important discoveries were made: the above mentioned variants of Art Gallery are all W[1]-hard with respect to k [Bonnet and Miltzow in 24th Annual European Symposium on Algorithms (Aarhus 2016)], the classic variant has an ({{mathcal {O}}}(log k))-approximation algorithm [Bonnet and Miltzow in 33rd International Symposium on Computational Geometry (Brisbane 2017)], and it may require irrational guards [Abrahamsen et al. in 33rd International Symposium on Computational Geometry (Brisbane 2017)]. Building upon the third result, the classic variant and the case where G consists only of all the points on the boundary of P were both shown to be (exists {mathbb {R}})-complete [Abrahamsen et al. in 50th Annual ACM SIGACT Symposium on Theory of Computing (Los Angeles 2018)]. Even when both G and C consist only of all the points on the boundary of P, the problem is not known to be in NP. Given the first discovery, the following question was posed by Giannopoulos [Lorentz Workshop on Fixed-Parameter Computational Geometry (Leiden 2016)]: Is Art Gallery FPT with respect to r, the number of reflex vertices? In light of the developments above, we focus on the variant where G and C consist of all the vertices of P, called Vertex-Vertex Art Gallery. Apart from being a variant of Art Gallery, this case can also be viewed as the classic Dominating Set problem in the visibility graph of a polygon. In this article, we show that the answer to the question by Giannopoulos is positive: Vertex-Vertex Art Gallery is solvable in time (r^{{{mathcal {O}}}(r^2)}hspace{0.55542pt}{cdot }hspace{1.66656pt}n^{{{mathcal {O}}}(1)}). Furthermore, our approach extends to assert that Vertex-Boundary Art Gallery and Boundary-Vertex Art Gallery are both FPT as well. To this end, we utilize structural properties of “almost convex polygons” to present a two-stage reduction from Vertex-Vertex <","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"73 1-3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138507780","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

Furstenberg Sets in Finite Fields: Explaining and Improving the Ellenberg–Erman Proof 有限域中的Furstenberg集:对Ellenberg-Erman证明的解释和改进

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-11-29 DOI: 10.1007/s00454-023-00585-y

Manik Dhar, Zeev Dvir, Ben Lund

引用次数: 3